Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
Subseries: USSR Adviser: L.D. Faddeev, Leningrad
1043
Linear and Complex Analysis Problem Book 199 Research Problems
Edited by V. R Havin, S.V. Hru~(~v and N.K. Nikol'skii
II II
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
Editors
Victor R Havin Leningrad State University Stary Peterhof, 198904 Leningrad, USSR
Sergei V. Hru~(3ev Nikolai K. Nikol'skii Leningrad Branch of the V.A. Steklov Mathematical Institute Fontanka 27, 191011 Leningrad, USSR
Scientific Secretary to the Editorial Board V.I. Vasyunin
AMS Subject Classifications (1980): 30, 31, 32, 41, 42, 43, 46, 47, 60, 81
ISBN 3-540-12869-7 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-38?-12869-? Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data. Main entry under title: Linear and complex analysis problem book, (Lecture notes in mathematics; 1043) 1. Mathematical analysis-Problems, exercises, etc. L Khavin, Viktor Petrovich. 11. Krushchev, S.V. IlL Nikol'skii, N.K. (Nikotai Kapitonovich) IV. Series: Lecture notes in mathematics (Springer-Verlag; 1043) QA3.L28 no, 1043 [QA301] 510s [515'.076] 83-20344 ISBN 0-387-12869-7 (U,S.)
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CONTENTS
List of Participants........... • • . . . . . . . x
Acknowledgements. . . . . . . . . . • • • • • • • • • • • .XIll
Preface. • . . . . . . . . . . • • • . • • • • • • • • • • • xvI
PROBLEMS
Chapter I. ANALYSIS IN FUNCTION SPACES . . . . . . . . . . . 2
1.1. Uniformly convergent Fourier series . . . . . . . . • 5
1.2. Compactness of absolutely summing operators.. • • • 7
1.3. When is D~(.X~,'~ ~) ~ h(X,~ ,~) ? . . . . . . . . . . . 10
1.4c. Local theory of spaces of analytic functions.. • • • 14
I. 5c. Complemented subspaces of A, ~ and H . . . . . . i8
1.6. Spaces of Hardy type . . . . . . . . . . . . . . . . . 22
1.7. Bases in H P spaces on the ball . . . . . . . . . . . 24
1.8. Spaces with the approximation property? .... . . . 26
1.9. Operator blocks in Banach lattices... . . . . . . . 27
1.10c. Isomorphisms and bases. . . . . . . . . . . . . . . . 29
1.11. Isomorphic classification of F-spaces . . . . . . . . 34
1.12. Weighted spaces of entire functions . . . . . . . . . 38
1.13c. Linear functionals and linear convexity. • • .... 41
1.14. Supports of analytic functionals . . . . . . . . . . . 46
Chapter 2. BANACH ALGEBRAS . . . . . . . . . . . . . . . . . 48
2.1. The spectral radius in quotient algebra . . . . . . . 50
2.2. Extremum problems . . . . . . . . . . . . • . . . . . 51
2.3. Naximum principles for quotient norms in ~ .... 53
2.4. 55 2.5. 58
2.6. 61
2.7. 65
2.8. 68
2 . 9 . 70
2.10. 72
Open semigroups in Banach algebras . . . . . • ....
Homomorphisms from 0*-algebras.... . . . . . . .
Analyticity in the Gelfand space of multipliers . • •
Homomorphisms of measure algebras . . . . • . • • . •
Separation of ideals in group algebras . . . . . . • •
Polynomial approximation . . . . . . . . . . . . . . .
My favourite algebra . . . . . . . . . . . . . . . . .
IV
2.11c. Sets of antisymmetry and support sets for~°°+ G . . • 75
2.12. Subalgebras of the disk algebra.... • • • • . . • • 78
2.13. Analytic operator families . . . . . ..... . . . . 81
Chapter 3. PROBABILISTIC PROBLEMS . . . . . . . . . . . . . . 82
3.1c. Some questions about Hardy functions . . . . . . . . . 85
3.2c. Analytic problems originating in stationary processes 87
3.3. Moduli of Hankel operators, Past and Future...... 92
3.4. Strong law of large numbers for stationary processes 98
3.5. Markov processes and contractions . . . . . . . . . . • 101
3.6c. Existence of measures with given projections ..... 104
3.7c. An indicator with a spectral gap . . . . . . . . . . . 106
Chapter 4. OPERATOR THEORY.. . . . . . . . . . . . . . . . . 108
4.1. Boundedness of continuum eigenfunctions...... • • 113
4.2. Scattering theory for Coulomb type problems..... • 116
4.3. Polynomial approximation and Hill's equation . . . . • 121
4.4c. Zero sets of dissipative operator functions...... 124
4.5. Point spectrum of perturbations of unitary operators 129
4.6. Spectral analysis of re-expansion operators . . . . . . 130
4.7. Non-negative subspaces of ~-dissipative operators . • 135
4.8. Perturbation theory and invariant subspaces...... 137
4.9c. Operators and approximation....... . . . . . . . 140
4.10. Spectral decompositions and the Carlescn condition . . 144
4.11. Similarity problem and the singular spectrum ..... 147
4.12c. Analytic operator-valued functions.... • • • • • • 152
4.13. Invariant subspaces of C10-contracticns . . . . . . . . 155
4.14. Titchmarsh's theorem for vector functions . . . . . . . 158
4.15. Operator functions and spectral measures of isometrics 160
4.16. ~-inner matrix-functions..... . . . . . . . . . . 164
4.17. Extremal multiplicative representations.. • . • • • • 169
4.18. Pactorization of operators on I~(@,~) . . . . . . . . 172
4.19. An infinite product of special matrices ..... . . . 17V
4.20. Pactorization of operator functions . . . . . . . . . 180
4.21. When are differentiable functions differentiable? . . 184
4.22c. Are multiplication and shift approximable? . . . . . . 189
4.23.
4.24.
4.25.
4.26.
4.27.
Extremal similarities. . . . . . . . . . • • • • • • . 197
Estimates of functions of Hilbert space operators... 199
Extimates of operator polynomials on ~p . . . . . . . 205
2x2-Matsaev conjecture ..... . . .... . .... 209
Diminishing of spectrum under an extension. • • • • • 210
v
4.28. The decomposition of Riesz operators.. • • • • • • • 211
4.29. Free inver~ibility of Fredholm operators . . . . . . . 212
4.30. Indices of an operator matrix and its determinant. , . 214
4.31, Compact operators with power-like s-numbers.... • • 217
4.32, Perturbation of spectrum for normal operators . . . . . 219
4.33. Perturbation of continuous spectrum.... • • • • , , 223
4,34. Almost-normal operators modulo ~p . . . . . . . . . . 22?
4,35. Hyponoz~al operators and spectral absolute continuity 231
4.36. Operators, analytic negligibility, and capacities... 234
4.37. Generalized differentiations and semidiagonality . . . 238
4.38. What is a finite operator?.. . . . . . . . . . . . . 240
4.39. Spectra of endomcrphisms of a Banach algebra.... • 244
4.40. Composition of integration and lubstitution . . . . . 2 4 9
Chapter 5. HANKEL AND TOEPLiTZ OPERATORS . . . . . . . . . . 251
5.1c. Approximation by elements cfH °°+ C . . . . . . . . . 254
5.2. Quasinilpotent Hankel operators... • • • . • • • • • 259
~.3. Hankel operators on Bergman spaces... • • • • • • • 262
5.4c. Similarity for Toeplitz operators . . . . . . . . . , . 264
5.5. Iterates of Toeplitz operators .... . . . . . . , . 269
5.6. Localization of Toeplitz operators . . . . . . . . . . 271
5.7. Toeplitz operators on the Bergn~n space... • • • • • 274
5.8. Vectorial Toeplitz operators on Hardy spaces.. • • • 276
5.9, ~actorization of almost periodic matrices.... • • • 279
5.10. Toeplitz operators in several variables.... • • • • 283
5.11. Around SzegB limit th~,orems...... • • • , • • • • 285
5.12. Moments, Toeplitz matrices and statistical physics.. 289
5.13. Reduction method for Toeplitz operators.. • • • • • • 293
5.14. Elliptizitat und Projektionsverfahren . . . . . . . . 298
5.15. Defect numbers of Riemann boundary value problem . . . 303 • #
5.16. Polncare-Bertrand operators in Banach algebras . . . 306
Chapter 6. SING~ INTEGRALS, BMO, H p . • • . . . . . . . . 308
6.1c. The Cauchy integral and related operators.. . . . . . 310
6.2c. Classes of domains and Cauchy type integrals . • • • • 313
6.3, Bilinear sin~alar integrals..... • • • • • • • • • 317
6.4. Weighted norm inequalities . . . . . . . . . . . . . . 318
6.5. Weak type substitute for Riesz projections on tcri • • 322
6.6. The norm of the analytic projection. • • • • • • . • • 325
6.7. Is this operator invertible? . . . • • • • • • • • • • 328
329 6.8, BMO-norm and operator norm. , . . • • • • , • • • • •
Vl
6.9c. Problems concerning H ~ and BMO . . . . . . . . . . . 330
6.10c. Two conjectures by Albert Baernstein . . . . . . . . . 333
6.11c. Blaschke products in ~o . . . . . . . . . . . . . ~ 337
6.12. Algebras contained within H =°" . . . . . . . . . . . . 339
6.13. Analytic functions in W I ~ . . . . . . . . ..... 341
6.14. Subalgebras of ~(T:) containing ~(T~) ...... 342
6.15. Inner functions with derivative in H P, 0<p<l . . . 343
6.16. Equivalent morns inN P . . . . . . . . . . . . . . . 345
6.17. A definition of H P . . . . . . . . . . . . . . . . . 346
6.18. Hardy classes and Riemann surfaces . . . . . . . . . . 347
6.19. Interpolating Blaschke products . . . . . . . . . . . 351
Chapter 7. SPECTRAL ANALYSIS AND SYNTHESIS . . . . . . . . . 353
7.1. Holomorphic functions with limited growth ...... 357
7.2. ~ -equation and localization of submedules ..... 361
7.3c. Invariant subspaces and differential equations .... 364
7.4. Local description of closed submodules ........ 367
7.5. Spectral synthesis for entire functions ....... 372
7.60. Spectral synthesis for differential operators .... 374
7.7. Two problems on the spectral synthesis ........ 378
7.8c. Cyclic vectors in spaces of analytic functions .... 382
7.9. Weak invertibility and factorization . . . . . . . . . 386
7.10c. Weakly invertible elements in Bergman spaces ..... 39O
7.11. Invariant subspaces of the backward shift ...... 393
7.12. Divisibility problems ina(~) and H~(~} ...... 396
7.13. A refinement of the corona theorem . . . . . . . . . 399
7.14. Invariant subspaces of the shift . . ........ 401
c; 7.15. Blaschke products and ideals in .... . . . . 403
7.16. Closed ideals in the analytic Gevrey class ...... 407
7.17. Completeness of translates in a weighted space . . . 409
7.18. Problems of harmonic analysis in weighted spaces. • • 414
7.19c. A closure problem for functions on ~+ ..... . . . 417
7.20. Translates of functions of two variables ....... 421
7.21. Algebra and ideal generation.. . . . . . . . . . . . 422
7.22. Harmonic synthesis and compositions . . . . . . . . . 426 % ° • .
7.23c. Deux problemes sur lea s~ries trlgonometrlques .... 429
Chapter 8. APPROXIMATION AND CAPACITIES . . . . . . . . . . 431
8.1c. Spectral synthesis in Sobolev spaces. ........ 435
8.2. Approximation by smooth functions in Sobolev spaces . 438 ~% 439
8,30. Splitting in -spaces . . . . . . . . . . . . . . .
VII
8.4. Trigonometric approximation in L~(~, ~A) . . . . . . 447
8.5. Decomposition of approxim~ble functions . . . . . . . 449
8.6. Approximation and quasiconformal continuation . . . . 451
8.7. Tangential approximation....... • • ° .... • 453
8.8. Integrability of the derivative of a conformal mapping457
8.9. Weighted polynomial approximation . . .... . . . . 461
8.10. Approximation in the mean by harmonic functions . . . 466
8.11. Rational approximation of analytic functions..... 471
8.12c. Pad6 approximation in several variables . . . . . . . 475
8.13. Badly-approximable functions .... . . . . . . . . . 480
8.14. Exotic Jordan arcs in O N . . . . . . . . . . . . . . 483
8.15. Removable sets for bounded analytic functions .... 485
8.16. On Painlev6 null sets . . . . . . . . . . . . . . . . 491
8.17. Analytic capacity and ration~l approximation .... 495
8.18o On sets of analytic capacity zero . . . . . . • • . • 498
8.19. Estimates of analytic capacity . . . . . . . . . . . . 502
8.20c. Regularitat fur elliptische Gleichungen . . . .... 507
8.21. Exceptional sets for Besov spaces . . • • • • • • • • 515
8.22. Complex interpolation between Sobolev spaces..... 519
Chapter 9. UNIQUENESS, MOMENTS, N~LITY . . . . . . . . . 520
9.1c. Representations of analytic functions . . • . • ° • • 522
9.2. Moment problem questions....... • • • • .... 529
9.3c. Uniqueness and finite Dirichlet integral.... • . • 531
9.4. Stationary functions, uncertainty, Jordan operators . 536
9.5. Problem in the theory of functions.... • . . • • • 541
9.6. Peak sets for Lipschitz classes . . . . . . . . . . . 544
9.7. A problem by R.Kaufman....... • • • • • • • • • 547
9.8c. Quasi-analyticity and differential operators..... 548
9.9c. Local operators on Fourier transforms . . . . . . . . 552
9.10. Density of exponentials on plane arcs . . . . . . . . 555
9.11. When i s ~ 1 ~ 1 ~ _ o o ? . . . . . . . . . . . . . . . . 557
9 .12. An a l t e z ~ a t i v e f o r a n a l y t i c Carleman c l a s s e s . . . • • 558
9.13. On a uniqueness theorem in ~ . . . . . . . . . . . . 561
Chapter 10. INTERPOLATION, BASES, ~JLTIPLIERS . . . . . . . 563
I0.1c. Interpolation by entire functions . . . . • • • . • • 566
10.2. Bases of reproducing kernels and exponentials .... 569
I0.3c. Multiplicative properties of $~ • • . . . . . . . . 572
10.4. Free interpolation in regular classes . • • . . • • • 575
10.5. Traces of ~-functions on hyperplanes. ..... . . 577
Vlll
10.6. Representations by exponential series . . , . . . . . 579
10.7. Restrictions of the Lipschitz spaces • • • • • • • • 8S3
10.8. Multipliers, interpolation, and ^(p~-sets . . • . 586
Chapter 11. ENTIRE AND SUBHARMONIC FUNCTIONS . . . . . . . , 589
11.1c. The inverse problem of best approximation . . .... 591
11.2. Derivatives of unbounded analytic functions . . . . . 595
11.3. Exceptional values of ~rious kinds .... . . . . • 597
11.4. Valiron exceptional values . . . . . . . . . , . . . 899
11,5c. Preservation of the completely regular growth .... 600
11.6¢. Zero-sets of sine-type functions . . . . . . . . . . . 605
11.7. An extremal problem for subharmonic functions .... 609
11.8c. A problem on exact majorants........ • . • , • 611 f
11.9. Entire functions of Laguerre-Polya class . . . . . . 614
11,106 Cluster sets and a problem of A.F.Leont'ev, . . . . . 617
Chapter 12. ~ . . . . . . . . . . , . . . . . • • .... 619
12.1. Polynomially convex hulls . .... . . . . . . . . . 620
12.2o. Positive plurlharmonic ftulotions . . . . . . . . . . 623
12.3. Proper mappings of classical domains...... • . . 625
12.4. On biholomorphy of holomorphic mappings . . . . . . . 629
Chapter 13. MISCELLANEOUS PROBLEMS . . . , , , . . . . . . , 631
13.1. Banach algebras and almost periodicity..... , . . 632
13.2. Support points of univalent functions . . . . . . . . 636
13.3. More problems by Albert Baernstein . . . . . . . . . . 638
13.4. Some extension problems . . . . • • . . . . . • • • • 639
13.5. Partition of singularities . . . . . . . . . . . . . . 641
13.6. Rearr~ngement-invariant hulls of sets . • • • .... 642
13.7. Norms and extremals of convolution operators . . . . . 646
13.8c. Algebraic equations with Bauach algebra coefficients 652
13.9. Holomorphic mappings and algebraic functions..... 657
I 3.1~ Singular points of plane algebraic curves . . .... 662
SOLUTIONS
S. Ic. Absolutely summing operators . . . . . . , . . . . . 665
S.2c. Golubev series and analyticity on continua...... 670
S,3c. The vanishing interior of the spectrum.... .... 674
S.4c. Uniquemess for mean periodic functions .... , . . . 677
S,5C. The Cauchy integral on Lipschitz graphs . . • . • . . 679
IX
S.6c.
S. 7c.
S.8c,
S.9c.
S.10c.
S.11c.
Sets of uniqueness for Q6 . . . . . . . . . . . . . 682
Another problem by R.Kaufman..... , . . . . . . 684
Rational functions with given ramifications..... 686
Asymptotic behaviour of entire functions . . . . . . 688
The inner function problem im b a l l s . . . . . . . . . 691
Homogeneous measures on subsets of ~ . . . . . . , 698
Subject Index . . . . . . . . . . . . . . . . . . . . . . . 700
Author Index . . . . . . . . . . . . . . . . , . . . , . . . V09
Standard notation........ . , . . . . . . . . . . . . 721
LIST OF PARTICIPANTS
AdamyanV.M. (Aha~H), 4.15, 5.1 Domar Y., 7.19 Adams D.R., 8.21 Douglas R.G., 5.6 Ahem P.R., 6.15 Duren P.L., 13.2 Aizenberg L.A. (A~seH6epP), 1.13 Dym H., 8.4 Aleksandrov A.B. (A~ezcaH~pOB)7.11, Dyn'kin E.M. (~MHBEHH), 7.22.
6.17 9. 6, S.11 Alexander H., 12.1 Anderson J.M., 6.12 Arov D.Z. (ApOB), 4.15, 4.16, 5.I Axler S., 5.3 Az~rin V.S. (AsapHH), ll.10, Azizov T.Ya. (AsHSOB), 4.7
Baernstein A., 6.10, 13.3 Bagby T., 8.10 Belyi V.I. (~e~), 8.6 Birman M.S. (B~pMaH), 4.6, 4.31 Boivin A., 8.7 Bollob~s B., 4.27 Bourgain J., 1.1 de Branges L., 2.9, 4.8, 9.9 Brenuan J., 8.8, 8.9 Brown G., 2.6 Brudnyi Yu.A. (Bpy~), 10.7 Bruna J., 7.16, 10.4
Calder6n A.P., S.5 Casazza P.G., 1.5, 6.19 Chang S.-Y.A., 6.13, 6.14 Clark D.N., 4.23, 5.4 Coburn L.A., 5.10 Coifman R.R., 6.1
Dales H.G., 2.5 Davis Oh., 4.32 Devinatz A., 9.2
Djrbashyan M.M. (~p6~), 9.1
Er~menke A.E. (Ep~MeHEo), ii.3, 11.4, 11.10
Faddeev L.D. ($a~heeB), 4.4, 4.80 Pel'dman I.A. (~e~B~MaH), 4.29,
4.30 Forelli F., 7.12, 12.2 Frankfurt R., 7.9
Gamelin T.W., 2.10 Gaposhkin V.P. (rssIOEEH), 3.4 Garnett J.B., 6.9 Gauthier P.M., 8.7 Ginzburg Yu.P. (I~Hs6ypP), 4.17 Gol'dberg A.A. (ro~B~6epr), 11.3,
11.4, s.8 Gonchar A.A. (roHqap), 8.11 Gorin E.A. (top,H) 4.39, 13.7,
' 13.8
Grishin A.P. (rp~mHH), 11.10 Gulisashvili A.B.
(l 'yJmca~,,~), 13.6 Gurarii V.P. (rypapm~), 7.17,
7.18
Haslinger F., 1.12 Hasumi M., 6.18 Havin V.P. (XaBI~H) 6.17, 9.3,
' 9 . 4 , S.2 Havinson S.Ya. (XaB~HOOH), 11.8 Hayman W.K., 8.16
Xl
Hedberg L.I., 8.1 Helson H., 4.14
Henkin G.M.(XeHEHH), 8.14, 12.3 Herrero D., 4.38
Hru$$~v S.V. (Xpy~eB)~ 3.3,12 9.3,
Ibragimov I.A. (MOpa2EmOB), 3.2 Igari S., 2.7 Iohvidov i.S. (H0XB~OB), 4.7 Ivanov L.D. (HBaHOB), 8.18
Jones P.W., 1.8, 6.3, 6.16, 8.2, 8.22
Joricke B., 9.4
Kadec M.I. (Ea~es), 11.1 Kahane J. -P., 7.2 3 Karlovich Yu.l. (Eap~oB~), 5.9 Kaufman R., 9.7, 13.5, S.7 Kisliakov S.V. (K~CJLKKOB), 6.5 Kitover A.K. (~TOBep), 4.26,4.39 Komarchev I.A. (EoMap~eB), 1.3 Koosis P., 9.5 Korenblum B., 7.10 Kr~l J., 8.19, 13.4 Krasiohkov-Ternovskii I.F.
(Kpac~moB-TepHoBc~), 7.4 Krein M.G. (Epe~H), 4.15, 5.1,
5.11, 13.1 Kriete T., 8.3
429 5.8, 6.6
Langley J., 11.2 Latushkin Yu.D. (~aT~), 5.15 Leiterer J., 4.20 Leontiev A,F. (~e0HTBeB), 10.6 Levin B.Ya. (~eBEH), 7.20, 11.6,
11.7, 11.9 Lin V.Ya. (/~H), 13.9, 13.10
Makarov B.M. (MsEapoB), 1.3
Makarov N.G. (MaEapoB), 4.5, 4.33, 9.4
Mark-as A.S. (~pzyc), 4.29, 4.30 Marshall D.E., I. 7 Matsaev V.I. (Ma~aeB), 9.8 Maz'ya V.G. (Mas~), 8.20 McKean H.P., 3.1, 4.3 Mel'nikov M.S. (MeJIBHEEOB), 8.17 Meyer Y., 6.1 Moran W., 2.6 Muckenhoupt B., 6.4 Murphy G.J., 2.1, S.3
Naboko S.N. (Ha6oEo), 4.11 Napalkov V.V. (Ha~a~EOB), 9.13 Nikol'skii N.K. (HEEOJlBCEI~), 4.9,
4.10, 4.33, 7.7, i0.2 Novikov R.G. (HOBHEOB), 12.3
0strovskii I. ¥. (0CTpOBCE~), II. 4, 11.5, 11.6
Palamodov V.P. (Ha~a~o~oB), 7.2 Pavlov B.S. (~OB), 4.4, 4.10 PeIczy~ski A., 1.2, S.I
Pellet V.V. (Heaaep), 3.3, 4.24, 4.25, 5.5
Power S.C., 5.2 PrBssdorf S., 5.14 Pt~k V., 2.2 Putnam C.R., 4.35, 4.36
Reshetihin N.Yu. (PemeT~H), 4.19
Rochberg R., 2.13, 6.8 Rubel L.A., 8.13, 11.2 Rudin W., S.I0
Sahnovich L.A. (CaXHOBE), 4.2, 4.18
Sapogov N.A. (CanoroB), 3.7 Litvinchuk G.S. (~TB~B~yE), 5.15 Sarasom D., 2.11, 6.11, S.6 L~abich Yu.l. (J~06~), 4.40, S.4 Sem~nov E.M. (CeMeHOB), 1.6, 1.9
Sem~nov-Tian-Shansky M.$. (CeM@HOB-TSa-~SaCE~), 5.16
Semiguk O.S. (Ce~mzg-~), 1.10 Sem~es S., 6.7 Shamoyan F.A. Shields A.L., Shirokov N.A. Shul'man V.S. Siddiqi J.A., Silbermamn B. Simon B., 4.1
(lllamO~H), 7.14
7.8 (I~OEOB), 10.5 (lllya~4), 4.37
9.10 , 5.13
Skita N.I. (CE~6a), 1.10 Smyth M.R.F., 2.1, 4.28 Solev V.N. (CoaeB), 3.2 Solomyak M.Z. (Coxo~E), 4.31 Spitkovskii I.M.
(Cn~TEOBC~), 5.9, 5.11
Stray A., 8.5 Sudakov V.N. (Cy~szoB), 3.6 Sundberg C., 5.7 Szokefalvi-Nagy B., 4.12
Taylor B.A., 10.1 Thomas M., 7.21 Tkachauke V.A. (T~a~eHEO), 7.5,
7.6 Trutnev V.M. (TpyTHeB), 1.14, 7.3 Tumarkin G.C. (TyMap~), 6.2 Teodorescu R., 4.13
XII
Vasyunin V.I. (BacD~), 4~10,13
Verbitsky I.E. (Bep6~J/E~), 5.8, 6.6
Vershik A.M. (BepmHE), 3.5, 4.22 Vinogradov S.A. (B~HoPps~oB), 10.3 Vitushkin A.G. (BHTy~EHH), 8.17 Vladimirov V.S. (Bas~E~/~pOB), 5.12
Voiculescu D., 4. 34 Vol'berg A.L. (Boa~6epr), 9.11
Volovich I.V. (BOaOBH~), 5.12
Waelbroeck L., 7. I Wallin H., 8.12 Wermer J., 2.12 West T.T., 2.1, 4.28, S.3 Widom H., 4.21 Williams D.L., 7.15 Wojtaszczyk P., 1.4, 1.7 Wolff T., 7.13
Young N.J., 2.3
Zafran M., 10.8 Zaidenberg M.G, (Sa~eH6epr), 13.10 Zaharyuta V,P. (Saxap~Ta), 1.10,
1.11
Zelazko W., 2.8
Zem~nek J., 2.4
AOKNOW ~EDGEN~NT S
This book was created by a very large body of mathematicians.
We were in touch with more than 200 colleagues, and approximately
twenty (mostly members of our seminar) assisted us in preparing this
volume. Nany of our correspondents will find their problems in the
pages of the book, and - regardless of whether they supplied us with
a mathematical text or with a criticism of our intentions - WE ARE
GRATEFUL TO ALL WHOSE PARTICIPATION CONTRIBUTED TO THIS BOOK.
The goodwill and enthusiasm of many colleagues were crucial for
our work. Pushing their own investigations aside they generously
rendered us invaluable help - invaluable both in its amount and its
skill. This help ranged from writing commentary to organizing the
material, from critical analysis of problems to linguistic consulta-
tions, to preparing of a huge mass of references and - last but not
least - to the technical scissors-and-glue toil (the proof-reading,
removing misprints, compiling indexes etc.), duties which, we dare-
say, are rarely allotted to mathematicians of comparable qualifica-
tions.
We list below in deep gratitude and respect our "informal
editorial board".
WRITING COM~ENTARY
The following colleagues put at our disposal valuable and some-
times very detailed information, used extensively in our commentary:
V,M.Adamyan (A~a~H) A.B.Aleksandrov (A~eEcs~7~OB)
L.A.Aizenberg (A~3eH6epr) D.Z.Arov (ApOB)
V.S.Azarin (AsapEH)
Ch.Berg
B.Bollob~s
L. de Branges
P.G.Casazza
D.N.Clark
A.A.Gol'dberg (ro~6epr)
A.Ya.Gordon (rop~OH)
E.A.Gorin (ropEH)
L.I.Hedberg
P.P.Kargayev (EapraeB)
R.P.Kaufman
S.V.Kisliakov (E~CJAKEOB)
M.G.Krein (Epe~H)
XIV
M.M.~lamud (~aMy~)
V.G.Maz'ya (Ma3~)
l.V.0strovskii (0CTpOBCEH~)
V.V.Peller (He~ep)
D.E. Sarason
F.A.Shamoyan (m~O~)
B.M. Solomyak (ColognE)
V.A.Tkachenko (TEa~eHEo)
l.E.Verbitskii (Bep6~E~)
A.M.Vershik (Bep~E)
A.L.Vol'berg (BoJIB6epr)
H.Wallin
V.P.Zaharyuta (SaxapDTa)
S.V.Znamenskii (SHaMeHcE~)
ADVICE AND MATHEMATICAL CONSULTATIONS
Helpful advice and consultations of the following colleagues
have been used on many occasions:
B.I.Batikyan (F~aTHF~V~I)
M. S. Birman (BHpMaH)
A. G. Chernyavskii (~epH~BCEH~)
E. M. Dyn'kin (~MHBEE)
Yu.B.Farforovskaya (~ap~opoBoEa~)
A.B.Gulisashvili (ryJIEC~BH~IE)
G.M.Henkin (XeHEEH)
S.V.Kisliakov (K~C~OB)
P.Koosis
Lee Lorch
A. S. ~rkus (MapEyc)
V.P.Palamodov (Ha~aMO~OB)
V.V.Peller (Hexxep)
V.A.Toloko~ikov (TO~OEOHHHEOB)
J.Zemanek
LINGUISTIC AID
We thank J.Brennan, E.M.Dyn'kin (~HBENH), A.B.Gulisashvili
(ry~Hc~Jl~), B.~6ricke, S.V.Kisliakov (K~CJL~EOB), N.G.Makarov
×V
(MaEapoB), V.V.Peller (He~L~ep), B.M.Solomyak (Co~oN~IE) and A.L.Vol'-
berg (~6epP)who helped us t~o translate about 100 problems from
Russian into English. We had also to write commentary and introduc-
tions in English, and A.B.Gulisashvili, S.V.Kisliakov and V.V.Peller
participated in solving many linguistic problems. P.Koosis, Lee
Lorch, S.C.Power and J.A.Siddiqi checked some parts of the text, and
Lee Lorch helped to translate the preface from Russian English into
English.
REFERENCES CONTROL AND INDEXES:
These were prepared by L.N.Dovbysh (~OB6~,n) and V.V.Peller
(nex~ep).
PROOF-READING AND CORRECTION 0F MISPRINTS
This task was allotted to L.N.Dovbysh (~OB6~), A.B.Gulisashvili
(I~JL~oamB~Z), S.V.Kisliakov (~CJLq~OB), V.Y.Peller (He~ep).
EDITORS
PREFACE
This volume offers a collection of problems concerning analytic
functions (mainly of one complex variable), linear function spaces
and linear operators.
The most exciting challenge to a mathematician is usually not
what he understands, but what still eludes him. This book reports
what eluded a rather large group of analysts in 1983 whose interests *)
have a large overlap with those of our Seminar . Consequently,there-
fore, the materials contained herein are chosen for some sort of mild
homogeneity, and are not at all encyclopaedic. Thus, this volume
differs markedly from some well-knov~n publications which aim at uni-
versality. We confine ourselves to the (not very wide) area of Ana-
lysis in which we work, and try- within this framework - to make
our collection as representative as possible. However, we confess to
obeying the Bradford law (the exponential increase of difficulties
in obtaining complete information). One of our purposes is to publish
these problems promptly, before they lose the flavour of topicality
or are solved by their proposers or other colleagues.
This Problem Book evolved from the earlier version published as
volume 81 of "Zapiski Nauchnyh Seminarov LOMI" in 1978 (by the way,
much of the work arising from the above mentioned Seminar is regular-
ly published in this journal). It is now twice the size, reflecting
the current interests of a far wider circle of mathematicians. For
*~i.e., ~ the Seminar on Spectral Theory and Complex Analysis
consisting principally of mathematicians working in the Leningrad
Branch of the V.A.Steklov Mathematical Institute (LOMI) and in Lenin-
grad University.
XVII
five years now the field of interests of the "invisible comm~n~ty"
of analysts we belong to has enlarged and these interests have drift-
ed towards a more intense mixing of Spectral Theory with Function
Theory. And the volume as a whole is a rather accurate reflection
of this process (see especially Chapters 4-7 below).
We are pleased that almost a half of the problems recorded in
the first edition, 50 of 99, have been solved, partly or completely.
This book contains a i I the problems of 1978 (we call them "old"
problems). They are sometimes accompanied with commentary reporting
what progress towards their solution has come to our attention.
Moreover, those "old" problems which have been c o m p i e t e i y
solved are assembled umder the title "SOLUTIONS" at the end of the
book (including information as to how and by whom they have been
solved).
When we decided to prepare this new edition we solicited the
cooperation of many colleagues throughout the world. Some two hund-
red responded withample and helpful materials, doubling the number
of collaborators of the first edition. Their contributions ranged
from carefully composed articles (not always short) to brief re-
marks. This flow it was our task to organize and to compress into
the confines of a single volume. To effectuate this we saw no alter-
native to making extensive revisions (more exactly, abbreviations)
in the texts supplied. We hope that we have succeeded in preserving
the essential features ef all contributions and have dome no injus-
tice to any.
At first sight the problems may appear very heterogeneous. But
they display a certain intrinsic unity, and their approximate classi-
fication (i.e. division into chapters) did not give us much trouble.
We say "approximate" because every real manifestation of life re-
sists systematization. Some problems did not fit into our initial
outline and so some very interesting ones are collected under the
XVIII
title "Miscellaneous Problems" as Chapter 13. We took the liberty to
provide almost all chapters with introductions. In these introducti-
ons we try to help the reader to grasp quickly the main point of the
chapter, to record additional bibliography, and sometimes also to
explain our point of view on the subject or to make historical com-
ments~
Chapters are divided into sections. They total 199 (in 1978
there were 99). We treat the words "section" and "problem" as syno-
nymous for the purposes of classification (though a section may con-
tain more than one problem). "Problem 1.25" means the 25-th section
of the first chapter; "Problem 1.26 old" means that Problem 1.26 is
reproduced from the first edition and has not been completely solved
(as far as we know); "Problem S.27" means the 27-th section of
"SOLUTIONS". Problems accompanied by commentary are designated in
the table of contents by the letter "c". Some notation (used some-
times without further explanations) is indicated at the end of the
book. A subject index and an author index are provided.
EDITORS
CHAPTER l
ANALYSIS IN FUNCTIONAL SPACES
Views on t h e p l a c e o f Banaoh F u n c t i o n a l A n ~ y s i s i n A n a l y s i s a s
a whole have undergone many changes d u r i n g i t s r e l a t i v e l y s h o r t h i s -
t o r y . Es~ ly s u c c e s s e s had p roduced the ex t reme (and i n c o r r e c t ) o p i n i -
on t h a t t h i s b r an ch would e v e n t u a l l y a b s o r b a l l ( o r a t l e a s t a l m o s t
a l l ) o f a ~ - l y s i s and t h a t e v e r y c o n c r e t e a n a l y t i c problem cou ld be
s o l v e d J u s t by i n v e n t i n g an a p p r o p r i a t e a b s t r a c t Banach f ramework .
L a t e r , when the f u n d a m e n t a l i d e a s o f t he t h e o r y o f Banach Spaces b e -
came cowoonplaoe ( a s n o r m a l l y happens w i t h a l l r e a l l y i m p o r t a n t i d e -
a s ) a n o t h e r ex t reme v iew emerged. Accord ing t o i t , t he m a t h e m a t i c a l
a c t i v i t y o f Banach space t h e o r i s t s was doomed t o mere t e c h n i c a l de -
t a i l s and i n s i g n i f i c a n t v a r i a n t s . The "go lden age" of the t h e o r i e s
o f L i n e a r T o p o l o g i c a l Spaces and D i s t r i b u t i o n s i n t he f i f t i e s and
e a r l y s i x t i e s a l s o c o n t r i b u t e d to shun t inK the Banaoh Space thsox 7
a s i d e .
But deve lopment s o f t h e l a s t %1o decades have shown t h a t t h e
second ex t r emism was a l s o u n j u s t i f i e d . The most s i g n i f i c a n t r e s u l t s
i n L i n e a r A n a l y s i s o b t a i n e d i n t h i s p e r i o d (and i n p a r t i c u l a r t h o s e
c o n n e c t e d w i t h "Hard A n a l y s i s " ) a r e u n d o u b t e d l y o f a B a n a o h - t h e o r e -
t i c n a t u r e . Avoid ing a n o t h e r commonplace, i . e . a mere s t a t e m e n t o f
mutua l b e n e f i t s b rough t by t h e i n t e r p l a y o f ~ C o n c r e t s " ( o r " C l a s s i -
c a l " , o r "Hard" ) and " A b s t r a c t " b r a n c h e s o f A n a l y s i s e we r e v i e w
b r i e f l y some i m p o r t a n t f e a t u r e s o f t h e p r e s e n t s i t u a t i o n r e f l e c t e d
i n t h i s C h a p t e r .
One o f them i s t h e r e m a r k a b l y i n c r e a s e d i n t e r e s t i n c o n c -
r • t e f u n o t i • n s p a c • s , and n o t o n l y i n t r a d i t i o -
n a l o n e s , l i k e L P ( ~ ) o r C~K) , b u t i n many o t h e r s ( v a r i o u s s p a -
c e s o f smoo th and a n a l y t i c f u n c t i o n s , o f F o u r i e r and power s a l l i e s ,
BMO e t c . ) . Some d e l i c a t e l n v a r i a n t s ( d / s c o v e r e d r e l a t i v e l y r e c e n t l y )
o f a Banach s p a c e X , n a m e l y , p r o p e r t i e s o f s p e c i a l c l a s s e s o£ o p e -
r a t o r s ( X - v a l u e d o r d e f i n e d on X ), p l a y a p r o m i n e n t r o l e t o d a y .
C l a s s e s o f p - a b s o l u t e l y summing o p e r a t o r s and o t h e r a n a l o g o u s c l a s -
s e s may s e r v e a s e x a m p l e s . We i n d i c a t e i n t h i s c o n n e c t i o n t h e I T o b -
l e a o f A . P e l e s y n s k i whose s o l u t i o n r e q u i r e d i n g e n i o u s a n a l y t i c a l
t o o l s a n d c u l m i n a t e d i n i n t e r e s t i n g new r e s u l t s o f a c l a s s i c a l c h a -
r a c t e r ( s e e S . 1 ) . I n t h e p r e s e n t C h a p t e r i t 18 r e p l a c e d ( i n a s e n s e )
b y P r o b l e m 1 . 1 . I t f i t s i n t o t h e same c i r c l e o f i d e a s b u t d e a l s w i t h
t h e s p a c e U o f u n ~ f o l ~ l y c o n v e r g e n t F o u r i e r s e r i e s ( i n s t e a d o f t h e
d i s c - a l g e b r a ) . The " a b s t r a c t " t h e o r y o f p - summing o p e r a t o r s i s p r e -
s e n t e d i n P r o b l e m s 1 . 2 and 1 . 3 .
A n o t h e r i m p o r t a n t f e a t u r e o f modern B a n a c h - t h e o r e t i c i n v e s t i g a -
t i o n s i s t h e s p e c i a l a t t e n t i o n p a i d t o t h e q u a n t i t a t i v e
r e f i n e m • n t s o f q u a l i t a t i v e r e s u l t s c o n c e r n i n g c o n c r e t e
s p a c e s . These r e f i n e m e n t s a r e o f t e n b a s e d on e s t i m a t e s o£ c e r t a i n
q u a n t i t i e s a s s o c i a t e d w i t h f i n i t e d i m e n s i o n a l s u b s p a c e s o f a s p a c e
( t h e s o - c a l l e d " l o c a l t h e o r y " o f Banach s p a c e s ) . T h i s t e n d e n c y i s
well illustrated by Problem 1.4 (now almost completely solved
although it arises in new fol~ms in the context of other spaces, e.g.
G+(T) or U ).
I t i s q u i t e u s u a l nowadays f o r " c o n c r e t e " f u n c t i o n - t h e o r e t i c
problems to appear in connection with general ideas of "abstract"
Linear Analysis. So, for instance, general problems of Banach Geo-
metry read in the context of a concrete function space become fas-
c i n a t l n K q u e s t i o n s on "individual" f u n c t i o n s : for example, problems
on the d e s c r i p t i o n of extreme p o i n t s o f b a l l s l i k e i n l~oblem 1.6
o r on oo lp l emen ted subspaoe8 l i k e i n l~ob l e~ 1 ,5 , t he i somorph ic c l a s -
s i f i c a t i o n o f spaces r e s u l t i n K i n a deepe r comprehens ion o f p ropa~-
t i e s o f f u n c t i o n s fo ru inK the space under c o n s i d e r a t i o n ( e . g . P rob-
lems 1.10 and 1.11 r e l a t e d to non-normed s p a c e s ) . The same can be s a i d
f o r the problems o f f i n d i n K a b a s i s o r i n v e s t i g a t i n g the a p p r o x i -
~ a t i o n p r o p e r t y i n c o n c r e t e (Banaoh and non-Banach) spaces (Problems
1 .7 , 1 .8 and 1 .12) and o f d e s c r i b i n g dua l spaces (Problems 1 .13 ,
1.14 and 3 . 2 ) . Even more t r a d i t i o n a l k inds o f spaces ( s a y measurab le
f u n c t i o n spaces ) s t i l l mapply i n t e r e s t i n g - ~ , o l v e d problems ( s ee e . g .
/~z'oblem 1 . 9 ) .
1.1. SOME QUESTIONS ON THE STRUCTURE OF THE SPACE OF UNIPOR~LY
CONVERGENT FOURIER SERIES
A= ~(~) denotes the disc algebra and
U---u(Ir)={~_A • ~ is uniform limit of #*D~t,
D~ O.<~.<S 'kO .
The norm on U is given by II~II U =~ II#~D~II~. Various analogies between the spaces A and U are known now. It was
shown by D.Oberlin ~3] that measure-zero compact subsets of T are
peak-interpolation sets for U , an improvement of the Rudin-Carle-
son theorem. Using related techniques, I obtained the following
PROPOSITION I: Let K be a compact subset of T and 8 • 0 •
Then there exists I ~ U satisfyin~
(i) II~ II U ~C ( i i ) lTl<s on K
(iii) II 4-#II~ <C(~)IKI 4/~,
where C is a constant.
Fixing a finite sequence ~,.. . ,~,~ in U, ll~sllu~M (~.~s.~), condition (i) can be strenghtened by requiring in addition
ll{.~sU U ~ C(~,M)
Here are some corollaries of Prop. 1 for the Banach space theory of
U (see[2]). PROPOSITION 2: The dual space U ~ of U is weakly complete.
In fact, bounded sequences in U have either a i~complemented
5 4 -subsequence or a weaklyconver~ent subsequence.
2. Reflexive subspaces of U* are isomorphic to subspaces of ~ .
In particular, they are of cotype 2.
Recall that a normed space X has cotype $ > ~ provided
following inequality holds for all finite sequences X
where 0 is a fixed constant, and (~) is the usual Rademacher
sequence,
No results seem to be knows as far as the finite dimensional
properties of U , U ~ are concerned. In particular, the following
problems can be posed.
PROBLEM I: Does
2?
PROBL~ 2: Assume
U ~ have an~ c ot,ype ~<o0? I_~s U of ootype
E ~ h-complemented subspace of U , o~f
dimension ~ . Is it true that ~ contains ~-subspaces for
FF~,~ ? How well can E be embedded as a complement gd subspace of
These questions are solved for the disc algebra A (see [I] ).
Their solution for the space U probably requires different techni-
ques,
RE~ERENCE S
I. B Q u r g a i n J. New Bauach space properties of the disc a!geb-
re and H ~, %o appear in Aota Math.
2. B a m r g a i n J. Quelques propri~t~s lin~aires topologiques t .
de l'espace des serles de Pourier uniform~ment convergentes. -
C.R.A.S. Paris, 1982, 295, S~r.1, 623-625.
3. O b e r 1 i n D.M. A Rudin-Carleson theorem for uniformly con-
vergent Taylor series. - Michigan Math. J., 1980, 27, N 3,
309-314.
J. BOURGAIN Vrije Universiteit Brussel
Dept, Mathematics
Pleinlaan 2
1050 Brussel
Belgium
1.2. COMPACTNESS OF ABSOLUTELY SUM~IING OPERATORS
Every concrete absolutely summing surjection with an infinite
dimensional range allows to prove the classical Grothendieck theorem
on absolutely summing operators from ~ into ~ . On the other
hand, given a Banach space ~ so that every absolutely summing ope-
rator from every ultrapcwer of X to ~$ is compact, one can con-
sider new local characteristics of operators on X (e.g. in spirit !
of Problem 3 below).
Our knowledge of what concerns the existence of non-compact ab-
solutely summing operators with a given domain space is however less
than satisfactory.
PROBLEM ]. Let ~ be a Banach space, Are Sh e followin~ condi-
tions equivalent:
(a) there is an absolutel,y summin~ non-compact operator from X
into a Hilbert space ~
(b) there is a n a bsolutel,y s~n~ s ur~ection from X onto ~ ?
Observe that if one replaces in (a) and (b) "absolutely summing"
by "2-absolutely summing" then the "new (a)" is equivalent to the
"new(b)" and is equivalent to the fact that ~ contains an iso-
morph of ~ (of . [ I ] , [2] ). An obvious example of a space satisfying (a) and (b) is any
~ -space of infinite dimension (by the Grothendieck theorem).
Another example is the disc algebra A . A well-known example of a
]-absolutely summing surjection from A onto ~ is the so called
"Paley projection"
PROBLEM 2. Are the followin~conditions equivalent~
(a) ever 2 absolutely summin~ operator from X to ~ . . . . . . factors
through a Hilbert-Schmidt operator;
(b) X is isomorphic to a quotient of a CCM) -space?
It is a well-known consequence of the classical result of Gro-
tendieck that (b) implies (a).
Observe that if ~ in (a) is replaced by ~ then the modi- fied property (a) is equivalent to (b), cf. J.Bourgain and A.Pe~-
czy~ski (in preparation). Every ~-space satisfies (a). The ~o~ -space constructed by
Bourgain [4] which does not contain C o is not isomorphic to any quotient of a C~)-space. so ,,~**" in (b) can not be replaced by tt ~ ~t •
Let L(~) (respectively K(~) ) stand for the spaces~ all bounded operators (respectively all compact operators) from in-
to itself.
PROBLEM 3. Is every absolutely summing operator from L(~)
into a Hilbert space compact?
Obviously every absolutely summing operator from K(~) into
~ is compact because the dual of K(~) is separable. However
Problem 3 has a local counterpart for K ($~) •
PROBLEM 3'. Do,ether, exist a "modulus of capacity" ~*N(8)
such that if ~(~:K(~) L ~) ~ ~ then the $-capacity of
~(~K(~)) does not exceed N(~) (here ~K(~)=ITEK(~):JJTJJ~} )$
The positive answer to PROBLEM 3 will follow if one could estab-
lish the following structural property of L(~) : ~ let X be a subspace of h(~ ~) isomorphic to ; then
there exists a subspace Y of h (~) isomorphic to a ~(~space such that Y ~ ~ is infinite-dimensional.
Our last problem concerns spaces of smooth functions.
PROB~ 4. I s every absolutely summir~ opera to r from ~K(~)
into a ,Hi!ber~ s~ce compact?
We do not know whether there exists a "Paley phenomenon" for ~(T ~) ~ i.e. whether there is a absolutely summing surJection
from ~(~ )onto $~ (of. comments to PROBLEM I). It seems to be
unlikely that there exists an i n v a r i a n t absolutely summing surjection, as in %he case of the disc algebra A .
The author would like to thank Prof. S.V.Kisliakov for a valuab-
le discussion.
REFERENCES
I. 0 v s e p i a n R.I., P e • c z y ~ s k i A. On the existen-
ce of a fundamental total and bounded biorthogonal sequence in
every separable Banach space, and related constructions of uni-
formly bounded orthcnormal systems in L ~ . - Studia ~th., 1975,
54, 149-159.
2. W e i s L. On strictly singular and strictly cosingular opeB-
tots. - ibid., 285-290. /
3. P e • c z y n s k i A. Banach spaces of analytic functions and
absolutely summing operators. Regional conference series in mathe-
matics, N 30. AMS, Providence, 1977.
4. B o u r g a i n J., D e 1 b a e n ~. A class of special
~o@-spaces. - Acta ~ath., 1980, 145, N 3-4, 155-176.
/
A° PE&CZYNSKI Institute of L~athematics
Polish Academy of Sciences /
S~adeckich 8,
00-950 Warsaw, Poland
10
1.3. WHEN IS -- . - --- - - - - - - ~ ( X , Zh=L(X,Z~)~
Let X and ~ be two inf in i te dimensional Banaoh spaces and let LCX, Y) denote the space of all continuous linear operators
from X to Y . An operator TELeX,V) is said to be p- a b-
s o 1 u t e 1 y s u m m i n g if there exists a positive constant
C such that
K-4 K=4
for each ~ in ~ and ~4, ~,''''~EX . The set of all
p-absolutely summing operators from X to Y is denoted by
~ <X ,Y) . The conditions for ~p<X,~) to coincide with
~$ <X,Y) or with U(X, Y) have been the subject of a great num-
ber of publications (see [1] -[~). The results obtained are not only
of their own interest but also are widely used in problems connected
with the isomorphic classification of Banach spaces.
It is easy to see that ~p(×,V) ~ n~C×,V) for ~<$ .
The Dvoretzky theorem on almost Euclidean sections of convex bodies
shows that the equality ~p~X ,V) = L(X ,Y) has the highest chance
to hold if V is iso~orphlc to a Hilbe~ space, i.e. that RpCX,~
= LLX, 6s) provided RpCX,Y)= LCX,Y) at least for one i n f i n i - te dimensional space y . Besides, i t is well-known that ~p(X, ~ ' )= = n~CX,~ ~) for p ~ . ~hus ths investigatio~ of the problem whe- ther RF(X , y) coincides with LCX,Y) leads immediately to the question of conditions ensuring the equality
n~ ~x, ~) = ~-(.x, ~,) (1)
A space X satisfying (I) will be called 2-t r i v i a I
(cf.[~). Obviously a space X and its dual X* are 2-trivial (or
not) simultaneously.
A GENERAL QUESTION we want to raise is to find out conditions
(in particular the conditions of geometrical nature) under which a
Banach s~ace X is (or is not) 2-trivial.
It is known~ [6] that (1) is impossible for the space X not
containing ~ uniformly (for example if X is uniformly convex).
11
On the other hand it is easy to verify that this condition is not
sufficient for the 2-trlviality. Indeed, the sequential Lorentz space
ACc) not only contains 4 ~ uniformly but is even saturated by
subspaces isomorphic to ~ (to wit every infinite dimensional sub-
space of A ~C) contains a subspace isomorphic to ~ ). Neverthe-
less A CC) fails to be 2-trivial and moreover it is a space of
type ~) (see the definition below).
It can be proved that X is not 2-trivial provided X satis-
fies the following condition: there exist two sequences ~ A ~
and I~l~ ~ of operators such that
A~: ~--X B~: X--e~ B~A~=~2
and ~ ~-~I A~I" IB~I= 0 . A space satisfying ~hese con- ditions is said to be of type (~).
~4 It has been essentially proved in ~6] that a space not containing
uniformly is the space of type ~) . However the condition
of being of type ~) is also not necessary for the non-2-triviality.
As S.V.Kisljakov has pointed out, the reflexive "non sufficiently
Euclidean" space built in ~ fails to be of type ~) and simulta-
neously it can be proved that this space fails to be 2-trivial.
What has been said above indicates that the class of all 2-tri-
vial spaces cannot be too large. The following conjecture looks there-
fore rather plausible.
CONJECTURE 1. No infinite-dimensional reflexive Banac h space , is
2-trivial.
An equivalent statement: t h e r e e x i s t s n o i n -
finite-dimensional reflexive Banach
space ~ ~ s~ch that each operator
from L(~,~) , which c~n be factored
through X , is a Hi 1 b e r t-S c hmidt
o p e r a t o r. We note that a positive solution to CONJECTURE I
would obviously imply the solution (in the class of reflexive Banach
spaces) to the GROTHENDICK PROBLEM on the coincidence of the spaces
of nuclear and compact operators.
The following QUESTIONS arise naturally.
I. Under wha t conditions does 2-triviality oif a I space X imply
the e,qualit2 ~ (X, B~) = L(X, B ~) ?
12
2. Which of the spaces of anal~tic or smooth , functions are
3. Is it true that in an~ space X df type (~) there exists
a sequence of subspaces { X~} ( ~ X~= f~) with one of the follow-
t.,o properties:CO 0 x(X ,X)<oo, (t,) s o. <oo,
(Here ~ X n, X) is the relative projection constant).
The assumption that a 2-trivial space has an unconditional ba-
sis apparently rather drastically diminishes the class of such spaces.
For example,each reflexive Banach space with an unconditional basis
is not 2-trivial [8]. On the other hand, as it is shown in [9], the
space (~Co)~ also fails to be 2-trivial (more precisely it is
of type (~) ). These results give some ground to the following
CONJECTURE 2. If a 2-trivial infinite dimen@ional Banach sp~ce "
X has an unconditional basis, then X is isomorphic to either ~o
£4 o__.r qe •
To illustrate conjecture 2 we mention a result which follows
from Theorem I in [8] : If X has an unconditional basis, if ~ is
not isomorphic to a Hilbert space, and if [-]~(,,X,Y) --L(X,Y) then X is isomorphic to O o .
RE FERENCE S
I. L i n d e n s t r a u s s J., P e ~ c z y ~ s k i A. Abso-
lutely summing operators in ~p-spaces and their applications. -
Studia ~ath., 1968, 29, 275-326.
2. K w a p i e ~ S. On a theorem of L.Schwartz and its applica-
tions to absolutely summing operators. - ibid., 1970, 38, 193-201.
3. D u b i n s k y E., P e £ c z y ~ s k i A., R o s e n t -
h a I H. On Banach spaces X for which ~2 (~ ,X)=~(~o°'X)
-ibid., 1972, 44, 617-648.
4. M a u r e y B. Th~oremes de factorisation pour les op~rateurs • • • LP r . llneazres ~ valeurs darts les espaces . - Asterzsquep 1974, 11,
I-I 63.
5. M o r r e 1 1 J.S., R e t h e r f o r d J.R. p-trivial
Banach spaces.- Studia Math., 1972, 43, 1-25.
6. D a v i s W.J., J o h n s o n W.B. Compact nonnuclear ope-
rators. -Studia Math., 1974, 51, 81-85.
13
7. J o h n s o n W.B. A reflexive Banach space which is not suffi-
ciently Euclidean. -ibid., 1976, 55, 201-205.
8. K o H a p ~ e B H.A. 0 2-a6coJ~0TMO c y ~ ~ oIiepaTopex B
6alaxoBHx pemeT~aX. - BeCT~ /[FY, cep.MaTeM., ~ex., acTpo~.,
I980, .~ 19, 97-98. 9. ~ i g i e I T., L i n d e n s t r a u s s J., M i 1 -
m a n V. The dimension of almost spherical sections of convex
bodies. - Acta Math., 1977, 133, 53-94.
I.A. KOMARCHEV
(H. A. KO~H~IEB) B. M. NAKAROV
(B.M. MAEAPOB)
CCCP, 198904, HeTpo~Bopen,
~M6~OTeqHa~ ~. 2, ~aTeMaT~o-Mexa~Eec~
~aEyx~TeT ~eB~cl~oro
Y~Bepc~TeTa
14
1.4. old
FINITE DIMENSIONAL OPERATORS ON SPACES
O~ ANALYTIC E/~CTIONS
Let A be the B~n~ch space of all functions continuous in
~ ~ and analytic in ~ , equipped with the supremum norm and let
H~ me the Hardy space. W consider A as a subspace of C(~) and HI
as a subspace of ~I(~) ~ We would like %o know the relation between
finite dimensional subspaces and finite dimensional operators in
and those in O(~) . This question is of importance in the theory of
the Banach space A . we feel also that such connection, when expres-
sed in precise terms, can lead to some new isomorphic invariants of
Banach spaces. Let us start with the following
PROBL~W~ ~. ~et X be an ~-d~ensional subs~ace of A . Doe,s
every projection P: A c~% ~ extend to a DrojectionP: CC~) ~---~'~X
May be we ~ve only
This problem is obviously a special case of the following
PROBLE~ 2. Le,t ~ ,be a Bs~ach space, and le$, T" ~ ~,,~ ~ be
op, ,e ra tor o f r a n k ~ . Does t h e r e e x i s ~ a n e x t e n s i o n T : C(~) ~'
Another particular case of this problem is also of interest-
such tha~ ~ o ~ ~X~ where ~ is ,,,the canonical quotient,, ~ p f ~
onto
It seems that the estimates of ~he above type can be useful in
proving the non-isomorphism of spaces of analytic functions of diffe-
rent numbers of variables. There are also some problems of this type
connected with Schauder bases. Let us recall that a system (~')~>/4.--
of elements of a Banach space ~ is called a S c h a u d e r
b a s i s if for every ~ , ~e~ , there exists a unique sequence O@ of scalars (~)~ such that the series ~ a ~ converges to
in the norm o~ ~ . If it is so, then there exists a cons%ant
15
N such that for every ~ I ~ % ~ . Jl ~ K ~}J The best such constant is
~=i called a basis constant of the basis (~) . S.V.So~kariev [I] has
proved that the disc algebra A has a Schauder basis. On the other
hand it was proved in [2] that A does not have a Schauder basis
with constant 1. So the question arises.
PROBLEM 4. Does ther e ~xist a constant ~, ~ ~ I , such that
every basis for the disc alEebra A has the basis constant > ~ ?
It was proved by P.E~!o [3] that there exists a B~uach space which
has the propert~ described in the problem ~
Our last problem is connected with the space of polynomials.
~et W J denote the l~ear span of ~ ~ ~'~ ~ "~ oo~Idered in the l,P(1~t) norm. It is known [4] that the norm of the best projection
A -~W~ and from ~4 onto W~ is of the order ~0~__ . from onto
If X and ~ are two ~-dimensional Banach spaces then we define
the B a n a c h - M a z u r d i s t a n c e between X and
by
Y, = {11 ' 11" II'P-'II: T • x ' -L Y }
PROBLEM 5. (a) Let X be an ( ~ + t ) -dimensional subspace of
[4' H' • IS it true tl~,, for eve,riz projection P fr~ onto X w_~e
(b) Let X be an (~+i) -dimensional sub spac e of ~I . Is it
true that for every pro.lection P from ~ ont O X we have
A positive solution to Problem 5(b) immediately yields that ~I
and ~1(~x~, ~x~) are non-isomorphic Banach spaces.
RENARK. In the above problems ~ means an absolute constant.
REPERENCES
I. B o ~ ~ a p e 13 C.B. Cy~ecTBoBa~e das~ca B npocTpa~cTBe ~ ,
aHa~T~ecE~x B ~pyre, z EeEoTopHe CBO~OTBa C~CTeMH ~ p ~ . --
MaTeM. cd., I974, 95, ~ I, 3-18.
16
2. W o j t a s z c z y k P. On projections in spaces of bounded
analytic functions with applications. - Stud.Math., 1979, 65,
N 2, 147-173.
3. E n f 1 o P. The Banach space with basis constant >
fSr Mat.,1973, 11, 103-107.
4. Z y g m u n d A. Trigonometric series, v. 1, Cambridge
Press, 1959.
• - Arch.
Univ.
P.WOJTASZCZYK Institute of Math.
Polish Academy of Sciences
~niadeckich 8,
00950 Warsaw, Poland
COMMENTARY
The AFPIRNATIVE answer to PROBLEM 2 (and therefore to PROBLEHS I
and 3) has been obtained by J.Bourgain (cf. the references mentioned
in the Commentary to S I)
PROBLEM 4 seems to be open.
PROBLEM 5 has a NEGATIVE solution• Namely, there exist a sequen-
ce ~ v~} of subspaoos of CA (of H ~ ~ and, sequence {P~} of H 40~to projections P~; C A 0~,0 V~ (resp. P~: -, V~ ) with
I This has been observed by J.Bourgain and A.Pelczynskz. Let us
SKETCH THE CONSTRUCTION for the disc-algebra (the H 4 -case is con-
sidered analogously).
Replaq~ C A by the direct sum OA@~ C A which is isomo hic
to CA(~CA~--~:~CA}). Then define I:W~---~CA~zcr~
a~d Q: C A ~ ~A - W~ by [(p) : (p, g(~÷t)p) ;
where ~, stands for the ~-th Pejer kernel. It is easy to verify
17
that I is an isometry, QI=~ and ~ ~Q~ <+~ •
The spaces (T) and H (T ~ are non-isomorphic [5]. More-
ever no two of the spaces ~ (~) are isomorphic. The last result
has been also proved by J.Bourgain.
REFERENCE
5. B o u r g a i n J. The non-isomorphism of H ~ -spaces in one
and several variables. - J.Funct.Anal., 1982, 46, p.45-57.
18
H 4 H ~ 1.5. COMPLEMENTED SUBSPAOES 01~ A , AND old
Per Enflo's counterexample to the approximation problem [I], and
subsequent results by Davie [2] and Figiel [3], indicate that an iso-
morphic classification of all closed subspaces of a Banach Space X
( X net isomorphic to a Hilbert space) is probably impossible in the
near future. An important and difficmlt, but not impossible, problem
is the classification of the complemented subspaces of X . Because
of the recent advances in the study of the Banach space properties
of the Disc Algebra A • ~i , and~ (see [4]), I think we can now
give serious consideration to classifying their complemented sub-
spaces. As a first step in the process, I make the following con-
jecture:
c o w , o H m . A and H ~ _ _ are prlmar~.
A Banach space X is p r i m a r y if wheneverX~ ~ @
then either X ~ ~ or X ~ ~ . In support of the conjecture we
will prove that if A~ ~ @ ~ and if ~ is isomorphic to a comple-
mented subspace of C[0~I] then . We first use an observa-
tion of S.V.Kisljakov which states tha~ if ~ ~ • ~ and if ~/
is isomorphic to a complemented subspace of O [0, J] then ~ is
non-separable, To see this, we let P be a projection of A onto
and use an argument similar to the proof of corollary 8.5 (e) of [4]
to show thatP*l ' ' 4~/Ho ~ maps weakly Oauchy sequences tO norm conver-
gent sequences. If ~= is separable, it is known that weak and norm
convergent sequences in ~* coincide, and hence it follows that the
same i~ true in 11/~ - which is a contradiction. It now follows by
coro l la ry 8.5 (b) of [4] that G[O,~] is isomorphic to a complemented ~bspace o f Z , i . e . Z ~ GLQIJe W for s o m e s p a c e W . since C[0.I] i s primary [5] , i t f o l l o w s t h a t ~e 61%1] ~ C [0,1] . Hence
A ~ Y ® Z ~ Y m C [0.1] • W ~ c [ 0 , 1 ] ~ W ~ Y . Bochkariev [6] has shown that A has a basis consist ing of the
Franklin system in I.~ [0, ~ ] . (Here we are identifying A with
the subspace of C[-~',sT] spanned by the characters {e$~}~ 0 ).
If we let ~ be the span of the first ~ elements of this basis,
Delbaen has recently announced that ( ~ e ~ ) Co is isomorphic %o a
complemented subspace of A This subspace is particularly interest-
i~E because it is not isomorphic to A and it is also not isomorphic
to a complemented subspace of C[0~I] . The complement of this sub-
space is .-W-own and identifying it should be the first step in prov-
ing (or disproving) the conjecture.
19
We now outline one approach you might use to try to prove the
conjecture. If X~X @ X , then X is primary if and only if X
satisfies: (I) If X ~ ~ , then either ~ or Z has a complement-
ed subspace isomorphic to X ; and (2) If ~ is a Banach space an~
if X and Y are isomorphic to complemented subspaces of each # .,
other, then X ~ . By Pe~czynski s decomposition method, if
~P 4~ p ~oo them property X ~ ( ~ e X ) E , where E i s Go o r ,
(1) implies property (2) . Therefore, you should f i r s t consider the question of MitJagtn [7] : Is A isomorphic t o ( ~ e A ) ~ ? TO give
a positive answer to this question, it suffices to show that(~@
is is~orphlc to a complemented subspace of A . In this case then,
you need to carefully examine the construction of Delbaen. Next, you
should try to generalize the technique of [8] to the basis of A , or
produce a new basis of A for which the technique works. This app-
roach to the problem has the advantage that it may ~mmedlately imply
that ~ is primary. Since Wojtaszczyk [9] has shown that ~
~(E~ ) ~ ( ~ e ~ ) ~ . , i f the above approach proves that A
is primary, then the technique of D O ] should show that ~ is p r i - mary. As a word of warning concerning the naivety of the conjecture, l e t us mention that the only complemented subspaces of A which are known are e i the r isomorphic to ~ , ~ Y , or to X ~ ~ ~ where X is isomorphic to a complemented subspace of C ~0,I] and
Y~ ( Zle E~)co with dim for all ~ 1,2, ....
luch less seems to be known about the subspaces of ~ . It is
also not known if ~ is primary. If you want to try to prove that
~i is primary, you should first consider the question: Is ~I iso-
morphic to (~ @ ~ )~i ? Next, you should look at Billard's basis
for ~" ~. Since this basis is even more directly related to the
Haar system than the basis for A , this question could actually
prove to be easier than ,Me others. (Again using the techniques of
[8] Do] ) .
REFERENCES
I. E n f I o P. A counterexample to the approximation property in
Banach spaces. - Acta.Math.~1973, 130, 309-317.
2. D a v i e A.M. The approximation problem for Banach spaces.
- Bull.LondonMath.Soc.~1973, 5, 261-266.
3. F i g i e 1 T. Further counterexamples to the approximation
20
problem, dittoed notes.
4. P e i c z y ~ s k i A. Banach spaces of analytic functions
and absolutely summing operators. - CBMS Regional Confer.Ser.
in Math.31977 , N 30. /
5. L i n d e n s t r a u s s J., P e ~ c z y n s k i A. Con-
tributions to the theory of classical Banach spaces. - J.~unct.
Anal.~1971, 8, 225-249.
6. B o q ~ a p e B C.B. CymecTBO~H~e 6aBHca B npocTpaHcTBe SyEK--
n~, a~a~IT~eCF~X B Epyre, m HeEOTOpHe CBO~CTBa C~CTe~g ~pa~E-- ~m~a. -MaTeM.c6., 1974, 95, ~ I, 3-18.
7. M H T a r Z H B.C. POMOTOn~xecKa~ cTpy~Typa ~e~o~ rp~ da-
HaxoBa npoc~paHcTBa.-Yc~ex~ MaTeM.~ayx, I970,25,~ 5, 63-I06. 8. A I s p a c h D., E n f 1 o P., 0 d e 1 1 E. On the struc-
ture of separable ~ spaces (4<~<~).. - Stud.Math.~1977,60,
79-90.
9. W o j t a s z c z y k P. On projections in spaces of bounded
analytic functions with applications. - Studia N~th., 1979, 65,
N 2, 147-173.
10. C a s a z z a P.G., K o t t m a n C., L i n B.L. On some
classes of primary Banach spaces. - Canadian J.Math., 1977, 29,
N 4, 856-873. 4
11. B i 1 1 a r d P. Bases dans H et bases de sous espaces de
dimension finie dans A • - Proc.Confer.0berwolfach, August
14-22, 1971, ISN~ Vol.20, Birkhauser Verlag, Basel and Stutt-
gart, 1972.
P. G. CASAZZA Department of Mathematics,
The University of Missouri-
-Columbia, Columbia, Missouri 65211
USA
COmmeNTARY
P.Wojtaszczyk proved that (Z~ A)~O~' A (>"®H4)u H 4 when the first edition of the Collection was in preparation. (Now
his result is published, of. 512] ). Nevertheless the problem of pri-
mariness of A and H I seems to remain open.
J.Bourgain [13] has proved that H ~ is primary (and the same
21
is true for ~-'H~(B ~ ), M~ being arbitrary). It is worth mentioning @@
that in Bourgain's proof the relation H~ (~ W~ )~ is
... ) , with the sup-norm), rather used = ,z
H" _ )~ , as proposed in the text of Problem. The
decomposition ~ ~(~ W~ )~@ is due to Bourgain and Pelczy~-
ski. The reason why this is valid rests just on the observation about
the complemented imbeddings of ''W ~ A into ~ made in Commentary S
to Problem I 4
We end with a quotation from the author's letter to the editors:
"The primariness ~f ~ ~4 seems to be still unknown but with the myriad
of new results here the last few years by J.Bourgain, this may be
"almost obviously" true~
REFERENCES
12. W o j t a s z c z y k P. Decompositions of
Duke M~th.J., 1979, 46, N 3, 635-644.
13. B e u r g a i n J. On the primsrity in H H~
tint.
H P -spaces. -
-spaces. - Prep-
22
1.6. SPACES OF HARDY TYPE old
A Banach Space E of measurable functions on [0,~] is called
a symmetric (or rearrangement invariant) space iff the norm of E is
monotone and any two equimeasurable functions have equal norms. (~],
chapter 2). The ~ -spaces (~ p~ ) , the Orlicz spaces and the Le-
rentz spaces can serve as examples. Remind that if the function
is non-decreasing and concave on [0,~] ,~(0)~0 , then the Lorentz
space A(~) consists of functions ~ such that
4
o
where ~* is the function non-increasing on[0,~g] and equimeasur-
able with ~ .
A symmetric space ~ gives rise to a space of complex functions
on ~ consisting of functions with moduli from ~ . This space is
also denoted by ~ . By ~(E) we denote the set of all functiom~
analytic in the unit circle D and satisfying ll~llH(E)~Oo,.
II # IHcE):
Thc the classical Eardy s ces H H(L p) have b.e studied rather well, the theory of general spaces H(E) is only fra@men-
tary.
The set of extreme points of the unit ball of A(~) is con-
%e.~ , where I £(~)I~--- ~ and tained in the set of functions ~(~)
~ is the characteristic function of a measurable set 6,eG[0,~].
In the case when ~ is strongly concave these two sets coincide. The
following PROBLEM arises natru~lly: describe the set of extreme points
of the unit ball of H(A(~)) . Some partial results are contained
in [2]. The space H(A(W)) is nothing but ~4 if ~(t)~-~ , and
coincides with ~00 , if ~(~)=5~?t~ . In these two cases the set of
extreme points of the unit ball is well-known, see [3], part 9.
We believe that the solution of the above-mentioned problem will
possibly be useful for describing all isometric operators on H(A(~))
Some interesting results on isometric operators on a symmetric space
are contained in [4].
23
RE~?ERENCE S
I. KpefiH C.r., HeTyHEH D.E., CeMeHOB E.M.
MHTepno~ ~ e f i ~ x onepa~opoB. M., HayEa, 1978. 2. Bp ~ c E E H Hob., C e~ a e B A.A. 0 reOMeTpz~ecEEx CBOfi--
CTBaX e~EHE~HOrO ~apa B ~pocTpSHCTBaX TEna F~aCCOB Xap~E. - 8an.
HayXH.ceMEH.~0MH, 1974, 39, 7--16o
3. H o f f m a n K. Banach spaces of Analytic Functions.Prentice-
Hall,Englewood Cliffs, New Jersey, 1962.
4. 8 a Pl ~ e H d e p ~ M.~. K EBOMeTpE~eoEofi F~accE@EE~ C~M--
MeTpE~L~X ~pocTpS~CTB. --~OF~.AH CCCP, I977, 234, #~ 2, 288-286°
E.M. SE~NOV
(E.M.C~0B)
CCCP 394000 BopoHem,
BOpOHe~OE~ r0cy~apcTBeHH~fl
Y~NBep CET e T
EDITORS' NOTE. Here are some more articles connected with iso-
metries of ~-spaces of analytic and harmonic functions:
I. H ~ o T ~ ~ H A.H. Hpo~o~meH~e L P -~30MeTpv~. - 8an.Hay~H.
ceM~H.~0~, 1971, 22, 103-129.
2. H ~ o T E Z H A.H. HSOMeTp~eoEEe oHepaTopH B ~ --LrpocTpSHCT--
BaX aHaJfi~TE~ecE~x E rapMoH~ecEHx ~d~. - Ibid. , 1972, 30,
130-145.
3. H ~ o T E E H A.M. A~redpa, nopo~eHHa~ oneps~opaM~ O~BEra,
~--HOpM~. B EH. : "~HZn~oHaz~m~ aHa~z3. BB~ycE 6. I~e~By3oB--
CE~ c6opHEE", YJ~OBCE, I976, II2-I2I.
4. H x o T E ~ H A.M. 06 ~SOMeTp~xecEEx onepaTopax B npocTpaHcT--
Bax cy~pyeMNX a~a~ETEecEEx E rapMOHE~ecEEx ~yHEn~. -- %OEX.
AH CCCP, 1969, 185, ~ 5, 995-997.
24
1.7. BASES IN H P SPACES ON THE BALL
£. By H[8) we will mean the natural Hardy space of analytic func-
tions on the unit ball of C" . A(B) denotes the ball algebra
of all functions continuous in ~ and analytic in ~ . We are in-
terested in construction and existence problems for Schauder bases in
these spaces. CO
Let me recall that a sequence of elements (;~)~=0 in a Ba-
nach spaces X is a Schauder basis for X if for every3~ in
)'~ such that the there exists~ a unique sequence of scalars (~n ~=0
series ~ ~n~ converges to ~ in the norm of X . The basis
is called unconditional if for every ~ in X the corresponding cO
series ~ n ~ is unconditionally convergent. ~-0
Per the ball algebra A(~) the question of the existence of a
Schauder basis is a well known open problem (cf. [I] ). This seems to
be the most concrete separable Banach space for which this question
is still open today. It is known that A(~) does not have an un-
conditional basis.
Por ~ ~ ~ <oo the situation is a little more intriguing. It
is a relatively easy task to check that for 4<P <oo the monomials
in a correct order form a Schauder basis for HP<~) . However this
basis is not unconditional for p~&~ . It was proved in [6] that
for 4 4 ~ <co H~8) is isomorphic as a Banach space to HP(~)~
the classical Hardy space on the unit disc. Since unconditional ba-
ses for are well known, of, [2], [4], [5], we get the exis-
tence of unconditional bases in HP(B) for 4 ~ ~ <oo . This argu-
ment however has one drawback, i% is non-constructive, me we pose
the following o
PROBLEM. Construct an unconditional basis in the space H~)
p %<oo.
The most interesting case is p ={ . There is also an auxilia-
ry question related to this:
Does there exist an orthonormal unconditional basis in H~(B) ?
The case p < ~ is less clear. In HPc~) we have uncondi-
tional bases, cf. [3] ,[5]. However in ~everal variables the very exis-
tence of an unconditional basis in Hr~ B), p < ~ is still open.
The proof of isomorphism between H{(8) and ~4(~)) given in [6]
25
can be extended to ~ < ~ (after some technical modifications)
provided the following question has positive answer.
QUESTION. I~s~P~), ~ < ~ isomorphic to a complemented sub-
HP(B) ?
REFERENCES
!
I. P e E c z y n s k i A. Banach spaces of analytic functions and
absolutely summing operators. CBMS regional conference series
N°30.
2. ~ 0 q K a p e B C.B. CymecTBoBa~e 6a3~ca B npocTpaHcTBe ~yH~--
I~, 8HS~T~qecENX B EpyDe, H HeEOTOpNe CBO~OTBa C~CTe~paHE--
/n~Ha. - ~Te~.c6OpH~E, I9V4, 95 (I87), B~n.I, 3-18~
3. S j 8 1 i n P., S t r o m b e r g J.-O° Basis properties
of Hardy spaces. Stockholms Universitet preprint No 19, 1981.
4. W o j t a s z c z y k P. The Franklin system is an unconditio-
nal basis in H ~ . - Arkiv f~r Mat.,1982, 20, No 2, 293-300.
5. W o j t a s z c z y k P. H P -spaces, ~ and spline sys-
tems. - Studia Math. (to appear).
6. W o j t a s z c z y k P. Hardy spaces on the complex ball are
isomorphic to Hardy spaces on the disc, ~ ~<oo . - Annals of
Math. (to appear)
P.WOJTASZCZYK ~ath.lnst.Polish Acad.Sci°
00-950 Warszawa, /
Sniadeckich, 8
POLAND
28
1.8. SPACES WITH THE APPROXIMATION PROPERTY?
Recall that a Banach space ~ has the approximation property
(a.p.) if for all compact Ec~ and for all 8 > 0 there is a bounded
linear operatorT : × • × ~oh that IJ~-T~H < 6 when ~ E .
and such that T has finite rank. Not every Banach space has the a~p. ~]
Does ~ have the a~p.?
Some mild evidence that this might be true comes from the recent
result that L~/ ~ (i.e.B~O) has the a°p, K2~ • Another interes-
ting space £or which the a.p. is unknown is ~'~(~)~ ~ ~.
(When ~ I the answer is easy and positives) Here
REPERENCES
I. E n f I o P. A counter-example to the approximation problem in
Banach spaces. - Acta Hath., 1973, 130, 309-317.
2. J o n e s P.W. BM0 and the Banach space approximation problem. -
Institut Mittag-Leffler report No,2, 1983.
PETER W.JONES Institut Mittag-Leffler
Aurav~gen 17
S-182 62 Djursholm
Sweden
Usual Address:
Dept~of Mathematics
University of Chicago
Chicago, Illinois 60637
USA
27
1.9. OPERATOR BLOCKS IN BANACH LATTICES
The operator Q 8 of multiplication by the characteristic func-
tion of a measurable subset 6c ~0,I~ has the unit norm in every
functional Banach lattice ~ on ~0,i~ (see ~ for a definition).
Associate with every continuous linear operator T : E
the number
and let
~ c E ) : { T c Z ( E , E ) : ~'(.T,E):-O } •
PROBLEM. Under what conditions on E th~ set ~ (E) i..ss
empty, i.e. g(T, E)= 0 fo r ever~ linear operator?
This question arose for the first time in ~2~ (for concrete
spaces) in connection with the contractibility problem of linear
groups in Banach spaces. In particular an isometry T of L~ sa-
tisfying • (T, L z) > 0 was constructed there. On the other hand~
it has been proved [3~ that ~(L~)~ ~(L ~) ~ and that
~(tS=~ , for 1<p~ [~. Recall now the definition of the Lorentz space ~'~ (see
[~ for their properties.) ~or a mea~ble f~ction ~ on ~O,g let 0C* denote the non-increasing rearrangement of j xJ . Then
4 4
lb 0
It is well-known that (LP'~) *: hp,r¢[ f ~ , where I '
,6< . Therefore without loss of generality i% can be assumed that V--p~ . It is also known that ~(L P'~) ~ ~ , ~<p<£ ~ d t ~ t n ~ ( L ~ "~') ~ c~ , l ~ p ~oo r . ~ .
F o r t h e c a s e p < $ < • oo t h e s i t u a t i o n r e m a i n s u n c l e a r .
28
is known about the set ~ ( h~' $ ) except $~ Nothing
when ~([,~) = ~(~) =~= ~ •
The problem of non-emptiness of ~( ~ ) remains open for
Orlicz spaces.
The operators T~ T(~)~ ~(~P) constructed in ~] depend on
and this is not a mere occasion. The set ~( l'P1)~ ~(-- P~) is not empty (let ~ "~ ~I "~ ~ < ~ for the definiteness) iff ~
~ ~ ~, [6]. However it is not clear what conditions provide
~(~4 ~ ~(~)~ ~ in the general case.
REg~NCES
I. L i n d e n s t r a u s s J., T z a f r i r i L. Classical
Banach Spaces II . Berlin, Springer Verlag, 1979.
2. M ~ T ~ r ~ H B.C. roMo~on~ecza~ cTpyETypa X~He~Ho~ rpym~ 6a~a-
xoBa npocTpsaorBa.-YcnexE ~eTeM.HayE,I970,25, }~ 5, 68--I06. 3. E d e 1 s t e i n I., M i t y a g i n B., S e m e n o v E.
The Linear Groups of ~ and ~4 are Contractible. - Bull.Acad.
Polon.Sci., Set.Math., 1970, 18, N 1.
4. C e M e H O B E.M., ~ ~ p e a ~ c o H B.C. 88~a~a o MaaOCTH
onepaTopm~x 62IOEOB B HpOCTRaHCTBaX ~.-- Zeit.Anal. und ihre
Anwend., 1983, 2, N 4.
5. K r e i n S.G., P e t u n i n Ju.I., S e m • n ~ v E.M.
Interpolation of Linear Operators. AMS Providence, 1982.
6. C e M e H o B E.M., ~ T e 2 H 6 e p r A.M. 0nepaTopm~e daOEE
B L~pOCTpaHCTBaX ~p,$ • -~OF~.AH CCCP. 1983, to appear.
E.M.S OV
( E. M. CE 0B)
CCCP, 394693
BopoHem, BopoHemcEE~
rOCy~&pCTBeHH~I~ yHE~BepCETeT
2g
1.10. SPACES OF ANALYTIC FUNCTIONS (ISOMORPHISMS, BASES) old
0(~) wil~ denote the space of all functions analytic in the
domain ~,~ C~ . A domain ~ is called standard if ~(~) is
isomorphic (as a linear topological space) to one of three (mutually
non-isomorphic) spaces
In [1] the class ~ of all standard domains was completely des-
cribed. Moreover in [ 1] the properties of ~ ~ were found out de-
termining to which particular one of these three spaces the space
~(~) is isomorphic. These properties involve the structure of
the set of all irregular points of the boundary $~ (see [2] for
notions from the potential theory). The isomorphic classification of
spaces ~(D) for ~ not in ~ remains unknown.
Any domain ~(~ ~)~\( ~ ~ (~i ~i) U {0}),
where ~(@,~)={~:l~-~l<~I, Sec0,~), ~= C~i)i,,4 i s a monotone sequence of pos i t ive numbers wi th
does not belong to ~ (and ~ ( $ , ~ ) ~ whenever the series i n ( l ) diverges).
CONJECTURE. There exists a continuum of mutu~ll,7 non-isomorphic
spaces ~(~(~)) . This conjecture is stated also in [6] (problem 63).
Let us mention - in connection with the open question on the
existence of a basis in the space ~(~) of all functions analy-
tic on a compact set K , KC ~, that this question is open for
K=C\ ~(~,~) as well (under condition (1))~though it was
proved [7] for such K that ~(K) has no basis in common with
~(~), ~ being any regular (in the potential - theoretic sense)
neighbourhood of ~. ~rom this fact it follows that ~C ~) has
no basis of the f ~=Q,
Let ~ be ~ -dimensional open Riemann surface.
30
(a) We say that ~ is regular iff there exists the Green func-
tion ~(~,E) with ~ G(~, Z~) = O, ~ e ~ , for any
sequence ( ~ ) with no limit point in ~ . Under additional rest-
rictions (for example if ~ is a relatively compact subdomain of
another Riemann surface ~4 ) it has been proved that ~(I~) and
~ are isomorphic if ~ is regul~r (cf.[8] and references therein).
Is this true in the general case? The necessity of this condition
follows from general results for Stein manifolds ([3] and references
therein)•
(b) Let ~ be a Riemann surface with the ideal boundary of capa-
city zero. Is then ~(~) isomorphic to, ~? The condition is
necessary even in the multidimensional case (unpublished).
(c) The QUESTION about the existence of a basis in ~(/i) is
solved only under some additional restrictions (even for surfaces
satisfying (a) and (b) above [8], ~9~).
Clearly ~ ( ~ ) and ~ (K) are non-isomorphic whenever
is open and K is a compact set (~, K c ~) .
QUESTION. Which other d%fferences in topological properties of
sets ~, E£ ( c C) imply that ~ (~4) and ~(Ez) are non-
isomorphic?
Here ~(E) denotes the inductive limit of the net {~(V)}
of countably normed spaces ~(V) , V running through the set
of all open neighbourhoods of E
that ~(.~ U&) 7@- f~(]~ uj~') ,j closed subarcs of the unit circle
non-degenerate arc of S D •
• V.P.Erofeev proved (unpublished)
and ~ being an open and a
~D • It is not known whether
are isomorphic if ~ is a closed
In [4] a method was proposed to construct common bases for
,(.9(~) and ~¢, K) , K c ~ . ~his method uses a special ortho-
gonal basis common for a pair of Hilbert spaces ~, , ~4 and for the
Hilbert scale x) ~ generated by ~0 and ~4 essentially genere- J ,,,,, ,,, ,
X) The notion of a Hilbert scale introduced by S.G.Krein has a number of important applications to problems of the isomorphic classi- fication of linear spaces and to the theory of bases• We refer to the paper by B.C.M~THD~H, r.M.XeHEEH , "~He~e sa~a~ EOMIRIeECHOPO aHa- x~3a", Ycnex~ MaTeM. HayE. 1971, 26, N 4, 93-152 containing many results concerning spaces" of analytic functions, a list of unsolved problems and an extensive bibliography. - Ed.
31
lizing well-known results of V.P.Erohin about common bases (see,e.g.
[4], [3], [7],[8], [9] ),
THEOREM ([4],[12],[8]). Let KCO, K: {££Q ; 11(£)1~'~1, V~ C ~(£)} and suppose ~\K is a regular domain in C
(or a relativel~ compact domain on a Riemann surf~ce~t Then there
exist Hilbert spaces ~0 , ~ with
H ~ (~(~) c_,. O(K)c_, FI, (2)
and, for a,,~l spaces ~ of the,,,,,,,correspondin~ scal,e
(O(c/x~ ~),~) c_, I-I~' c_,. 0 ( £ ~ ) , (3)
where ~ = ~ E ~ ; ~ ( ~ , K , ~ ) < ~ UK , ~(~, K,z) is the harmonic measure of ~ with respect to ~\ K ([3],
p.299). All embeddings in (2) and (3) are continuous. The common or-
thogonal basis ~6~) ~0 of the spaces N0, H 4 is a common basic
A QUESTION arises: hOT "far" is it possible to "move apart" the
sDaces ~0 , ~ sat isfyin~ (2) with0u ~ breakin~ (~)?
Let ~w(O) be a Banach space of all bounded functions analy-
tic in ~ . We consider Hilbert spaces ~4 with
The well-known Kolmogorov's problem about the validity
totic relation
for the ~ widths %~ (~) of the compact set
is the Green s capacity of the compact set K
can be reduced to the following
PROBLEM. Describe all domains ~ with (4) > (3) (for a
suitable Ho ) ( [8], see also ~11]).
of the asymp-
with respect to ~))
32
RE~ERENCES
I. 3 a x a p ~ T a B.II. IlpOCTpaHCTBa ~yHE~ o~oro nepeMe~oro,
aHSJH~TI~qecE~X B OTEpNTRX MHo~eCTBaX ~ Ha KOMIIaETaX. -- "~T.C6."~
I970, 82, J~ I, 84-98.
2. ~ a H ~ ~ O ~ H.C. 0CHOB~ coBpeMeHHo~ Teop~ noTeHn~a~a. M.,
"HayEa", I966.
3. 8 a x a p ~ T a B.II. 3EcTpeMa~H~e n~op~cy6rapMoH~ecEHe ~yH~--
L~L~, r~B6epTOB~ mE~ ~ ESoMop~EsM npocTpaHCTB SHS~T~eCENX
~ MHOE~X nepeMeHH~x, I, H. - B c6.TeopM ~y~L~, ~yHE~.
s~ax~3 ~ ~x npy~oz., Xap]s~OB, 1974, ~ 19, 133-157, ~ 21, 65-83.
4. 8 a x a p ~ T a B.H. 0 npo~oxmaeMHx 0a3Hcax B npocTpaHcTBax
a~a~T~ecEmx ~yHE~Z~ O~HOZO ~ MHOr~X nepeMeHR~x. - C~6~pcE.~a-
TeM.~., 1967, 8, ~ 2, 277--292.
5. ~ p a r ~ e B M.M., 3 axap ~Ta B.H., Xan~a-
H 0 B M.F. 0 HeEoTop~x npo6~eMax 6aszca aHS~mT~ecE~x ~yHE~.
-- B C6. : "AETya~e npo6~e~m HayE~", POCTOB--Ha--~OHy, I967,
9I--I02.
6. Unsolved problems. Proceedings of the International Colloqium
on Nuclear Spaces and Ideals in Operator Algebras, Warsaw, 1969.
Warszawa- Wroc~aw, 1970, 467-483. 7. 3 ax ap ~ T a B.H., E a~ a M n a T T a C.H. 0 cy~ecTBo-
Barite npo~oJ~aeM~x 6as~coB B npocTpaHCTBaX ~yHEL~, aHa~T~ec-
K~X Ha Eo~a~TaX. -- ~T.saMeTE~, I980, 27, ~ 5, 701-713.
8. S ax ap ~ T a B.H., C E ~ ~ a H.M. 0neHE~ ~-nonepe~-
HI~EOB HeEOTOpNX E~aCCOB ~yHE~E~, aHS~I~TEqecE~X Ha p~OBRX HO-
BepxHocT~X. --MaT.saberEd, 1976, I9, ~ 6, 899-9II.
9. C e M E r y E 0.C° 0 cy~ecTBOB~ O6m~X 6as~COB B npocTpa~cT-
Be aH~TEeCEEX ~ Ha EOMIIaETHO~ p~aHOBO~ HoBepxHOCTE,
POCTOB.yH--~, POCTOB-Ha--~oRy, I0 C, 6~0X.7 HaSB. (PyEon~c~ hen. B
BMHHTM I5 ~eBp. I977 ~ 620-77 ~en.) P~MaT I977, 6 B I38 ~en.).
I0. W i d o m H. Rational approximation and ~-dimensional dia-
meter. -J.Approximation Theory, 1972, 5, N 2, 343-361.
II. C E H 6 a H.H. 06 o~eHEe cBepxY ~-nonepe~EoB o~o~o F~ac-
ca rO~OMOp~X ~yHE~. -- B c6."Tpy~ mo~o~x y~eH~X Ea~e~pN
BMcme~ MaTeMaTHEE", P~M, POCTOB--Ha--~OHy, I978. ~eno~poBa~o B
BMHMTM, ~ 1593-78 ~en.
12. N g u e n T h a n h V an. Bases de Schauder darts oertains
espaces de fonctions holomorphes. - Ann.lust.Fourler (Grenoble),
1972, 22, N 2, 169-253.
V, P ZAHARIUTA
(B. If. SAXAHOTA) o, s. SEMIGUK
N. I SKIBA
(H.H.CEHBA)
33
CCCP 344711 POCTOB-Ha-AOHy
POCTOBCE~ PocyAapCTBeHH~
yHEBepCHTeT
POCTOBCE~ Mm~eHep HO--CTpOHT ea~m~2
EHCTMTyT
* * *
COMMENTARY BY THE AUTHORS
The problems a), b), c) (in a more general situation~namely for
Stein manifolds) have been solved in [13] by a synthesis of results
on Hilbert scales of spaces of analytic functions [3] and last re-
sults on characterization of power series spaces of finite or infini-
te type [14], [15].
We formulate one of this results as an example.
Let ~ be a connected Stein manifold. ~ is said to be P-regu-
lar if there exists a plurisubharmonic function ~(E) such that
~(~) < 0 in 0 and ~(Z~)-* 0 for any sequence (~) with-
out limit points in I~I --a
THEOREM. A(~)-~ ~(D ) if an d only if ii is P-regular.
REFERENCES
13. 8 a x a p ~ T a B.H. ESOMOp~MBM npocTpaHCTB 8aSZ~TM~ecEEx ~yHE-
sm~. -~oKa.AH CCCP, 1980, 255, ~ I, 11-14.
14. v o g t D. Eine Charakterisierung der Potenzreihen~'ume vom
endlichen Typ und ihre ~olgerungen, preprint (to appear in Stu-
dia Math. ).
15. V o g t D., W a g n e r M.J. Char~kterisierung der Un-
terr~ume der nuk!earen stabilen Potenzreihenr~ume vom unendlich-
en Typ, preprint (to appear in Studia Math. ).
34
~ . ~ . o~ z s o ~ o ~ c OZaSSZ?ZOA~O~ o~ F - s ~ o s s
I. For a given family of positive sequences {~p~ let K(%p) ~e a ~Sthe space i.e. F is the space of a l l sequences ~ = { ~ 4 satisfying
l~Ip~---~, l~ t~p <+~, p=~,~, .... (1)
The space ~(@{p) is endowed with the topology defined by the fami-
ly of semi-norms (I). It is called a p o w e r ( K o t h e )
s p a c e if ~p-kp (i) ~ where -o0 <~p(~) ~<kp+ 4 (~),
kp(~)-k~(h.<C(p)<+~, ~,pe~. For e~p le %he so-called power series spaces
are power spaces in our sense,
(infinite) type if
Consider two
I) the class
E& (~) is said to be of finite ~ < + ~ (~= +~).
classes of power spaces:
S of power spaces of the first kind [!], [2]:
2) the class ~ of power spaces of the second kind:
p
in both oases. If we consider isomorphic spaces as identical, then ~ n
consists of spaces (2) and also of their cartesian products;
contains spaces E 0 (~) @ E~ (~) [2] and ~ contains spaces of a l l analytic functions on unbounded n-circular domains in C ~ [3].
35
PROBLEM 1. Give criteria of isomorphisms:
'-" E (,p, 6) ( .3)
F( k, ,,, FI , (4)
in terms of (k,&) , (J~,~) .
Articles [I], [2] co~tain a criterion of isomorphism (3) but un- der an ~ddition~l requirement on ~(~,~) (note that in A~S trans-
lation of [I] in Lemma 4 the important chain of quantifiers ~p/ ~p~ B~/ ~v / ~ ~5 ~5'~ C ~,Z, 6"~0' has been omitted).
Let us formulate one result on isomorphism (4), which somewhat
generalizes the result of [3]. Denote by ~ the set of all sequen- ces A=(~%), 0<~%~<4 such that there exist limits
and ~(~) is strongly increasing in ~ .
TEEORE~ I. Let ~ ~ ~ and ~i ~" ~ .0Then (4) implies
1) - ,~ 4, ,
Note for a comparison that isomorphism (3) takes place for ar-
bitrary ~,~ A whenever the condition h i ~ ~ is fulfilled. Class ~ is also of a great interest because the following con-
Jecture seems plausible.
CONJECTURE 1. There exists a nuclear powe r space of the ' second
k indwithout the bases quasiequivalence property ~).
2. Let X be an F -space with the topology defined by a sys- tem of semi-norms [ l'Ip, pE ~I and ~(~) be a convex increas-
ing function on [~, ~) . Denote by ~ the class of all spaces X
such that J ~ ~¢ 3 M'I,, ~ , C :
,J ,,
*) The definition see e.g. in the article M~T~rBH B.C. AnUpoEcm- mBTNBHSM 1085MSIOHOOTT= ~ (~85,~C 1~ B 8~SpHHX HIDOCTI08HCTSSX- - Yonexa UaTeM. HayE, 1961, I6, ~ 4, 63-I32.
36
C, I,ll , (:f(t,D lllp IIl, , -t,
Classes ~ ~ being invariant with respect to isomorphisms,
are a modified generalization of Dragilev's class ~ [4] (see
also ~5]); similar dual classes ~ were considered in Vogt'Wag-
net [6S .
Classes ~ have been used in [7] to give a positive answer
to a question of Zerner. Consider the family ~ of all domains
with a single cusp:
being a non-decreasing C1-function on [0,{] , ~(0) =0 .
Then there exists a continuum of mutually non-isomorphic spaces
C~Q~) with ~ E ~ . The following theorem clarifies the
role of classes ~ in this problem.
THEOREM 2. C (~.) ~,0 i f f there ex is t ~ , ~ >0 ~ ....
e a t i s fy in~ ~/~(~)%C~ (~ , ) ) J~ for 0(0~ <~o •
PROBT.~ 2. Are all spaces C'( ~ ) from the same class
~ isomorphic or there exists a more subtle (than in Theorem ~)
classification of these s~aces?
CONJECTURE 2. There exists a modification (apparently very essen-
tial one) of VoEt-Wagner's 01asses ~ which allows to prov e the
oon.~ecture on the exist~noe of a continuum of mulually non-isomorph~o~
spaces ~ (~ (~,$)) (see this Oollection, Problem ~ 10)
REFERENCES
I. S a x a p ~ T a B.H. 06 ~SOMOp~sMe ~ EBaSEaEB~Ba~eRTHOOTE 68--
B~COB ~ cTene~x HpOCTpaHOTB E~Te.'~AH CCCP, 1975, 221, ~ 4, 772-774.
2. 8 a x a p ~ T a B.H. 06 EBOMO~I~BMe E EBaSEgEBEBSJIeRTHOCTE 6a- s~COB ~s cTe~eBR~x npocTpaRCTB E~Te.-Tpy~ 7-~ ~porod~cEo~ Ma-
TeM.mEox~ no ~yB~.aRax~sy, M., 1974.
S. S a x a p ~ T a B.H. 06o6~ea~He ~H~ap~aaTN M~T~Z~Ha ~ EORT~--
37
RyyM gonspRo ReEBoMop~RHX npOCTpaHCTB aHa~ETE~ecEEx ~yREL~2.- ~yREL~EOHa~H~ a r a b 3 E ero n p ~ o ~ e R ~ , 1977, I I , ~ 3, 24-30.
4. 3 a x a p m T a B.H. HeEoTopNe x~He~RNe TOnO~Or~ec~e ~mBa-
pEaRTH E I~BOMOp~ESMN TeHBOpHNX npOESBe~eR~L~ HeRTpOB mEa~.-HsBec-
TES CeBepo-KaBEascEoro Hay~Roro ~eHTpa BHcme~ meow, 1974, 4,
62-64
5. V o g t D. Charakterisierung der Unterraume yon S.-Math.Z.,
1977, 155, 109-117.
6. V o g t D., W a g n e r M.J. Charakterisierung der Quo-
tienter~raume von S und eine Vermutung yon Martineau.-Studia Math.
1980, 67, 225-240.
7. r o H ~ a p o B A.H., 3 a x a p m T a B.H. HpOCTpHHCTBO 6ec-
~oRe~Ro ~epeHn~pyeMHx ~yRmm~ Ra o6nacT~x c yr~aME (to appear)
V.P.ZAHARIUTA
(B.H. 3AXAPIOTA) CCCP, 844 711, POCTOB--Ha-~oHy,
POCTOBCEEI] rocy~apcTBeRH~
yH~BepCETeT
38
1.12. WEIGHTED SPACES OF ENTIRE PUNCTIONS
Le¢ p: C-~ be a continuous function and define
where ~ E ~ + . We suppose that [ ' ~ % II'll s for ~>$>~ and
that ~ is not t r i v i a l . I t i s eas i ly seen that ~ i s a ~r~chet space, the ~opology of which is strictly stronger than the topology
of uniform convergence on the compact subsets of C . With the help
of the Riesz representation theorem the dual space of ~ can be
identified with the space of all complex valued measures ~ on C
such that
f o r an %>~ (see ~5~). As an example cons ide r p ( z ) = l ~ l ~ , f o r %>0 , then i s
the space o f a l l e n t i r e f u n c t i o n s o f o rde r ~ and type ~ (see
t h i s =onomials fZ"t 0 c o n s t i t u t e = So uder basis in ~ and ~ is topologically isomorphic to the space
of all holomorphic functions on the disc ~ if ~>0 and to the
space ~(0) of all entire functions if ~=0 , both spaces ~)
and ~L,(.~) endowed with %he topology of compact convergence. Here, ~ is also a nuclear Pr~chet space (see ~7~) and the dual space can be
identified with a space of germs of holomorphic functions (Kothe dua-
lity [43 ). All this can be used to find a solution for interpolation
problems such~as for instance I (~) (~ • ) = ~ • , ~ = 0,4, ~ ,...
in the space ~ by means of methods from functional analysis (see~3], [I] ).
If we take ~(~)= [~z I = e x , ~=X+t~ (here p is net a function
of I~I ) , then the corresponding spaces ~ do not contain the poly-
nomials and properties s~m!lar to the above example are not known,, Ano%her example of in%erest is due to Gel'land and Shilov ~2~:
z ~
K,OI,"O,4,~,,.,., where &,J~,A,~ >0 and ot+}~4.
39
Tu_fact, each function ~ ~:~ can be extended to ~,~ and ~,A coincides with the space of all entire functions
such that
l
where 0< / < ~ , >~ >0 (see ~2]). The following problems are of special interest if the weight
is not a function of IEI .
PROBLEM I. Is it possible to find a representation of th e dual
space of as a space of certain ho!omorphic functions or
~erms of holomorphic functions, analogous to the so called "K0the-
d~lit~" ~4] for the space ~C~) o_~r ~(~) ?
PROBLEM 2: For which weights ? is the spac e ~ nuclear?
Nityagin ~6] proved the nuclearity of the spaces ~'~ .
PROBLEM 3. Existence of Schauder bases in ~R .
This problem seems to be quite difficult. If ~ is nuclear
and has a Schauder basis, then ~ can be identified with a Kothe
sequence space (see ~7]). If the monomials {E~I~w0 constitute a
Schauder basis in ~ , as in the first example, then ~ is a so
called powe~series-spaoe(see ~ 7~). Let ~ ~ ~ be an entire func-
tion , which is not of the form ~+~ 6b~ ~ , then$~@~(~|)~
W ~ : by our assumption on ~ , there exist two points Z~ , Z~ ~,
Z~ E~ with ~Z~)= ~Z~) , now set ~= ~Z~- 4~ , where
~ denotes the Dir~c measures (~=.~ ~) ; then ~ ~ ~J~ and
/~ )~ ,~ ) = 0 ~ E ~ , t he re fo re , by the Hahn-Banach theorem,
~ ( ~ : ~ > ~ 4 ) ~ . SO, i f ~ does not contain the monomials, ~ cannot have a Schauder basis of the form { ~ }~ . B.A.Taylor
[8] constructed an example of a weighted space of entire functions
containing the polynomials and the function ~(E) , but where
~(~) cannot be approximated by polynomials
REFERENCES
I. B e r e n s t e i n C.A. and T a y I o r B.A. A new look
40
at interpolation theory for entire functions of one variable. -
Adv. of Math.,1979, 33, 109-143.
2. G e I ' f a n d I.M. and S h i 1 o v G.E. Verallgemeinerte
Funktionen II, III. VEB Deutscher Verlag der Wissenschaften,
Berlin 1962. ~z
3. H a s i i n g e r P. and M e y e r M. Abel - Goncarov
approv~mAtion and interpolation. - Preprint.
4. K o t h e G. Topologische lineare Raume. Berlin, Heidelberg~
New York, Springer Verlag, 1966.
5. M a r t i n e a u A. Equations diff~rentielles d'ordre in-
fini. -Bull.Soc.Math. de France, 1967, 95, 109-154.
6. M Z T ~ r Z H B.C. H~epRocT~ ~ ~p~e C~O~CTBa npocTpa~cTB
THHa S . --Tp.MocEB.~mTeM.O-Ba, I960, 9, 817--328. (Amer.Nath.
Soc.Transl., 1970, 93, 45-60).
7. R o 1 e w i c z S. Metric linear spaces. Warsaw, Monografie
~atematyczne, 56, 1972.
8. T a y I o r B.A. On weighted polynomial approximation of en-
tire functions. -Pac.J.Math., 1971, 36, 523-539.
F. HASLINGER Institut f~r ~thematik
Universit~t Wien
Strudlhofgasse 4
A-1090 Wien
AUSTRIA
41
1.1.3. old
LINEAR ~UNCTIONALS ON SPACES OF ANALYTIC FUNCTIONS AND
THE LINEAR CONVEXITY IN ~ ~
A domain ~ in ~ is called 1 i n e a r 1 y c o n v e x
(1.c.) if for each point ~ of its boundary ~ there exists an
analytic plane{~C~:~1+°°. +~+~=0~passing through ~ and
not intersecting ~ . A set E is said to be a p p r e x i m a b-
I e from inside (from out s i de ) by
a sequence cf domains ~K , K =~, ~,..,if ~ ~k ~ ~K+~ (resp.,
~K+4 c~O K ) and ~ =~K ~K (resp., ~ = ~IK ~K ). A com-
pact set ~ is called 1 i n e a r 1 y c o n v e x (1.c.) if
there exists a sequence of 1.c. domains approximating ~ from the
outside. Applications of these notions to a number of problems of
Complex Analysis, similar concepts introduced by A.~Tartineau and
references may be found in [1]-[5].
If ~ is a bounded 1.c. domain with ~-boundary then every
function continuous in ~0~ ~ and holomorphic in ~ has a simple
integral representation in terms of its boundary values. The repre-
sentation follows from the Cauchy-Pantapple formula [7] and is
written explicitly in ~8], [I] ,[2]. It leads to a description of the
conjugate space of the space 0(~) (resp. 0(M) ) of all fun-
ctions holomorphic in a 1.c. domain ~ (resp. on a compact set M )
which can be approximated from inside (from the outside) by bounded
1.c. domains with ~2-boundary (see [9] for convex domains and com-
pacta and [2] for linear convex sets ; the additional condition on ap-
proximating domains imposed in [2] can be removed). Such an approxi-
mation is not always possible [6]. This description of the conjugate
space is a generalization of well-known results by G.Kothe, A.Gro-
tendieck, Sebasti~o e Silva, C.L. da Silva Dias and H.G.Tillman for
the case . . henE=
for every ~ ~ F ~is Called the c o n j u g a t e s e t and
plays the role of '~he exterior" in this description. Let
be the approximating domains specified above, ~ C ~
on ~ . Consider a differential form
H
K='I
42
A 8u,,,, A . . . A gu,, A ~%~ A . . . A %~, , , where < . , ~> : 9 ~ + , . . + . , . ~ . . ~,et ~r (4 ) ) : (~ ( ( I ) ) , ...,'%@)) where ~K ('(I)) : CI)~I< < ~ ~ ( '~ ) ' :~ >-'1. Every linear cemtinuous func- t iona l F on 0(9 ) (on 0(M) ) has a representation
F(J~)--I ~c~)~c~:c~.~))~(~%~,.(~), ~), (~)
where ~0(~) (respectively, ~e 0(M) ), ~$ depends omly
on ~ . Formula (I) establishes an isomorphism between the linear to-
spaces 0' (~) and 0 (~) (respectively 0I<M) and polo~gical
OLM) ). PROBLEM I. Describe !.c. domains and compact sets which can be
approximated from inside , (from outside) by bounded l,c. domains
with C ~ -boundary. Let ~ ~a~, 0 C ~ and let F(~) denote the set of
~e C n such that the plane { ~" < $~, ~> : ~} passes through
and does not intersect ~ .
CONJECTURE I. A bounded l,c, domain ~ , 0 ~ with the
piecewise smooth boundary ~ a~nits the approximation incicated
inPROBLEM I if and only if the set s
~ ~ . !
Let F~0 (~) (respectively,
~(~) are connected for all
l
F~ 0 (,M) ). The function
[~ [(~-<z,m>y~] is called the Fantappi~ indi-
c a t o r; here ~ , ~/C~ , 0C~ (respectively, ~ M ,
tOG M , 0c M )- The function ~ in (I) is the Fantappi~ indi- cator of the functional F . A 1.c. domain ~ (a compact M ) is
called s t r e n g I y 1 i n e a r 1 y c o n v e x , if the
mapping which establishes the correspondence between fumctionals and
their Fantappi6 indicators is an isomorphism of spaces 01(~) and
0(~) (respectively 0'<M) and 0(M) ). similar definition
has been introduced by A.Martineau (see references in [9] ). Every
convex domain or compact is strongly l.c. (see, for example, [9] ).
At last,the result from [2] discussed ~bove means that the existence of approximation indicated in PROBLEM I is sufficient for the strong
linear convexity. Strongly l.c. sets have applications in such prob-
43
lems of multidimensional complex analysis, as decompositions of ho-
lomorphic functions into series of simplest fractions or into gene-
ralized Laurent series, the separation of singularities D~,E2~,E5~.
That is why the following problem is of interest.
PROBLE~ II. Give a ~eometrica! description of strongly linearly
convex domains and compact sets•
CONJECTURE 2. A domain (a compact set~ is a strongly 1.c.
set if an d only if there exists an approximation of this set indicat-
ed in PROBleM I
It was shown in E5] that under some additional conditions, the in-
tersection of any strongly 1.c. compact with any analytic line con-
tains only simply connected components. The next conjecture arose in
Krasnoyarsk Town Seminar on the Theory of Functions of Several Comp-
lex Variables.
CONJECTURE 3. A domain (a compac,t set~ is a strongly l~c. set if
and only if the intersection of this set with any analytic line is
connected and simpl 2 connected.
Let ~ be a baunded 1.c. domain with the piecewise smooth boun-
dary ~ . The set ~= {C~) ~C2~ : ~ ~ , t~ c ~(~)}
is called the L e r a y b o u n d a r y of ~ . Suppose that
is a cycle. In this case it can be shown that for any func-
tion ~ holomorphic in ~ and continuous in C ~ , we have
I (2}
This representation generalizes the integral formula indicated at the
beginning of the note to the case of 1.c. domains with non-smooth
boundaries. If a 1.c. domain ~ (a compact set M ) can be approxi-
mated from inside (from the outside) by 1.c. domains whose Leray boun-
daries are cycles then every linear continuous functional on 0~)
(0CMD can be desc bed by a fo ula analogous to (I) with
instead of ~ . Note that such a domain ~ (a compactum M )
is strongly 1.c. Therefore the following problem is closely connected
with PROBLEI~ II.
PROBLEM III. Describe bonnded 1.c. domains whose Lera 2 bounda~
44
is a cycle.
This problem is important not only in connection with the desc-
ription of linear continuous functionals on spaces of functions ho-
lomorphic in 1.c. domains (on compacta). ~ormula (2) would have other
interesting consequences (cf° D] , ~ )-
CONJECTURE IV. The classes of domains in problem~ I I-Ill coincide.
REFERENCES
I. A ~ s e H 6 e p r ~.Ao 0 pasxo~eHH rOXOM0p~H~X ~yHE~ MHO--
r~x EOM~XeEc~x nepeMeHH~x Ha npocTe~e ~po6H. - C~6.MaT.x.
I967, 8, ~5, II24-II42.
2. A ~ s e H 6 e p r ~.A. JI~He~HaH BH~OOTB B C ~ ~ passe,fe-
te OCO6eHHOCTe~ rO~OMOp~H~X ~y~EL~. - Bull.Acad.Polon.Sci.,
Ser.mat., 1967, 15, N 7, 487-495.
S. A ~ s e H 6 e p r ~.A., T P Y T H e B B.Mo 0d O~HOM Me--
TO~e C~oBaH~ nO Bopam~ ~--KpaTH~X CTeHe~ p~OB. -- CE6.
MaT°Z. 1971, 12, ~ 6, 1898-1404.
4. A ~ s e H 6 e p ~ ~.A., ~ y 6 a H O B a A.C. 06 O6~aCT~X
IDJ~OMOp~HOCT~ ~ C ~e~OTB~Te~H~m~ ~ HeoTp~aTe~ Te~-
~OpOBCE~ EOS~eHTa~. -- Teop.~, ~yHE.S2aX~S ~ ~X np~-
xo~., 1972, 15, 50-55.
5. T p y T H e B B.M. 0 CBO~CTBaX ~ , ID~OMOp~H~X Ha c~o
Jl~He~n~o B ~ MHo~eCTBaX. -- B C6."HeEOTOp.CBO~OTBa rO~o~op~.
~yHE.MHO~.EOMI~.HepeM.", Kpa0Ho~pcE, 1978, 18~--155.
6. A2 s e~ 6 ep ~ ~.A., D~aEOB A.H., MaEapo-
B a ~oH. 0 ~He~Ho~ B~UyF~OCT~ B C~ . -- CH6.MaT.~. I968,
9, ~ 4, 7~I-746. 7. ~ e p e ~. ~epe~s~_BHoe ~ NHTeI?pS~Hoe ~C~C~eH~ Ha EOMn--
~e~CHOM aH~T~ecEoM M~O~OO6pas~. M., ~, 1961.
8. A ~ s e ~ 6 e p r ~.A. ~Te~ps~oe npe~cTam~eH~e ~y~, IY~OMOp~X B B~FJ~X o6~aOT2X npocTpaHoTBa C ~ • -- ~AH CCCP,
196S, 151, 1247-1249. 9. A ~ s e H 6 e p ~ ~.A. 06~ B~ ~G~He~HOPO He~pepHBHOrO SyHE--
E~oHs~Ia B HpOOTpaHCTB~X ~yHE~, I~O~O~Dp~K~X B B~H~ O6~aCT~X
C~ .- ~H CCCP, IR66, I66, IOIS-I018o
L.A.AIZENBERG
(~.A.A~SEHBEPr)
CCCP, 660086, KpacKo~pcE AF~eMropo~oE
MHCTeTrr ~eS~ CO AH CCCP
45
COMMENTARY BY THE AUTHOR
A solution of Problem II given in ~0~, ~I] shows that Conjec-
ture III is true. The definition of a strong linear convexity (s,1 c )
due to Martineau differs from the definition in the text only by the
power (-I) (instead of (--~)) in the indicatrix formula The two de-
finitions turned out to be equivalent
Yu.B.Zelinsky has shown in ~2], D3] that the second conditions
of Conjecture III and Conjecture I are equivalent They mean the acyc-
licity of all sections of the domain by analytic planes of a fixed di-
mension ~, ~ K<~, and coincide with the s,l.c- ~] These condi-
tions form a precise "complex analogue" of the usual convexity ~I] ,
[12]. But the standard convex machinery cannot be generalized to this
context. In particular, an &-contraction of a s.l.c, domain is in
general no more s.l.c.
Using the results of ~0] one can show that the sections of s.l,c.
domains are not too tortuous This observation yields examples of un-
bounded s.l.c, domains non-approximable by bounded 1.c- domains with
smooth boundaries,
REFERENCES
I0. 3 H a M e H C K ~ ~ C.B. FeOMeTpHMecKH~ Kp~Tep~ C~X~HO~ aHHe~-
HO~ B~nym~ocT~. - ~yH~.aHaa. Hero npH~., 1979, T.I3, ~% 3,
83-84.
II. 3 H a M e H C ~ H ~ C.B. BKBMBa~eHTHOCTB paS~H~X onpe~eaeHH~
CM.~BHO~ ~I4He~Ho~ BB~IyI~OCTM. Mex~yHapo~Ha~ EoH~epeH~H2 no ~OMn-
~eNCHOMy aHaaHsy H np~omeHHm~. BapHa, 20-27 CeHT26p~ 1981 r.,
30. 12. z e i i n s k y Y.B. On the strongly linear convexity Interna-
tional conference on complex analysis and applications Varna,
September 20-27, 1981, 198
IS. 3 e ~ ~ H C ~ Z ~ D.B. 0 reoMeTp~ecEHx EpHTep~x CH~HO~ m~He~-
HOB B~yK~OCTM. ~o~a~M AEa~e~M Hay~ CCCP, 1981, T.261, ~ I,
II-13.
46
I. 14. ON THE UNIQUENESS OF THE SUPPORT OF AN ANALYTIC PUNCTIONAL old
The symbol H(E) will denote the space of all functions analy-
tic on the (open or compact) set ~ , ~ C C~ , endowed with the
usual topology. Elements of the dual space ~(~) (here and below
stands for an o p e n set) are called analytic functionals
(= a.f.). A.Martineau has introduced the notions of the carrier
~orteur) and of the support of an a.f.
A compact set ~ , ~C~ , is called a c a r r i e r of
an a.f. ~ if T admits a continuous extension onto ~(~) , or
equivalently [2] if T is continuously extendable onto ~(~) for
an arbitrary open 0J , ~ D~JD ~ . Every a.f. has at least one
carrier.
Let ~ be a family of compact subsets of ~ such that if ~A)
is a subfamily of % linearly ordered by inclusion then ~ ~AC'~ ".
A compact set ~ , ~ ~ is called an ~ -s u p p o r t of
the analytic functional T if ~ is a carrier of T and ~ is mi-
nimal (with respect to the inclusion relation) among all carriers of
T in ~.If D has a fundamental sequence of compact sets from
then any analytic functional has an ~ -support but in general the
-support is not unique. It is possible to consider various fami-
lies of compact subsets of ~, e.g. the family of all compact subsets
of ~ , the family of ~(~) -convex compact sets, the family of
all convex compact sets (in this case an ~ -support is called a
convex support or a C -support).
Any analytic functional T on C 4 has a unique C-support but
can have many polynomially convex ( = e~ ) supports. (If for
examp I •
0
then any simple arc connecting 0 and I is a pC -support of T .)
PROBLEM. Describe convex compact sets K ( C C ~ ) such tha t
K is the unique ~-support of any a nal2tic functional ~-su~ort-
K. C.O.Kiselman [3] has obtained for ~=~ necessary and suffi-
cient conditions for a compact set to be a unique pC -support.
For ~> ~ a compact set with a C~-boundary is a unique
pC -support ~4]. Kiselman has proved in ~4] that a convex com-
47
pact set with a smooth boundary is a unique 0 -support. A stronger
result is due to Nartineau [5]: a convex compact set K is a unique
-support if any extreme point ~ of ~ (~ ~ ~ ~) belongs
to a unique complex supporting hyperplane (:with respect to the comp-
lex affine manifold V(K) generated by K ).
Our problem is stated for the above two families of compscts
only, though it is interesting for other families as well. Using the
ideas of Martina~u one can prove the following
THEOREM. A convex compact set K~c C ~) is a unique ~,sup-
port if the set of all its supporti~ hyperplanes is the closure of
the set of hyperplanes ~ with the fq!!owin~ oro~erty: ~ KN
contains a point lyin~ in a unique complex supporting hyperplane ' wit h
respect t O V(K) •
It is probable that the sufficient condition of the theorem is
also necessary.
REFERENCES
1. M a r t i n e a n A. Sur lee fonctionneles analytiques et la
transformation de Fourier-Borel. - J.Analyse Math., 1963, 9,
1-I 64.
2. B j S r k J~E. Every compact set in C ~ is a good compact
set. -Ann.Inst.Fourier, 1970, 20, 1, 493-498. . • r
3. K i s e 1 m a n C.0. Compact d'unlclte pour lee fonctionnelles
analytiques en une variable. - C.R.Acad.Sci°, Paris, 1969, 266,
13, A661-A663.
4. K i s e 1 m a n C.0. On unique supports of analytic functio-
rials. -Arkiv for Math. 1965, 16, 6, 307-318.
5. M a r t i n e a u A. Unlcite du support d'une fonctionnelle
analytique: tun th~oreme de C.O.Kiselman. -Bull.Soc.Math.France,
1968, 92, 131-141.
V. M. TRUTNEV
(B.M.TPYTHEB) CCCP, 660075, KpaoHo~poE,
y~e Maep~aEa 6,
EpaoHo~p0EH~ IDcy~ap OTBeH~
yH~BepcHTeT
CHAPTER 2
BANACH ALGEBRAS
Thirteen sections of this Chapter can be conventionally divided
into three groups. General theory of Banach algebras is represented
by Problems 2.1-2.5. This group of problems is connected mainly with
the spectral structure of elements of an abstract Banach algebra.
The second group consists of a couple of problems concerning
Convolution Measure Algebra followed by a problem on harmonic synthe-
sis in group algebras. The convolution algebra M($) of all finite
Borel measures on a locally compact abelian group G is interesting
from many points of view and among them from the spectral one. The
subject originates in the classical paper by Wiener and Pitt (Duke
Math.J. 1938, 4, N 2, 420-436) and has been intensively studied so
far. However, the bulk of all publications on the theme has revealed
only different pathologies in the structure of M(~) , and the num-
ber of "positive" achievements here is not large. J.L.Taylor has
calculated the cohomologies of the maximal ideal space of M(~)
(see [2~ in references of Problem 2.6). G.Brown and W.Moran have
described the structural semi-groups of important su~algebras of
M(~) (Acta ~th., 1974, 132, N 1-2, 77-109). Some years ago B.Host
and M.Parreau solved a problem of I. Glicksberg (C.R.Ac. sc. Paris,
1977, 285, 15-17 and Ann.Inst.Fourier, 1978, 28, N 3, 143-164). They
described all measures ~ whose ideals, ~(~) is closed in M($) .
49
The question of description of Shilov boundary for ~ Q$) , which
is the subject of Problem 2.6, undoubtedly, is the core problem of
the theory. It has been posed by J.L.Taylor and still remains un-
solved. Problem 2.7 considers a description of homomorphisms of
L-subalgebras of ~ (6) in the spirit of the well-known Cohen-Rudin
theorem. The last problem of the second group, Problem 2.8, deals
with the structure of ideals in group algebras.
The third series of questions concerns more visible, but not
less mysterious algebras such as the algebra H~(V) of all bounded
and analytic functions in a domain Vc C . Note that Problem 2.9
contains an interesting conjecture about the axiomatic description
of H~(~) in the category of all uniform algebras. Eleven problems
are formulated in 2.10 and among them the Corona Problem for H~(V)
which remained unsolved untill now. We would like to complete the
list of references to 2.10 by the following ones. M.F.Behrens (Trans.
Amer.~ath.Soc., 1971, 161, 359-379) has shown that it is sufficient
to solve the Corona Problem for a special class of domains. It is
also known that for some of these V the algebra Ha(V) does not
have a corona. New progress has been obtained in a recent paper by
L.Oarleson (Proc.Conf.Harm.Anal.in Honour of A.Zygmund, Wadsworth
Inc., Belmont, California, 1983, 349-372). The classical Hardy al-
gebra ~-~- ~(~) does not have a corona but nevertheless the
structure of its maximal ideal space remains puzzling. Problem 2.11
is important for the understanding of this structure.
The last two problems of the third group concern the disk al-
gebra, though the question posed in Problem 2.13 is considered in a
more general setting.
50
2. I. THE SPECTRAL RADIUS F O ~ IN QUOTIENT ALGEBRAS old
If A is a complex Banach algebra and ~ e A let @(X) denote
the spectral radius of ~ . If l is a proper closed two-sided ide-
al of A , ~ + I denotes the coset in the quotient algebra contain-
ing ~O . Clearly, by spectral inclusion,@Cx÷I)~ ~ .(x~) A is called an S R - a I g e b r a if equ~l~t;lholds in this
formula for each so , ~ A , and eac~ closed two-sided ideal I of
A . The algebra A(~) of all continuous functions on the disk ana-
lytic on its interior is not SR [1]. The following algebras are
C* 1-31, E l l g,b s Zgebras algeb s oo=paot or Riesz operators, semi-simple dural algebras, semi-simple ann~lator
algebras and algebras with a dense socle. If A is commutative and
has a discrete structure space then ~ is ~ .
QUESTION. Is this true for n onco~tative A ?
Let ~ be commutative and let A be the Gelfand transform al-
gebz'~, of A and let Z(A) denote the spectrum of A • Then [2]
if ~ is dense in $o(~(A)) , A is an ~-algebra, Conversely, if A is a regular ~-alge'bra, ~, iS dense an Co(~(~)) .
QUESTION. Can the condition of re&~larSt~ be omitted frgm this
~ V ~ o t h e s i s ?
L e t A be a ~ - a l g e b r a and l e t ~ be a n y e l e m e n t i n A a n d
let I be any closed %we-sided ideal of A •
QUESTION. ,!,s it true that the,r~,, ,~lwa.ys exists ~j ~61
(dependS- K on o0 ), such t~t ~(~÷I)-~-~(X+~)?
This result is true if @(X+I)=~ 0 and it is a corollary of
~] Theorem 3.8. The case @(~+I)~- 0 is opera.
REFERENCES
I. S m y t h M.R.P., W e s t T.T. The spectral radius formula in
quotiemt algebras. - Math. Zeit. 1975,145, 157-I 61.
2. M u r p h y G.J., W e s % T.T. Spectral radius formulae. -
Proc.Edimburgh Math.Soc.(2), 1979, 22, N 3, 272-275.
3. P e d e r s e n G.K. Spectral formulas in quotient algebras. -
Math. Zeit. 148. 4. A k e r m a n n O.A., P e d e r s e n G.K. Ideal perturbations
of elements in 0*-algebras. - Math.Scand.1977,41, 137-139
G.J.~ 39 Trinity college
M.R.P.SMYTH T.T.WEST Dublin 2 Ireland
51
2.2. EXTREMU~ PROBLEMS
1. THE GENERAL MAXII~ PROBLEM. Give n a unital Banaoh algebra
A , a compact set ' F in the plane and a function ~ holomorphic
in a nei~hbourhoodpf F , to find
of
su@{l ~(o~)I : I~I..< I, ~(o~)cF}.
The problem was formulated and solved first [1,2] in the case
2. THE SPECIAL MAXi~,~J~ PROBLEM. Let f~ be a natural number,
a positive number .!ess than one, to find, amon~ all contractions T
o_~n ~-dimensional Hilbert space whose spectral radius does not e x-
ceed ~ those (it turns out that there is essentially only one)
for which the norm I Tal assumes its max~um.
This is a particular case of the general problem for
A-- F = { ~ ' l ~ l ~ t ] , ~(~)=~. The solution of problem 2 was divided into two stages. The first
step consists in replacing the awkward constraint that the spectral
radius be ~ ~ by a more restrictive one which makes the problem
considerably easier: the operator is to be annihilated by a given
polynomial. This is
3. THE FIRST MAXIMU~ PROBLEM. Le t ~ be a natural number, p a
pn]v~omial of d~aT~ e ~ with all roots inside the unit disc. Let
A(p) be the set of a!l contractions T~ B(H.) such that
(T) =0. To find the maximum of I Tml , more ~enerally of
l CT)I as T ranges A cp) • We call this maximum C(.p,~) •
Having solved the first maximum problem we have te solve
4. THE PROBLEM OF THE WORST POLYNOMIAL. To find, amon~ all ~ol,y-
nomials with roots ~ ~ i n modulus that one for which ~(p, ~)
is maximal.
Per the case ~(~)=~ the result of [2] shows that the
worst polynomial is p(~)-- (~-t) ~ . The method used in F2] is
based on some fairly complicated algebraic considerations and does
52
not extend to functions other than 0~ ~ . It is not known whether the
worst polynomial for ~ other than ~ has a root of multiplicity
whether the roots have to be concentrated on the boundary of the
disc I~1 ~ ~ . Thus it seems useful to study C CP,~) as a func- t ion of the roots of p ; a recent contribution to Problem 4 isE~.
A list of references up to 1979 is contained in the survey paper [4-
RETERENCES
I. P t ~ k V. Norms and the spectral radius of matrices. - Cze-
chosl.Math.J. 1962, 87, 553-557.
2. P t ~ k V. Spectral radius, norms of iterates and the criti-
cal exponent. - Lin.Alg.Appl. 1968, 1, 245-260.
3. P t ~ k V., Y o u n g N.J. F~nctions of operators and the
spectral radius.- Lin.Alg.Appl. 1980, 29, 357-392.
4. Y o u n g N.J. A maximum principle for interpolation in H ~ ,
Acta Sci.~th.Szeged.
VLASTIMIL PT/~K Institute of Mathematics
Czechoslovak Academy of Sciences
~itn~ 25 11567 Praha I
Czechoslovakia
2.3.
53
M~B~JM PRINCIPLES FOR QUOTIENT NORMS IN H '~'
A surprising variety of starting points can lead one to study
norms in quotient algebras of H=by a closed ideal. Well known examp-
les are classical complex interpolation [4], canonical models of ope-
rators [2] and some problems of optimal circuit design [I], It also
arises from a maximum problem for matrices to which V.Pt~k was led by
considerations relating to numerical analysis. The problem is to esti-
mate the maximum value of ~ ~ (~)~ , where ~ a ~ , over all con-
tractions A of spectral radius at most %< ~ on ~-dimensional
Hilbert space (Ptak was mainly concer~ed with the case ~(A)=~,
some ~ ). An account of this problem is given in [3].
The only known way of handling the spectral constraint is to rep-
lace it by the condition ~(A)=o , for some polynomial p , and then
to vary ~ among the polynomials of ~egree ~ having all their zeros
in % ( 6~S ~) • After some calculations one is led to study the fun-
ctional
F(p) - ti pH'tl H'/pH
as a function of ~ ~ for fixed ~ . In particular, as p vari-
es over the class of polynomials described above, at what ~ does
attain its maximum? The result we should like to prove is that
attains its maximum at a polynomial of the form p(~)=
for some ~ ~C with I~] =~4 . Pt~k proved this was so
(~) = ~ , and I proved that if ~ is a Blaschke
ree ~ then all the zeros of an extremal polynomial
: in fact, F is then the composition of a strictly
ction and a plurisubharmonic function of the zeros of
is known, however, about the most interesting case,
in the case
product of deg-
have modulus
increasing fun-
p [5]. oth ,
(~) =E~ with ~>~
To formulate the problem concisely, let us say that the m a x i-
mum p r i n c i p 1 e h o 1 d s f o r ~" IlCC-- ~ if
for any compact set K C ~ , the supremum of on K is attained at
some point of the boundary of K relative to ~ , And if M is a com-
plex manifold, we shall say that the maximum principle holds for
F: M --~ ~ if, for any open set i~c C and any analytic functi-
on ~ : ~ -'=" ~ , the maximum principle holds for F o ~ .
54
He:@ PROBLEM. Let ~ ~ and le t F" 0 ~-'~" R be defined by
where
F ...,oct)-: * FI='II
9 ( z } = F] (~-~0
H~/gH ~
Does the maximum principle hold for F ?
REFERENCES
I. H e I t o n JoW. Non-Euclidean functional analysis amd electro-
nics. - Bull.Amer.Math.Soc., 1982, 7, 1-64
2. H z K o m ~ c K ~ ~ H.K. ~e~ o~ onepaTope c~zra. Uoc~a,
Hanna, 1980.
3, P t a k V,, Y o u n g N.J. Functions of operators and the
spectral radius. - Linear Algebra and its Appl,, 1980, 29, 357-392
4- S a r a s o n D. Generalized interpolation in H ~ . - Trans.Amer~
Math. Soc., 1967, 127, 179-203.
5o Y o u n g N.J. A maximum principle for interpolation in H~ -
Acta Sci.Math°, 1981, 43, N I-2, 147-152~
N. J. YOUNG Mathematics department
University Gardens
Glasgow GI28QW
Great Britain
55
2.4 OPEN SEMIGROUPS IN BANACH ALGEBRAS
Let A be a complex Banach algebra with identity, not necessa-
rily commutative. Let S be some open multiplicative semigroup in
A . For an element ~ in A let ~(~) denote the distance from
the point ~ to the closed set A k S (in other words, it is the
radius of the largest open ball centred at ~ and contained in S ),
and let $(~) be the supremum of all ~0 such that the elements
Cb-~'~ belong to ~ for I~I <~ (that is the radius of the largest
open disk centred at ~ and contained in the intersection of
with the subspace spanned by ~ and I). So we clearly have $(~)~
~C~). For a ~riet 2 of particular semigroups S we know that
the formula )~
is valid for every ~ i_~n A - We list below the most important
cases.
~irst of all, if S=G(A), the group of invertible elements,
the result follows from the spectral radius formula. Second, the
formula is true when S is the semigroup of left (or right) in-
vertible elements of A , cf. E6 ]. Third, it also holds when S
is the complement of the set of left (right) topological divisors of
zero in A , cf.~].
Next, the formula is true for various semigroups of the algebra
A = B(X) of bounded linear operators on a Banach space. In the case
when S is the semigroup of surjective (or bounded from below) ope-
rators on X it was obtained in [1] by an analytic argument (in
fact, this is equivalent to the third case mentioned before). Using
an additional geometric device these results were applied in [6] to
prove the above formula for the semigroup S of (upper or lower)
semi-Fredholm operators on X , and hence it follows for the semi-
group of Fredholm operators as well. In these cases the distance
~CT) admits other natural interpretations, namely, it coincides
with (or is related to) certain geometric characteristics of the ope-
rator T (like the ~urjection modulus, the injection modulus, the
essential minimum modulus [~, etc.).
In each of the cases listed above an individual approach was
needed to find a proof. The difficult steps are of an analytic cha-
racter, based on the theorems of G.R.Allan (1967) and J.Leiterer
(1978) on analytic vector-valued solutions of linear equations de-
86
pending analytically on a ~rameter. The main idea is well demonst-
rated in [1] though in that case a combimatorial argu-
ment is also available [2]. So there seems to be some motivation
for investi~atin~ the problem in ~eneral to seek a theorem which
would contain all these oarticula r results.
Let us give some warnings. Let S =G~ CA) be the principal
component of the set of invertible elements in A . There are (non-
commutative) Banach algebras A for which the group GCA)/G4(A)
is finite, but not trivial, cf.[4],[3]. In such a situation for every
invertible element ~ not in G4CA) one can find a positive integer
k such that G) k is in G4~A ) . Then we have 5(0~)=0 but
~ k) >0 so that the formula cannot be true for this ~ and all
6~ in A . ~oreover, it is easy to see that ~ ~S~) 4/n
cannot exist for these G) .
Let A be a commutative Banach algebra and let ~ be the open
semigroup of all elements whose spectra are contained in the open
unit disk. In Ibis case we have ~(~)=~(~) for every ~ in ~ but ~I.i--
, and t h e l a s t i n e q u a l i t y may be s t r i c t .
We a r r i v e a t the same conc lus ion i f we rep lace the u n i t d i s k i n the
p reced ing d e f i n i t i o n by a % - m u l t i p l e o f i t , w i t h 0 < ~ <
Th is suggests some ana logy between our problem and the c l a s s i c a l f o r -
mula f o r the r a d i u s o f convergence o f a power s e r i e s ( the f o rmu la
does not g i v e the r a d i u s o f the d i s k where we are c o n s i d e r i n g the
function but the radius of the disk where the given function can na-
turally be defined).
Thus some additional conditions should be imposed on 5 in ge-
neral. ~or instance, the property "if ~= ~ belongs to ~ then
both ~ and 6 are in S " is shared by most of the semigroups for
which the problem is solved in the affirmative. This condition en-
sures, by the way, that 5(~) = 5(S)) ~ ~ for all ~ in A and
~= ~, ~,... (we note that ~(~) ~ ~SO) is always true). I s it
important that the identity element be in ~ , or that ~ be con-
nected, or "maximal"? Does ~ ~(~)[~ exist for ~ in an
arbitrary open semi~roup S ? %Vhat is th,,e ,m,eani,n~ of it in ~ene-
,,ra~?
I should like to thank Tom Ransford for a valuable discussion
on this topic.
57
REFERENCES
I. M a k a i E., Jr., Z e m ~ n e k J. The surjectivity ra-
dius, packing numbers and boundedness below of linear operators.
- Integr.Eq.Oper.Theory, 1983, 6.
2. M u 1 1 e r V. The inverse spectral radius formula and remov-
ability of spectrum.
3. P a u 1 s e n V. The group of invertible elements in a Banach
algebra. - Collo~.Math., 1982, 47
4. Y u e n Y. Groups of invertible elements of Banach algebras. -
Bull.Amer.~th.Soc.,1973, 79.
5. Z e m ~ n e k J. Geometric interpretation of the essential mi-
nimum modulus. - Operator Theory: Advances and Applications, vol.6
p.225-227. Birkh~user Verlag, Basel, 1982.
6. Z e m ~ n e k J. The semi-Fred_holm radius of a linear opera-
tor. - To appear.
JAROSLAV ZEN~NEK Institute of Nathematics
Polish Academy of Sciences
00-950 Warszaws, P.O. Box 137
Poland
58
2.5. HOMOMORPHISMS ~OM C*-ALGEBRAS
Let A and ~ be Banach algebras. A basic AUTOMATIC CONTINUITY
PROBLEM is to give algebraic conditions on ~ and ~ which ensure
that each hemomorphism from A into ~ is necessarily continuous.
An important tool in investigations of this problem is the se-
parating space: if ~:A--~ is a homomorphism, then the s e -
p a r a t i n g s p a c e of ~ is
~(@) = ~E~ : there is a sequence ( ~ ) C A
with a n d
Of course, 8 is continuous if and only if ~(@) = [0~ . The basic
properties of ~(8) are described in [6].
Consider the GENERAL QUESTION: i_~f ~ £~(8) , what can one sa x
abOUt 6"(~0) , %he spectrum of ~ in the Banaoh alsebra ~ ?
Pirst let us note that, if ~ 65(@) , then ~ ek~tq for
each character q on ~ . For such a character q is necessa-
rily continuous, the character ~ o 8 is continuous on ~ , and
commutative, it follows that ~(~) 40} .
Is the same result true in the non-commutative case? An element
of a Barmch algebra is a q u a s i - n i I p o t e n t if
~(~) ={0} , and so our question is the following.
QUESTION I. Let @ : ~ ~ ~ be a homomorphism , and let
£ % ( @ ) o I.~s ~ necessarily a quasi-~ilpotent element of ~ ?
It can be shown that ~(~) is always a connected subset of
containing the origin (see [6 , 6.16]), but nothing further
seems to be known in general.
The question was raised as Question 5' in [3], and it is shown
there that the question is equivalent to the following. Let @:A-~
be a homomorphism, and suppose that ~ is semi-simple. ~
necQssarily continuous?
It is shown by Aupetit in [2] that, if ~ and 5 are unital
Banach algebras, if ~ ~ ~B is a homomorphism, and if ~(@)
then 9($@) .4 ~(~ + ~@) for all ~ A (Here, $ denotes the spect-
ral radius.) Thus, if ~6~(@)~@(~),then 9(6)=0 and ~ is a
quasi-nilpotent. However, it is not in general true that the set of
quasi-nilpotentsin a Banach algebra is closed (see ~I~), and so we
cannot immediately conclude from this result that each ~ ~ ~(~)
is quasi-nilpotent.
Quite probably, there is a counter-example to Question S. How-
ever, let us concentrate on the case in which both ~ and B are
~* -a~gebras
QUESTION 2. If, in ~uestiQn I, both ~ and ~ are ~-al~ebras,
can we then0onclude that ~ i s necessaril E quasi~nilpotent?
The c o n t i n u i t y i d e a i of a homomorphism
~:~--~B is the set
It w~s proved by Johnson (~4~, see~6, 12.2~) that, if ~ is a C*-al-
gebra, then ~(8) is a two-sided ideal in ~ and that its closure
has finite condimension in ~ . Next, Sinclair (K5 , Theo-
rem 4.1]) showed that, if ~ and B are both C*-algebras, and if
e:~ ~ is a homomorphism with ~)=B then ~ I~ can be
decomposed as ~+~ , where ~ is a continuous homomorphism and
:(~ ~(e)is a discontinuous homomorphism (or ~ = 0 ). Now
~(~----~ and ~(e) are closed ideals in ~ and B , respectively~
and so both are C~-algebras. Moreover, the range ~(~) is a den-
se subalgebra of ~(~) and so, by our above remarks, consists of
quasi-nilpotent elements. Thus, ~(~) is a C~-algebra with a dense
subalgebra consisting of quasi-nilpotents. No such C* -algebra is
known, and I would like it to be true that no such C~-algebra
exists. So we come to the sharpest form of our original question.
QUESTION 3. Is there a C~-al~ebra (other than ~0~ )which ha S
a d~nse subalEebra consistin~ of quasi-hi!potent elements?
If no such C* -algebra exists, then the homomorphism ~ in
Sinclair's theorem must be zero, and so the element ~ in Question 2
must indeed be quasi-nilpotent.
REFERENCES
S. A u p e t i t B. Proprletes spectrales des algebres de Banach.
- Loot.Notes Math.~ 1979, 735, Springer-Verlag.
60
2. A u p e t i t B. The uniqueness of the complete norm topology
in Banach algebras and Banach-Jerdan algebras. - J.Functior~l Ana-
lysis, 1982, 47, I-6.
3. D a I e s H.G. Automatic continuity: a survey. - Bull.London
Math.Soc., 1978, 10, 129-183.
4, J o h n s e n B.E. Continuity of homomorphisms of algebras of
operators II. - J.London ~th.Soc~2)~1969, I, 81-84.
5. S i n c 1 a i r A.M. Homomorphisms from C* -algebras. - Prec.
London ~ath.Soc., (3), 1974, 29, 435-452; Corrigendum 1976, 32~ 322
6. S i n c 1 a i r A.~. Automatic continuity of linear operators.
- London Math.Soc.Lecture Note Series, 21, C.U.P., Cambridge,1976.
H. G. DALES School of Mathematics,
University of Leeds,
Leeds LS2 9JT.
Great Britain
61
2.6. ANALYTICITY IN THE GEL~AND SPACE
old oP THE ALGEBra OF ~CRJ mmTIPLIERS
We shall be concerned with spectral properties of the Banach al-
gebra of those bounded linear operators on ~(~) which commute
with translations. However, it is convenient to represent the action
of each operator by convolution so that the object of study becomes
the algebra M(~) of bounded regular Borel measures on~ . The gene-
ral problem to be considered is the classification of the analytic
structure of the Gelfand space, A , of M(~) despite the fact that
A is sometimes regarded as the canonical example of a "horrible"
maximal ideal space from the point of view of complex analysis (cf.[1]
p.9). Some encouraging progress has been made in recent years and it
will be possible to pose some specific questions which should be
tractable.
We refer to Taylor's monograph, ~], for a survey of work up to
1973 (Miller's conjectured characterization of the Gleason parts of
A has since been verified in [3]) and for further details concerning
general theory of convolution measure algebras. In particular, we
follow Taylor in representing A as the semigroup of continuous cha-
racters on a compact semigroup ~ (the so-called s t r u c t u r e
s e m i g r o u p of ~(~)) and in transferring measures in M(~)
to measures on % . In this formulation an element ~ of A acts as a
homcmorphism according to the rule
Every member ~ of A then has a canon ica l p o l a r decomposi t ion, , where and ~ has idempotent modulus. If ~ it-
self does not have idempotent modulus (a possibility which corres-
ponds to the Wiener-Pitt phenomenon and was first noted by ~reider
[4]) then the map~-~l~l~, for ~¢~)>0 , demonstrates analyticity in
A . ~rom that observation Taylor showed that the ~ilov boundary
of ~ (~) is contained in clos ~ , where
1 S A: I I= }. He posed the converse question which is still unresolved. Sub-
sequent work tends to suggest a negative answer so that we propose
It should be noted that the result ~ clos 8 remains valid
for abstract convolution measure algebras and that it is very easy
to find convolution measure algebras for which ~\ 8=#=~ . It is also
62
possible to find natural ~ -subalgebras of M(~) , itself, for which
the corresponding conjecture is true. (An ~ -subalgebra is a closed
subalgebra A which contains all measures absolutely continuous with
respect to any measure in ~ ). Thus, a disproof would depend on not
only a new phenomenon peculiar to M(~) but one which is specific to
the full algebra. In addition we established a weak form of the con-
jecture in E5] by showing that a certain idempotent ~ (see below)
fails to be a strong boundary point for ~(~) (although it is a
strong boundary point for the ~ -subalgebra of discrete measures).
It should also be noted that Johnosn, E6], proved t~tA\ ~=#=~ but
the techniques used to prove this result and its subsequent refine-
ments depend essentially on the use of elements lying outside e . A
natural strategy is to embed M(~) in a suitable super-algebra and
prove the impossibility of extension of an appropriate homomorphism,
It appears to be almost as difficult to exhibit, in the oppo-
site direction, large numbers of elements of ~ which DO belong to
. Before we describe some progress in this direction let us intro-
duce the notation ~ for the unit function in A and the notation ~
for the homomorphism given by
where #~ is the discrete p~art of ~ . ( ~ plays the role of the
unit function for the subalgebra of discrete measures, which can be
regarded as M(~) , where ~ is the discrete real line).
Let us define a partial order on A by saying that ~ $ if
We have shown in E7~ that maximal elements are members of the
~ilov boundary.
THEOREM ].(i) if ~ is maximal in A then ~ is a strong bounda-
rypoint.
(ii) If I~I i s maximal inA\~ then ~ belongs to S .
if, moreover, ~he ~ -subalgebra
is countabl~ ~enel~teld then ~ is a stron~ boundary point.
It is obvious that maximal elements belong to 9 but not entire-
ly trivial that there are many examples other than those homomor-
phisms induced by continuous characters of ~ (viz. extensions of
non-zero homomorphisms of ~(~) ). To see that this is the case con-
63
sider in connexion with (i), homomorphisms which are induced on the
discrete measures by discontinuous characters, and in connexion with
(ii), homomorphisms which annihilate some fixed member of ~(~) .
The additional hypothesis in (ii) does not correspond to a spe-
cific obstruction and merely reflects the constructive nature of our
proof. We have avoided a similar difficulty in (i) by an appeal to
Rossi's local peak set theorem and it seems plausible that a similar
device should be available here. A proof which reduced the uncountab-
ly generated case to the countably generated case by pure measure al-
gebra techniques would be particularly interesting since this species
of difficulty often arises. In any event we propose
CONJECTURE 2. If ~ is maximal in A \ I~I then ~ is a stron~
boundary point.
It would be useful to determine for specific subclasses of
whether or not the elements are strong boundary points. The result
that ~i is the centre of an analytic disc v~s extended in [8] to
cover the case of the idempotent corresponding to any single genera-
tor Raikov system. On the other hand we show in ~] that ~ is acces-
sible in the sense that it is the infimum of those maximal elements
of A \~} below which it lies. It is natural to expect that both re-
sults extend, although we feel that present techniques would require
substantial development to prove
CONJECTURE 3. The idempotents corresponding to proper Raikov
systems are accessible but fail to be stron~ boundary points.
We have chosen to present these problems from the standpoint of
the development of the general theory. Prom a practical position the
most useful results are those which exhibit classes of homomorphisms
which belong to the ~ilov boundary of ~ -subalgebras of ~(~) --be-
cause such results give information on spectral extension. In fact,
THEOREM I is of this type because it remains valid for arbitrary con-
volution measure algebras(provided the technical hypothesis that
is a critical point is added to part (ii)). Variants of that theorem v
with the weaker conclusion that ~ belongs to the Silov boundary but
valid for a larger class of ~ would be of considerable interest.
I. Gamelin
1969.
2. Taylor
REFERENCES
T.W. Uniform Algebras. New Jersey, Prentice-Hall,
J.L. Measure Algebras. CBMS Regional confer, ser.
64
math., 16, Providence, Amer.Math. Soc., 1973.
3. B r o w n G., M c r a n W. Gleason parts for measure algeb-
ras. - Nath.Proc.Camb.Phil.Soc., 1976, 79, 321-327.
4. m p e ~ ~ e p D.A. 05 O~HOZ npm~epe oSodm§m{oro xapazTepa. -
~TeM.c6., I95I, 29, ~ 2, 419-426. 5. B r o w n G., M o r a n W. Point derivations on M(G) . -
Bull.Lond.Math.Soc., 1976, 8, 57-64.
6. J o h n s o n B.E. The ~ilov boundary of M(G) . - Trans.Amer.
~th.Soc., 1968, 134, 289-296.
7. B r o w n G., M o r a n W. Maximal elements of the maximal
ideal space of a measure algebra. - ~th.Ann., 1979/80, 246, N 2,
131-140. 8. B r o w n G., ~ o r a n W. Analytic discs in the maximal
ideal space of M(~) . - Pacif. J.~ath., 1978, 75, N I, 45-57.
GAVIN BROWN University of New South Wales
Sydney, Australia
WILLIA~ MORAN University of Adelaide
Adelaide, Australia
65
2.7. ON THE COHEN-RUDIN CHARACTERISATION
old OF HO~O~OEPHIS~S OF ~EASURE ALGEBRAS
Let I.(~) be the Lebesgue space and ~(T) the set of all bound-
ed regular Borel measures on the unit circle ~ .~(T) is a commuta-
tive Banach algebra with the convolution product and the norm of to-
tal variation, and ~(T) is embedded in M(~) as a closed ideal. A
subalgebra N of M(~) is said to be L - s u b a 1 g e b r a if
it is a closed subalgebra of~(~ and#~N and~)<<~, that is, # is
absolutely continuous with respect to ~ , implies I)~ N .
Let Af(N)be the set of all homomorphisms of ~ to the complex v
numbers (which might be trivial). Then, by Yu.Sreider [I] , for every
, ~Ar(N), there corresponds a unique generalized character
I~/~:~N~ or zero system such that
In the following we shall use the same notation ~ for . A
g e n e r a 1 i z e d c h a r a c t e r~=I~:~Nl satisfies,
by definition,
(i) ~I,=~([~I) and ~-- ~6~ ~i~p I ~jal ~ 0
( i i ) ~ : ~ ~-a. e. i f ~<<~
Let ~ be a homomorphism of N to N Q~) . Then the mapping
~ (~)^(I~), 9~ N ~ defines a homomorphism for every integer i~ ,
where ,,A" denotes the Fourier-Stieltjes transform
? Thus there exists a generalized character ~( , )= I~ , (M,~) :~EN} or zero system such that
= I '
Let{~.}~ 0 be a sequence of integers such that~>~ and~>~
for infinitely many ~ . Put
r l
Let
4
~:4
66
be a Bernoulli convolution product, where ~ (~) is a Dirac measure
concentrated on a point 0~ . We fix such a ~ and denote by N {~)
the smallest L-subalgebra containing ~ .
"HEOREM ([2],[3]). Let M be IPsu~!~eb~ L(T) crNI~)s~ad~
( , ) for a l l tb in Z and ~ in. M. .mh~n w,e, ~ve
(a) a positive integer ~ and a finite s ubsetR=l~4,Bv~+~, ,Bt~ o~
(o)~'#eAIN}with [~}1~= I~;}I (}=%Z,..., ~) such that
t ~ . , ~ = n } ~ B
(2)
wher# 0 E d#notes the characteristic function of the set E .
Converse l F if {~(~)} is a sequence in A/(~) satisfyina (A),
(a), (b) and (c~, then the mappin~ ~ given by (I) is a homomor-
9hism O f M t~o M£~)-
When ~=-h(~) , thenAq~)={6 {~:Me~}U{0~ and the condition
(A) is obviously satisfied. For this case the theorem is due to W.
Rudin. The other case, when ~=N(~) , the theorem is proved by S.
Igari - Y.Kanjin. Since h(~) is an ideal, our theorem holds good for
M=b(]h • NV~). We remark that we cannot expect the conditions (a), (b) and (c)
without the hypothesis (A) (cf. [2~.
PROBLEm I ° For what kind of L-subal~ebra M does th e above
theorem hol d good?
PROBI~ 2. Let M=M(~) and ~ b e a hqmomorphism of ~ (~) t_£o
~ ( ~ . Let ~ ( ~ ) } be a sequence..o.f Af(M(~)) ~iven by (1) and
assume that {~(M)} satisfies the condition , (A) for a measure 9 .
Then~ characterize ~ such that the conditions (a), (b) and (0)
ho~d for {%(~)}.
67
REFERENCES
I. ~ p • ~ ~ e p D.A. CTpoe~e MaKC~R~X ~eaxom ~ ~ox~ax
Mep co cBepTKoI. -MaTeM.c6., I950, 27, 297-818.
2. R u d i n W. The automorphisms and the endomorphisms of the
group algebra of the unit circle. - Acta 5~ath., 1976, 95, 39-56.
3. I g a r i S., K a n j i n Y. The homomorphisms of the mea-
sure algebras on the unit circle. - J.5~th.Soc.Japan, 1979, 31,
N 3, 503-512.
SATORU IGARI Mathematical Institute
Toh~ku University Sendal 980, Japan
68
2.8. TWO PROBLENS CONCERNING SEPARATION OF IDEALS
IN GROUP ALGEBRAS
All algebras in this paper are commutative complex regular B~-
nach algebras. In such algebras all ideals consist of joint topolo-
gical divisors of zero, i.e. if I is a (not necessarily closed) ide-
al in a regular Banach algebra, then there is a net (~@) of ele-
ments of the algebra in question, which does not tend to zero, but
~ ~ =0 for all ~ in I (cf.[1]). In this case we say that @ the net (Z@) annihilates the ideal I. We say that an ideal I = A has the separation property if for each S~ in A \ I there is a net
(Z~)cA annihilating ~ and such that the net ~ does not tend
to zero. It can be shown that in this case there exists one net (~)
which works for all elements ~ in ~\I , and, in fact, I =
={~A:~-~0] (cf.[~). In case when there exists such a bound-
ed net we say that the ideal I has the bounded separation property.
An ideal with bounded separation property is necessarily closed. If
A is a regular Banach algebra and F is a closed non-void subset
of its maximal ideal space, then both the maximal and the minimal
(non-closed) ideal with the hull F have the separation property.
However the bounded separation property may fail for the minimal
closed ideal with the given hull, even if it possesses the separation
property. It is also possible to construct a closed ideal in a regu-
lar Banach algebra which has bounded separation property and it is
different from the intersection of all maximal ideals containing it
(the question whether it is possible was stated as a problem in the
paper [2], but the construction of suitable example is rather easy:
we take as the algebra A the algebra of all continuous functions
on the unit interval possesing the derivative at O, and provide it
with the norm II ~II = l~l~ + I~I(0)I . The ideal in question is
then I 0 = { ~ ~ A: ~(o)= ~ / (o )=o ] ) . Thus the nets provide a tool
for separation and description of ideals. It is particularly interest-
ing whether this tool works for the group algebras. In this context
we pose the following problems.
PROBLEM 1. Let I be a closed ideal in ~(~) fRr an LCA ~roup
, Does I possess the separation property?
PROBLEM 2. Does there exist an ~C& group ~ and a closed ideal
I i n m4(~) which has the bounded separation property and is not of
the form Q*) I= 0 {ME CA): I= M}?
69
In fact we do not know any example of a closed ideal in a group
algebra which has separation property and is not of the form (*) .
REFERENCES
1. Z e i a z k o W. On a certain class of non-removable ideals in
Banach algebras, - StudiaMath. 1972, 44, 87-92.
2. Z e 1 a z k o W. On domination and separation of ideals in com-
mutative Banach algebras. - Studia Math. 1981, 71, 179-189.
WIE S~AW ZELAZKO Math.Inst.Polish Acad°Sc.,O0-950
Warszawa, ~niadeckich 8
POLAND
70
2.9. POLYNOMIAL APPROXINATION old
Let A be a uniformly closed algebra of continuous functions
on a compact Hausdorff space ~ . Assume that A contains constants,
that ever~ continuous linear multiplicative functional on ~ is of
the form ~--~($) for a unique element $ of S , and that every
element of ~ whose reciprocal belongs to ~ is of the form ~
for an element ~ of A •
Let ~ be a positive measure on the Borel subsets of 5 ,whose
support contains more than one point, such that the closure of A
in ~(~) , considered in its weak topology induced by ~(~) ,con-
tains no nonconstant real element. Assume that the functions of the
form ~+~ with ~ and ~ in ~ are dense in ~(~) in the
same topology. It is CONJECTURED that the closure of A i~n ~(~)
is isomorphic to the al~ebra of functions wh%chare bounded and ana-
lytio in the unit disk.
Positive measures on the Borel subsets of S are considered in
the weak topology induced by the continuous functions on ~ . Two
positive measures ~ and ~ are said to be e q u i v a I e n t
(with respect to "~ ) if the identity
holds for every element ~ of A . The closure of the set of measu-
res which are absolutely continuous with respect to ~ and equiva-
lent to ~ is a compact convex set, which is the closed convex span
of its extreme points. Extremal measures are characterized by the
density of the functions of the form ~ ~ , with ~ and ~ in A ,
in Let be an extremal meas e and let B be the wea
closure in ~(~)/ of the functions of the form ~+~ with
and ~ in A . It is CONJECTURED that the quotient Banach space
L • (~) / ~ is reflexive.
Pot equivalent positive measures ~ and ~ , define ~ to be
1 e s s t h a n o r e q u a 1 t o Q if the inequality
holds for every element ~ of A . If ~ is an extremal measure,
71
it is CONJECTURED that a sreatest element ~ exists in (the closure
of~ the set of measures which are abselutel,7 ' continuous with respect
t_~o ~ and equivalent to ~ . It is CONJECTURED that the functions
of the form ~+~ with ~ and @ i__~n ~ are weakl,y dense i n
REFERENCES
I. d e B r a n g e s L., T r u t t D. Quantum Ces~ro opera-
tors. - In: Topics in functional analysis (essays dedicated to
M.G.Krein on the occasion of his 70th birthday), Advances in Math.,
Suppl.Studies, 3, Academic Press, New York, 1978, pp.I-24.
2. d e B r a n g e s L. The Riemann mapping theorem. - J.Math.
Anal.Appl., 1978, 66, N 1, 60-81.
L. DE BRANGES Purdue University
Department of Math.
Lafayette, Indiana 47907
USA
72
2.10 . old
PROBL~3 PERTAINING TO THE ALGEBRA OF BOUNDED
ANALYTIC ~JNCTIONS
Here is a list of problems concerning my favorite algebra, the
algebra H°°~V) of bounded analytic functions on a bounded open sub-
set V of the complex plane. Some of the problems are old and well-
known, while some have arisen recently. We will restrict our discus-
sion of each problem to the bares% essentials. For references and
more details, the reader is referred to the expository account [1],
where a number of these same problems are discussed. The maximal ide-
al space of H~(V) will be denoted by ~ (V) , and V will be
regarded as an open subset of ~(V) . The grandfather of problems
concerning H°°(V) is the following.
PROBLEM 1 (CORONA PROBLEM). I ss V dense in ~(V) ?
The Corona Theorem of L.Carleson gives an affirmative answer
when V is the open U~'it disc ~ . In the cases in which ' ~ ( V ) has been described reasonably
completely, there are always analytic discs in ~(V)\ V , but ne-
ver a higher dimensional analytic structure.
PROBLEM 2. !s there ' alwa,ys an a~l~ic disc ~ ~(V)\ V ? I s
there ~ver an anal~ic bidisc in ~(V) ?
The 3hilov boundary of Ho°(V) will be denoted by ~ (V) . A
There is a plethora of inner functions in ~(V) , but the follow-
ing question remains unanswered.
PROBLEM 3. Do ~he !n~er functions sepa~te t h~ points of ~(V) ?
An affirmative answer in the case of the unit disc was obtained
by K.Hoffman, R.G.Douglas, and W.Rudin ~2, p.316~.
The Shilov bemndary ~(V) is extremely disconnected. Its Dix-
mier decomposition takes the form ~(V)-----T U Q , where ~ ana
are closed disjoint sets, O(T) ~--- L~(~?) for a normal measure on T ,
and Q carries no nonzero normal measures. The next problem is to
identify the normal measure ~ . There is a natural candidate at hand.
Let ~ be the "harmonic mearures" on ~(V) . These are certain na-
turally-defined probability measures on ~ (V) \ V that satisfy
for .
73
PROBLEM 4. Can the normal measure ~ on T be taken to be ~he
restriction of harmonic measure to ~ ?
There are a number of problems related to the linear structure
of H°°(V) o It is not known, for instance~whether H~@(V) has the
approximation property, even when V is the unit disc. As a weak- ~t~r olosed s . ~ l g e b ~ of L ~ t V ) , H~°( V~ i . a ~ 1 space. The following problem ought to be accessible by the same methods
u s e d t o s%~dy ~ V ) °
pROBTRu 5. Does ~(V) have a unique predual?
T.Ando [3] and P.WoJtaszczyk [4] have shown that any Banach
space ~ with dual isometric to H°@(~) is unique (up to isometry),
However, Wo~taszczyk shows that various nonisomorphlc B 's have du-
als isomorphic to H@a(~). An extension of the uniqueness result is
obtained by JoChaumat [5].
The weak-star continuous homomorphisms in ~(V) are called
d i s t i n g u i s h e d h c m o m o r p h i s m s, The evalua-
tio~Is at points of V are distinguished homomorphisms, and there may
be other distinguished homomorphisms. Related to Problem 2 is the
following.
PROBLEM 6. Does each distinMuished homomorPhlsm lie on an anal~-
The coord inate f unc t i on ~ extends to a map ~ : ~ ( V ) = V . I f ~ ~V , then the f i b e r ~ ~ ( ~ } ) conta ins a t most one d i s -
tinguished homomorphism.
PROBLE~ 7- Suppose there is a distinA~ished homomorphism ~ ,
~ ,e ~ an~ Sup~o~ e ~ is ~ aroil n V tle~inatl,in~ at ~. I f
~,~H~v~as ~ Lmi t alon~ V, dge,~ that l imi t coino~de ~!th ~ (~ ?
J.Garnett [6] has obtained an affirmative answer when ~ is ap-
propriately smooth.
The next problem is related to Iversen's Theorem on cluster valu-
es, and to the work in [7]. Define ~----~(V)N~, Denote by ~(~,~)
the range of ~ , ~ ~e°(,V) , at ~ , ~ ~V , consisting of those
values assumed by # on a sequence in V tending to ~ . An abst-
ract version of Iversen's Theorem asserts that ~ (~) includes the
topological boundary of ~(~) , so that~(~)\~(~) is open in
. The problem involves estimating the defect of ~(I~ ~) in
74
PROBT.~ 8. Ifeve~ point of ~ V is an essential singula r-
it~ for some f~CtiiOnii,ii ~ ~( VJ l doe..._~s ~(~)\~$(~)U~(~)~
have zero lo~arithm~ccaDao~t~for each ~ , ~ ~( V~ ?
The remaining problems pertain to the algebra H~) , where
is a Riemann surface. We assume that H~(~) separates %he
points of ~ . Then there is a natural embedding of ~ Into~(~J .
PROBLEM 9. ~s the natural embeddin~ ~ r ~/~(~) a home om.or-
p hlsm of ~ and an QDen subset of ~(~) ?
PROBLEM 10. If ~ is a simple closed curve in ~ that separa-
te s ~ , doe_~s ~ separate ~(~) ?
The preeedi~problemarlses in the work of M.Hayashi Ea], who
has treated Widom surfaces in some detail. For this special class of
surfaces, Hayashl obtains an affirmative anger to the followlng
proble~
PROBLEM 11. I_~S ~II(~) extremel~ disconnected ?
REPERENOES
I. G a m e 1 i n T.W. The algebra of bounded analytic functions. -
Bull.Amer.Math.Soc., 1973, 79, 1095-1108.
2. D o u g 1 a s R.G., E u d i n W. Approximation by inner
functions. - Pacif.J.~ath., 1969, 31, 313-320.
3. A n d o T. On the predual of H ~° . - Special issue dedicated
to W~adis~aw Orlicz on the occasion of his seventy-fifth birth-
day. Comment.Nath.Special Issue, 1978, I, 33-40.
4. W c j t a s z c z y k P. On projections in spaces of bounded
analytic functions with applications. - Studia Math., 1979, 65,
N 2, 147-173.
5. C h a u m a t J. Unicit~ du pr~dual. - C.R.Acad.Sci.Paris,
S~r A-B, 1979, 288, N 7, A411-A414o
6. G a r n e t t J. An estimate for line integrals and an appli-
cation to distinguished homcmcrphisms. - Ill.Jo~th., 1975, 19,
537-541.
~. G a m e 1 i n T.W. Cluster values of bounded analytic functi-
ons. - Trans.Amer.Math.Soc., 1977, 225, 295-306.
8. H a y a s h i M. Linear extremal problems on Riemann surfaces,
preprint. Dept. of Nath.UCLA,
T.W.GAMELIN LOS Angeles, CA 90024, USA
75
2.1 I. SETS OP ANTISYMMETRY AND SUPPORT SETS FOR ~+ G. old
Let X be a compact Hausdorff space and A a closed subalgeb-
ra of 6(X) which contains the constants and separates the points
of X . A subset ~ of X is called a set of antisymmetry for A
if any function in A which is real valued on % is constant on ~ .
This no~ion was introduced by E.Bishop EI~ x) (see also E2~,), who es-
tablished the following fundamental results: (i) X can be written
as the disjoint union of the maximal sets of antisymmetry for ~ ;
the latter sets are closed. (ii) If ~ is a maximal set of anti-
symmetry for ~ then the restriction algebra A IS is closed.
(lii) If ~ is in ~(~) and ~I~ is in AI~ for every maxi-
mal set of antisymmetry ~ for ~ , then ~ is in ~ .
A closed subset of ~ is called a s u p p o r t s e %
f o r A if it is the support of a representing measure for
(i.e., a Borel probability measure on ~ which is multlplicative on
). It is trivial to verify that every support set for ~ is a
set of antisymme%ry for ~ . However, there is in general no closer
connection between these two classes of sets. This is illustrated by
B.Cole's counterexample %o the peak point conjecture E3, Appendlx~,
which is an algebra ~=~0(X) such that X is the maximal ideal space
of ~ and such that every point of X is a peak point of ~ . For
such an algebra, the only supper% sets are the singletons, but not
every set of antisymmetry is a singleton (by (iii)).
The present problem concerns a naturally arising algebra for
which there does seem to be a close connection between maximal sets
of antisymmetry and support sets. However, the evidence at this point
is circumstantial and the precise connection remains to be elucidated.
Let ~oo denote the I, °° - s p a c e o f Lebesgue measure on T . Le t ~oo
be the space of boundary' functions on ~ f o r bounded holomorphic functions in ~ , and let ~ denote C(T) . It is well known that
~oo+ C is a closed subalgebra of ~eo E4], so we may identify it,
under the Gelfand transformation, with a closed subalgebra of C(~(~,°°)) , where ~(~oo) denotes the maximal ideal space of ~oo
(with its Gelfand topology). In what follows, by a set of antisim~et-
ry o~ a support set, we shall mean these notions for the case X--~(]oo)
and A----- ( t he Gel fand t r a n s f o r m o f ) ~ 0 o C . A l so , we s h a l l i d e n t i f y the functions in ~.o with their Gelfand transforms.
See the note at the end of %he section. - Ed.
78
The first piece of evidence fo~ the connection alluded to above
the fonowi [5]" is in l," and is in
(H°°+C) I Z for each support set S , then ~ is in ~'+ 0 ,
This is an ostensible improvement of part (iii) of Bishop's theorem
in the present special situation. It is natural %0 ask whether i% is
an actual improvement, or whether it might not be a corollary to
Bishop's theorem via some hidden connection between mav~mal sets of
amtisymmetry and support sets. The proof of the result is basically
classical analysis and so offers no clues about the latter question.
The question is motivated, in part, by a desire to understand the re-
sult from the viewpoint of abstract function algebras.
A second piece of evidence comes from [6], where a sufficient
condition is obtained for the semi-commutator of two Toeplitz opera-
tors %o be compact. The condition can be formulated in terms of sup-
port sets, and it is ostensible weaker than an earlier sufficient con-
dition of Axler [7] involving maximal sets of an%isymmetry. Again, i%
is natural to ask whether the newer result is really an improvement
of the older one, or whether the two are actually equivalent in vir-
tue of a hidden connection between maximal sets of an%isymme%ry and
support sets. As before, the proof offers no clues.
As a final bit of evidence one can add the following unpublished
results of K.Heffman: (I) If two support sets for H °°+ O intersect,
%hen one of them is contained in %he other; (2) There exist maximal
support sets for~ O.
All of the above makes me suspect that each maximal set of anti-
symmetry for Hoe+ 6 can be built up in a "nice" way from support
sets. It would no% even surprise me greatly to learn %hat e a c h
maximal set of antisymme try is a
s u p p o r t s e t. At any rate, there is certainly a connection
worth investiEa%ing.
RE~CES
I. B i s h o p E. A generalization of the Stone-Welerstress theo-
rem. - Pacif.J.Math., 1961, 11, 777-783.
2. G I i c k s b e r g I. Measures or%hogonal %o algebras and sets
of antisymmetry. - Trans.Amer.Math. Soc., 1962, 105, 415-435.
3. B r o w d e r A. Introduction to Punction Algebras. New York,
W.A.Benjamin, Inc., 1969.
4. S a r a s o n D. Algebras of functions on the unit circle. -
Bull.Amer.Math.Soc.~1973, 79, 286-299.
77
5. S a r a s o n D. Functions of vanishing mean oscillation. -
Trans.Amer.Math.Soc.,1975, 207, 391-405.
6. A x 1 e r S., C h a n g S.-Y., S a r a s o n D. Products
of Teeplitm operators. - Int.Eqt~at.Oper.Theory, 1978, I, N 3,
285-309.
7. A x I e r S. Doctoral Disseration. University of California,
Berkeley, 1975.
DONALD SARASON University of California,
Dept.Math., Berkeley,
California, 94720, USA
EDITORS' NOTE: The notion of a set of antisymmetry was intro-
duced by G.E.Shilov as early as in 1951. He has proved the first theo-
rem about representation of a maximal ideal space of a uniform algeb-
ra as a union of sets of antisymmetry (see Chapter 8 of the monograph
M.M.rex~a~, r.E,~HXOB, ~.A.Pa~oB, "KOMMyTaT~BHNe HOpMZpoBa~He
Eo~a", M., ~ESMaTFEB, 1958).
COB~ENTARY BY THE AUTHOR
The structure of the maximal sets of antisymmetry for H~tC
remains mysterious, although a little progress has occurredt P.M.Gor-
kin in her dissertation [8] has the very nice result that M (~)
contains singletons which are maximal sets of antisymmetry for H~+C.
Such singletons are of course also maximal support sets. The author's
paper [9] contains a result which is probably relevant to the problem.
REFERENCES
8. G o r k i n P.M. Decompositions of the maximal ideal space of
L ~ , Doctoral Dissertation, Michigan State University, East Lan-
sing, 1982.
9. S a r a s o n D. The Shilov and Bishop decompositions of H~+C
- conference on Harmonic Analysis in Honour of Antoni Zygmund,
vol.II, pp.46]-474, Wadsworth, Belmont, CA, 1983.
78
2.12. SUBALGEBRAS OF THE DISK ALGEBRA old
Let A denote the disk algebra, i.e. the algebra of all functi-
ons continuous on ¢~@~ ~ ard analytic on ~ . Fix functions
and ~ in A • We denote by [~,~] the closed subalgebra of A gene-
rated by I and ~ , i,e. the closure in A of the set of all func-
tions
N
'i~,H1,=O
We ask: w h e n d o e s[~,~]= A ?
Necessary conditions are
I) ~ , ~ together separate points of 0~
2) For each @ in ~ , either~@)=~ 0 or ~Q~)=~= 0 .
and 2) together are not sufficient . some regu-
larity condition must be imposed on the boundary. We assume
3) ~ are smooth on ~ , i.e. the derivatives ~/ and ~/ extend
continuously to ~ .
I), 2), 3) are not yet sufficient conditions. We add
For each ~ on ~ , either~) =~= 0 or ~I(@)=t=0 . 4) In [I] R.Blumenthal showed
THEORE~ 1. I), 2), 3) and 4) to~ether are sufficient for
Related results are due to J.-E.Bjork, [2], and to Sibony and
the author, [3] •
Condition 4) is, however, not necessary, since for instance
[(~_~)% ,(~_~)3]= A and conditions 1), 2), 3) hold here while 4) is not satisfied.
The problem arises to give a condition that replaces 4) which is
both necessary and sufficient for~.q]~A . In the special case
~.__(~_~)3 this problem has been solved~'°~ by J.Jones in [4] and his re-
sult is the following: let ~+ and W- be the two subregions of clos~
w h i o h a r e i d e n t i f i e d by the map ( E - { ) 5 . P u t
~ * ~ 'I + ¢-X-C~ - ' / ) .
Then for ~ in W +, ~ lies in W- and (~--I) 3 identifies
. W + and ~* Let % be an inner function on whose only singularity
is at ~ ° Then for some t , t >0 ,
79
TH~ORE~ 2 ( [ 5 ] ) . Let ~ be a f u n c t i o n in, A such that ~= [2 - { )~
a~d ~ to~ethe= satis~.~ ~), ~). ~). Then IT, ~] ~ A i f and onl,y
if for some % of the form ~i),
(2)
for all ~ in W+ __ , where K is some constant.
We propose two problems.
PROBLEM I. Prove an analo6ue o~ Theore m 2 for the case when
is an arbitrarEfunctionanalytic in an ope n set which contains
C~ ~ b E findin~ a conditio n to replace (2)whac k t0getherwith
1), 2}, 3} ~neoessar~and sufficient ~or [ & ~ ] = ~ A •
Furthermore, condition (2) implies that the Gleason distance
from ~ to ~, computed relative to the algebra [&~] , approaches
O rapidly as ~--~ , and so is inequivalent to the Gleason distance
computed relative to the algebra ~ . Let ~ denote a closed subal-
gebra of ~ which separates the points of C~ ~ and contains
the constants. Let ~B denote the Gleason distance induced onC~
by ~ , i.e.
I1~1t-- I
Let ~ denote the Gleason distance on 6~X>5 ~ induced by A .
PROBLEM 2. Assume that (a) The maximal ideal space of B
(b) There exists a constant ~ ,
is the disk 6~O~
~>0 , such that
Show that then ~ A •
80
RE~EEENCES
1. B I u m e n t h a I R. Holomorphically closed algebras of ana-
lytic functions. - Math.Scand.,1974, 34, 84-90.
2. B j o r k J.-E. Holomorphic convexity and analytic structures
in Banach algebras. - Arkiv for Mat.,1971, 9, 39-54.
3. S i b o n y N., W e r m e r J. Generators for A(~) • -
Trans.Amer.Math.Soc.~1974, 194, 103-114.
4. J o n e s J. Generators of the disc algebra (Dissertation),
Brown University, June, 1977.
5. W e r m e r J. Subalgebras of the disk algebra. - Colloque
d'Analyse Harmonique et Complexe, Univ.Narseille I, Marseille,
1977.
J . ~ Brown University
Department of ~ath.
Providence, R.I., 02912 USA
81
2.13. A QUESTION INVOLVING ANALYTIC FAMILIES OF OPERATORS
Let A be a uniform algebra with Shilov boundary ~ . (The
case when A is the disk algebra and ~--T is an interesting
example for this purpose. ) Suppose we are given a linear operator
which maps A into C(~) and has small norm. Suppose further that
the image(I+S)(A)~ g(~) is a subalgebra (here I is the inclusion of
A into ).
QUESTION: Is there an analytic family of linear operators ~ (~)
defined for ~ i_~n ~ so that (I+$(~ (A) is a subal6ebra of C(~)
for each % i_nn D and so that ~(~)=~ ?
The hypotheses are related to questions of deformation of the
structure of A , see [2] for details and examples. In cases where
(~) can be obtained the differential analysis of ~ (~) connects
the deformation theory of A with the oohomology of A . (See [I]). !
For instance, ~ (0) would be a continuous derivation of the algeb-
ra A into the ~ -module C(~)/A • Such considerations lead ra-
pidly to questions about operators on spaces such as V~lO(=C(T)/disk
algebra). (See [3] for an example.)
REFERENCES
I, J o h n s o n B.E. Low Dimensional Cohomology of Banach Al-
gebras. - Proc.Symp.Pure ~/ath. 38, 1982, part 2, 253-259.
2. R o c h b e r g R. Deformation of Uniform Algebras. - Proc.
Lond.Math.Soc. (3) 1979, 39, 93-118.
3. R o c h b e r g R. A Hankel Type Operator Arising in Deforma-
tion Theory. - Proc.Symp.Pure Math. 35,1979, Part I, ¢57-458,
RICHARD ROCHBERG Washington University, Box 1146
St.Louis, M0 63130
USA
CHAPTER 3
PROBABILISTIC PROBLEMS
The problems assembled i n t h i s Chapter a r e o f p r o b a b i l i s t i o o r i -
g i n bu t a r e more o r l e s s c l o s e l y connec t ed w i t h S p e c t r a l F u n c t i o n
Theory. Nowadays, p r o b a b i l i s t i c methods a r e i n c r e a s i n g l y a p p l i e d i n
Harmonic A n a l y s i s . Many such examples can be found , f o r i n s t a n c e , i n
the book by J . - P . K a h a n e "Some random s e r i e s of f u n c t i o n s " .
The t h e o r y o f s t a t i o n a r y Gauss ian p r o c e s s e s b r i d g e s P r o b a b i l i t y
and l~anct ion Theory. Moreover i t s u p p l i e s F u n c t i o n Theory w i t h many
i n t e r e s t i n g problems. This does no t e x h a u s t , o f c o u r s e , a l l c o n n e c t i -
ons between the two t h e o r i e s . R e c a l l , f o r example, t he t r a d i t i o n a l
a p p l i c a t i o n of F o u r i e r i n t e g r a l s to the i n v e s t i g a t i o n of p r o b a b i l i t y
d i s t r i b u t i o n s , o r the m a r t i n g a l e t h e o r y o f Hardy c l a s s e s , o r t he Brow-
n i a n m o t i o n which i s a t r a d i t i o n a l sou rce of coun te rexamples i n c l a s -
s i c a l F o u r i e r a n a l y s i s as we l l as a powerfu l t o o l f o r the s t udy o f
boundary problems. However, we would l i k e to emphasize the c i r c l e o f
" s p e c t r a l " i d e a s a r i s i n g i n the t h e o r y o f s t a t i o n a r y Gauss ian p r o c e s -
ses ( s ee 3 .1 , 3 . 2 , 3 . 3 , 3 .4 be low) . These p r o c e s s e s a r e l i n k e d w i t h
S p e c t r a l F u n c t i o n Theory by t h e concep t o f f i l t e r ( i . e . t he convo lu -
o p e r a t o r ~ ~ K*~ ) . N.Wiener s y s t e m a t i c a l l y and s u c c e s s f u l l y t i o n
a p p l i e d t h i s c o n c e p t , o r i g i n a t i n g i n e n g i n e e r i n g , to v a r i o u s p u r e l y
ma themat i ca l problems. Almost any s t a t i o n a r y Gauss ian p r o c e s s can be
considered as a r e sponse of some filter to a "white noise". All sta-
83
t i s t i c a l i n f o r m a t i o n b e i n g c o n t a i n e d i n t h e " w h i t e n o i s e " , t h e p r o b -
l em i s r e d u c e d t o t h e s t u d y o f i t s r e d i s t r i b u t i o n u n d e r t h e a c t i o n
of a given filter. This is, maybe, one of reasons why Hard~ classes,
entire functions, etc. - in other words almost all the tools used
now in Function Theory - are so important for some purely probabilis-
tic papers.
The questions posed in 3.1 can hardly be considered as concrete
problems. Rather, they indicate possible directions of investigation
and the interested reader may consult the excellent book [I~ (see
References in 3.1).
In contradistinction t o Problem 3.1 the "old" l>roblem 3.2 con-
tained a series of analytic questions of which one is solved (see the
Commentary).
The first part of Problem 3-7 has also been solved, whereas,
all aspects of Problem 3.~ remain open.
In this edition this Chapter has been enlarged by three Problems.
Problem 3.3 deals with the Hilbert space geometry of Past and Futu-
re. The questions posed there concern also the theory of Hankel ope-
rators. Problem 3.5 outlines a new field for dilation theory in the
theory of Markov processes. Problem 3.4 deals with limit theorems~
We conclude by indicating some Problems from the remaining ("de-
terministic") part of the Collection. In 8.4 approximation by trigo-
nometrical polynomials with bounded spectra is discussed. The proba-
bilistic interpretation is well-know~ (see e.g. the above metnioned
book by Dym and McKean, ~IJ in 3.1). Chapter 6 has some relations
with Probability as mentioned above~ Sarason's result cited as Theo-
rem I in 3,2 has very much in common with the contents of that Chap~
ter.
The years since the first edition have been marked by closer
and clearer connections among the elements of the triad "Punctlon
Theory - Operator Theory - Probability". This tendency (partly ref-
84
l e c t e d i n t h e o p e r a t o r - t h e o r e t i c i t e m 3°5) i s w e l l i l l u s t r a t e d b y t h e
[5] , [9] cited in the Com,,entary to Problem 3.2 papers t 3 L ~
85
3. I. SOME QUESTIONS ABOUT HARDY FUNCTIONS old
The theory of Gaussian-distrib,ated noise leads to a v~riety of
substantial mathematical questions about Hardy functions. I will put
the questions in a purely mathematical way; the reader is referred to
[I] for the statistical interpretation and/or additional information.
I. Let A , A~ 0 , be summable on the line and let
~ X ~ i×~----~ .
for fixed T> 0 e f f i c i e n % I Z
tions)~ See [1], § 4.2.
let
Then the exponentials
, but how is ~xT
approximated (by these func-
2. ~et ~ , ~ ~ , be ~ter, let T , T > 0 , be fi~ed, =d
~($) being the inverse transform ~
cannot be <oo for all small T ; w.at ,c~. be s a i a about K ? I1 z aloe, ~ can be eno=c~ly s~ar; see Ell- ~4.4.
3. Let ~ , ~6~ , be outer. The QUESTION is to explai n ~t
makes , the phase function ~*/~ the ratio" of two inner functions
or the reciprocal of an inner function; see [I], §4.6.
65~T~*/~ is itself an inner function if and only if ~ is integ- ral of exponential type ~<~ .
4. Let ~ , ~ , be outer. When does ~*/~ belon~ %o the
span, of ~i~$H °°, ~ 0 , .-.in ~oo ? see El], ~4.12.
5. The following conditions are equivalent for outer ~ ,
~ H~" ~ c ~ ~ /~ i s the ~ t i o e~ ~ ~unot ion of c1~ss
H~ and a function of class H~ ;RI ; b) ~ l ~ l ~ m < oo f o r
86
some integral function ~ of exponential type ~< T
~ ~
for some ~, ~ 0 , ~6~; see [1], §4o13o ~ f cad be said about
such function s ~ ? Note that b) is a problem of "multiplying down"
the function 4/~ in the style of [2]. What outer funct$cn satisfy
a), b), c) for every T , T~0 ? for no T ,T>0 ? Note that
cazmot satisfy c) for T ,T=O .
6. ~he phase ~-~ctlo~ ~/~ is ublq~tous. What c~ be ~id
abOUt it for the ~eneral outer f%mction ~ , ~ ~ ?
REI~ENCES
I. D y m H., M c K e a n H.P. Gaussian Processes, l~anction
Theory, and the Inverse Spectral Problem. New York, Academic Press,
1976.
2. B e u r 1 i n g A., M a 1 1 i a v i n P. On Fourier trans-
forms of measures with compact support. - Acta Math. 1962, 107,
291-302.
H.P.MCEEAN New York University,
Co~t Institute of Mathematical Sciences,
251 Mercer Street,
New York, N.Y. 10012, USA
COMMENTARY
In connection with section 4 see the COMMENTARY to problem 3.2.
The question discussed in section 5 is related to the paper [3].
REFERENCE
3- K o o s i s P. Weighted quadratic means of Hilbert transforms. -
Duke Math.J., 1971, 38, N 3,609-634.
87
3,2. old
SO~E ANALYTICAL PROBLEMS IN TI~ THEORY OF STATIONARY
STOCHASTIC PROCESSES
I. Let ~ (~) be a stationary Gaussian process with discrete
or continuous time (see [I] for the definitions of basic notions of
stochastic processes used here). Denote by ~(~) the spectral
density of ~ (in case of discrete time ~ is a non-negative inte-
grable function on the unit circle ~ ; in case of continuous time
is replaced by the real line ~ ).
Let ~(1) be a Hilbert space of functions on T (or ~ )
with the inner product
For ~>-0 , let ~+~(~) (resp. [,-~(~) ) be the subspaces of
(~) genetated by exponentials ~t~ with ~ >/~ (resp.
~.<-g ). Let ~j and ~- be the orthogonal projections onto
~+~(~) and ~-~(1) . Consider the operators
These positive selfadjoint operators were introduced into the
theory of stochastic processes in [2]. Many characteristics of pro-
cesses can be expressed in their terms. In particular, important
classes of Gaussian processes correspond to the following conditions
on ~ :
a) B~ is compact for all (sufficiently large in case of con-
tinuous time) ~ ;
b) ~ is nuclear for all (sufficiently large in case of con-
tinuous time) ~ .
Since the finite-dimensional distributions of a Gaussian pro-
cess are completely determined by its spectral density ~ , i t
would be desirable to describe pro-
perties of ~ in terms of ~ .
2. Processes with discrete time.
THEOREM I (~). The operators ~g are compact if and onl 2 if
the spectral density ~ can be reoresented in the form
88
Here P is a pol,Tnomial with roots on the unit circle and the fu,'~c-
tions ~ and ~ are continuous.
THEOREM 2 (I.A.Ibragimov, V.N.Solev, cf.[I] ). The operators B~
belon~ to the trace class if and only if
I(x) : I P(e ~x )1 ~ e°'(~),
Here ~ stands for a polynomial with roots on T and
~(~) T ~,ie ~'ix }]l~,il~lil < ~
PROBLEM 1. Under what conditions on spectral density ~ do the
operators ~ belon~ to the class T~ ' ~ ~ ~ ~ co , i.e. for
what
~ ~,I~ < co,
where ~i~ are ei6en-values of ~ ?
Theorems I and 2 deal with the extreme cases ~=co
THEOREM 3 (l.A.Ibragimov, see [I]). The estimate
, I .
I:B I- 0 f o r
(% is an integer, 0 < o~ < ~ ) holds if and only if
~(x) =lP(e~x) I ~ e~ (x~ ,
where P is m polynomial, ~ is an
tion and ~(~) satisfies the Lipschitz condition of order
• he val~e 15~I is expon~ntiall7 sm~ll for C ~
9n12 if the spectral density ~ is an analytic function.
3. Processes with continuous time.
-time differentiable func-
A •
if and
Nothing similar to
89
theorem 3 is known in that case.
PROBLEM 2. Under what conditions does the value IB~I
with power or exponential rate as ~ ~+ co ?
THEOREM 4 (I.A.Ibragimov, [I~). Let ~ C~) = I~(~)I -z
is an entire function of exponential type with roots
"~ -.--~ + oo
decrease
, where
E~ El~.., .
2. _co<~<oo t ~ <oo.
PROBLEM 3. Investigate the case when 0 ~(~)= I~ (A)I ~ . . . . . .
an_~d ~ , ~ are entire functions0f exponential t~pe.
This problem is essential for the analysis of the operator ~
in the multivariate case ~4~.
Note in conclusion that Problem I can be easily reformulated for
continuous time.
RE PEREN CE S
I. H 6 p a r ~ M o B M.A., P o S a H o B D.A. I~ayCCOBCE~e c~-
~ e nposeccH, M., "Ha~-Ka", 1970.
2. r e x ~ ~ a H ~ E.M., H r x o M A.M. 0 B~HCXeH~ Eox~e-
CTBa ~m~opM~ O c~y~a~o~ ~yHEn~, co~epma~e~c~ B ~pyro~ TaEo~
~ . --YcnexH MaTeM.HayE, I957, XH, I, 3-52.
3. S a r a s o n D. An addendum to Past and Future. - Math.Scand.,
1972, 30, 62-64.
4. M 6 p a r ~ M o B E.A. 0 no~Eo~ peryxapHOCTH MHOroMepH~x cTan~-
oHapH~x npo~eccoB. -~oEx.AH CCCP, 1962, 162, ~ 5.
I.A.IBRAGIMOV (H.A.HHPAI~0B) CCCP, I9IOII, ~eH~H~a~,
V.N, SOLEV (B.H.C0~EB) ~OHTaHEa 27, ~0~ AH CCCP
90
CON~ENTARY
Problem I has been solved by V.VoPeller in ~5 S :BTe ~p if f~=
=1~1~'~ ~ where ~ is a p ol,ynomial with roots on T an__d_d ~ belongs to
~l~,p the Besov class ..~,p .
Things are more complicated in case of continuous time Let ~ be
the outer function in ~+ satisfying ~ ~ I~I ~ on ~ and let
AI~-~- ~(~} (U~0~-~" ~ ) Then it is easy to show that
I I % 1 t j I = .H ] , where ~@o
stands for the usual Hardy algebra in ~T , and therefore a process
satisfies the strong mixing condition iff ~/~ ~ A I . The
structure of the Douglas algebra A 4 in contrast with that of
H ~ C(T) is very complicated, This is the main reason of troub-
les arising in the investigation of the continuous time case~. Many re-
sults valid for processes with discrete time could be easily extended
to processes with continuous time, if the following factorization
were true
~=~-I, where ~ is an invertible unimodular function in A I and I is an
inner function in ~ o Being valid in ~+ C , see ~], this fao-
torization, unfortunately, does not hold in AI ~]..
The factorization theorem in ~ + ~ can be applied to a des-
cription of ~-r e g u 1 a r G a u s s i a n p r o c e s -
s e s with continuous time, i.e. the processes with compact B~ ,
~ ~ > 0 . Let C denote the space of all continuous functions
on ~ having a finite limit at infinity.
THEOREM (Hru~8~v -Peller). A stationary. Gaussian proces..s.
[X~} ~6~ with spec.tral measure A is ~ -regular iff
where $~,v6C,
that I ~6 t~+~E~4
is an e.ntire function .of exponential, type such
91
PROOF (SKETCH). Let ~ be an outer function in~ satisfying
(see e~g. [9])~ which in turn by the factorization theore~ of T,Wolff
is equivalent to~6$~-6 ~(v~). B , where ~ ~6 ~ and
is a Blaschke product in ~ . Consider an auxiliary outer function
~@ defined b~ ~ . I~o(~)1~ -J-~{~(3) -- ~(~)} , ~ e ~ . Let~=~.
Then clearly ~ 65~= B , which implies that
is a restriction to ~ of an entire function of exponential type°
Conversely, s ppose { w th en-
tire function ~ of exponential type Clearly, ~ belongs to the
Cartwright class and it can be replaced by an entire function of ex-
ponential type being an outer function in C$ of the same modulus
on ~ . It follows that on
Since ~ r is of exponential type, ~ V 6 ~ ~@@ for some ~ 0
Therefore~ ~ ~ ~+ C . •
Some sufficient conditions for the strong mixing were obtained
in [8]. See also [9] for a brief introduction to the subject
REFERENCES
5. fI e ~ a e p B.B. 0nepaTop~ F a H K e ~ E~acca ~p H ~x npH~o~eHH2
(pa~HoHa~bHa~ annpoxcHMa~, rayccoBcK~e npo~eccH, npo~eMa Ma~o-
pa~HH onepaTopoB). - MaTeM.c6opH., 1980, I13, ~ 4; 588-581.
6. W o I f f T. Two algebras of bounded functions~ - Duke Math J ,
1982, 49, N 2, 321-328,
7. S U n d b e r g C. A counterexample in H@°+ B~C - 1983
(May-Jtme), preprint.
8. H a y a s h i E. The spectral density of the strong mixing sta-
tionary Gaussian process. - 1981, preprint.
9. H e 2 ~ e p B.B., X p y A 6" B C.B. 0nepaTopM FaHKe~, H a ~ e
HpH52~eHH2 H CTa~HoHapHMe rayCCOBCEHe npo~ecou. - YcnexH MaTeM.
Hays, 1982, 37, ~ I, 53-124.
92
3.3. MODULI OP HA/~KEL OPERATORS, PAST AND FUTURE
A discrete, zero-mean, stationary Gaussian process is a sequence
{X~}~e ~ in the real U ~ space of a probability measure i),
such that ~X~ O, i.e. depends only on ~--~ ; every function in the linear span of the
functions X~ has a Gaussian distribution. In prediction theory
the Past is associated with the closed linear span (over ~ ) ~p
of X k , ~<0 , and the Future with the span ~ of X k ,k>~O. These closed subspaces are usually considered as subspaces of the com-
plex Hilbert space ~ spanned by the whole sequence { Xk ~ keZ •
Our problem concerns the description of all possible positions
in ~ of the Future ~ with respect to the Past ~p .
The sequence {Q(~)~6Z being positive definite, there exists
a finite positive Borel measure /~ on ~ satisfying ~(~)~ A
~(~), ~Z . The measure ~ is called t h e s p e c t r a 1
m e a s u r e of ~X~}~6 ~ . Clearly the mapping ~ defined
by ~)X~l,-~-~ ~ , ~ 1 4 ~ ~ can be extended to a unitary operator from
to ~(~) . To avoid technical difficulties we consider henceforth
all stationary sequences I ~eZ in ~ , not necessarily real. A
stationary sequence is unitarily equivalent to a Gaussian process iff
the spectral measure /~ is invariant under the transform ~ ~
of T. Consider the set of all triples (~ ,~, ~) where A and
are closed subspaces of the complex separable infinite-dimensional Hil-
bert space ~ such that ~0~ (A + ~.] ~ ~ . The triples
(~,~,~i) and (A$, ~, ~) are said to be equivalent if there
exist isometr V o to tis ying =
V~ ~- ~ . Let ~ be the set of all equivalence c l a s s e s w i , h
respect to the introduced equivalence relation.
PROBLEM I. Which classes in ~ contain at least one element
(~, ~ ~) corresponding to a stationar,y sequence ~ X~}~ ?
The class ~ admits a more explicit description. Let ~A denote
the orthogonal projection onto the subspace A . Each triple ~
~-(A,~) defines the selfadjoint operator ~A ~ ~A and the numbers
93
L]~I:NA. % triple ~ = (A~,B i, ~ ) is equivalent to ~ =
unitaril 2 equivalen~and ~± (~) = ~±(}~).
SKETCH OF THE PROOP. We may assume without loss of generality
(note that ~ i = ~ under the assumption of the lemma). Given a subspace C in ~ let 6 I= ~@ C Consider the partial
isometrics V~,Vz determined by the polar decompositions
Let ~ be an operator on ~ defined by
~I ~in~ 1 is an arbitrary unitary operator from A nB onto A~N~I • It is easy to see that ~ is defined correctly (if ~
~A~n B then V~ V ~ z ). Also clearly ~ maps isometrical- ly A~ onto A~ • it remains to verify that ~ is a unitary opera- tor on ~ . Clearly it suffices to show that (~'~,~)---~(~,}) for
• , this is evident• Now we can con- sider only vectors Z of the form ~---J~A E , ~ ~ ~ It follows
i from (I) that
Pot every ~ ~+ m { ~ J and fo r every se l fad jo in t ope- r a to r T on a Hilbert space T~ such that ~ T ~ I ~$~T~{~ there exists ~ (A, ~)~ ~-(~)=~ and such that ~B ~A ~B I~eA1 satisfying ~+(~)=~
is unitarily equivalent to~ Indeed, without loss of generality ~± (~] ~ 0 By the well-
known Naimark theorem in ~i ~ ~4 there exists a projection ~A defined by
T ?{ (z- T)}) ¢
T (I- I - T
g4
PutB=~le {@} and ~=666~ (A+B) . Then clearly
a ternative e ther %= or C- s/Id then , where ~ is an outer function in
~ and ~ is a singular measure on ~ o We have cP'~(h~(~o)
c ~p ~ ~ and therefore only the case ~ ~ 0 is interes-
ting. Remall that for a bounded function
tor H~ on ~ is defined by ~ ~-
the orthogonal projection from ~ onto
%0 on ~ the Hankel opera-
C I- P÷) ~ , where ~+ is
H ~ ii
Consider the Hankel operator r1~,/h ' with the unimodular sym-
bol ~---6/~ • It is easy to see that ~G~ ~p~@~ I ~$ is
unitarily equivalent to ~%1% H~/~ (see [I], Lemma 2.6). The modulus of an operator ~ on Hilbert space is the selfad-
joint non-negative operator (T*~)4/~ . Problem I is therefore
intimately connected with the problem of description of the moduli of
Hankel operators up to the unitary equivalence.
PROBLEM 2. Which operator can be the modulus of a Hankel ooera-
tor?
There are two n e c e s s a r y c o n d i t i o n s for an
operator to be the modulus of a Hankel operator which imply evident
restrictions on triples equivalent to (~p, ~ ~). For any ~ the operator (H~ ~ /~ iLS not invert ible
because ~ II ~ ~II = 0 It follows that ~ does
contain an orthogonal basis { 8~ }~o with ~ 6~ ~ = 0 (i.e. an orthogonal sequence "almost independent" with respect to the
Past) provided Gp =9 = $& . • ~/~
The kernel ~e$(H~ ~) is either trivial or infinite
dime n.s.ional. Indeed, being invariant under multiplication by ~ by
Beurling's theorem, it is either trivial or equal to ~H ~ for am in-
ner function 0. Note that for ~ === (~p, ~ CT) we always have ~(~)-~--~-[~).
Indeed passing to the spectral representation ~ we see that
Y: i~*~ is an isometry of ~ (over ~ ) with ~ @p=== G~
Y~= ~ . SO we have either ~÷C"~) ~-(~) ~--- 0 or ~+(~)~-~_(~)=~ .
Under the a priori assumption that the angle between ~p and ~
is positive Problem I can be in fact reduced to Problem 2.
Indeed, if the angle between ~rp and ~ is positive then by
the Helson-Szeg~ theorem the spectral measure ~ is absolutely con-
95
tinuous and ~ ~& ~@~ is unitarily equivalent to H~/k~t HK/~, = I I * . . . . . . . . . .
• On the other hand if ~= H~ ~ and II~ ~ <
then there exists an outer function ~ in ~ with H~-H~/~
(see [2]). Put ~==[,~(I~l~ , B= :~pa~,L~(t~,l~ ) ~2~:~0} ,
A~__~h:(I~I:)I ~: ~< o} . Clearly [ ~ is a stationary
sequence with the Puture B and the Past A .
It follows from the above considerations that in the case of non-
zero angle between A and ~ the problem of the existence of a
stationary sequence with the ~uture ~ and the Past A can be redu-
ced to the existence of a Har~kel operator whose modulus is unitari-
ly equivalent to ~B ~A ~B •
In connection with Problems I and 2 we can propose two conjectures.
CONJECTURE I. Let ~ ~ } ~ o be a non-increasin~ sequence of
positive numbers and let ~,~ ~ = O. Then there exists a Hankel
operator whose singular numbers ~) ~ ( ~ ) satisf,y
Q ~ c H ~ ) = ~ , ~ 0
CONJECTURE 2. Let T be a compact selfad~oint operator such
that ~eg T is either trivial or infinite dimensional. Then there * )~
exists a Hankel opera to r U @ satisfyina T~- ( ~ ~ •
It can be shown that the last conjecture is equivalent to the fol-
lowing one.
CONJECTURE 2'. Given a t riPl e ~ (~,B,~)6 ~ such that
~B J~A is compact and ~+(~)-~- ~-(~) is either 0 o_~r~
there exists a stationary sequence ~ X~ }~6~ i__nn ~ whose
Future is B and Past is ~.
We can also propose the following qualitative version of Conjec-
ture I.
THEOREm. Let $ > 0 and ~ ~}~ o be a non-increasir~ sequence
of positive numbers. Then there exists a Hankel operator H~ satis-
fyin~
4
*) See def. of singular numbers in [3~,
08
PROOE. Let ~ be an interpolating Blaschke product having zeros
~} ~o with the Carleson constant ~ (see e.g, ~). Consider
the Hankel operators of the form ~i~ , ~ H ~ Then we have (see
[4], ~h. VIII)
where ~ is the compression of the shift operator ~ to ~ d¢__j~
= H~e~, ~ ~PG~ ~6~, P~ is the orthogonal projection onto ~ , ~ is multiplication by ~, ~ (~) &e~ 2~ ~# ~ ~.
Since ~ is an interpolating Blaschke product, there exists a func- tion ~ in H @0 satisfying $(~)~ ~ 1,1,~0.
It follows from (~) that ~(~)~- ~(~(~)), ~0.
Consider the ~ectors6 ~. t I~ (~ IZ;-I~) ~t~" We have~(~. )6~,.--~(~ )e_,~ (see L~J, Ch. VI) and there exists an invertible operator V on ~6
such that the sequence I V¢~} ~ 0 is an orthogonal basis of
6 , moreover if ~-- I is small enough then we can choose V so that
I] V If' IIV411 ~ ~ +8 (see ~], Ch. VII). The result follows from the obvious estimates
¢ NvlI.Hv"II • llv II. llv"ll
Conjecture I can ~e interpreted in terms of rational approximation.
It follows from the theorems of Nehari and Adamian-Arov-Krein (see
[I]) that for a function ~ in ~0 A ~P# ~$ I. °° we have
~ ^ where ~@ is the operator on with the matrix{~(~+~)}~,k$ 0 ,
~ is the set of rational functions with at most ~ poles outside
~0~ ~ (including possible poles at co ) counting multiplicities,
Conjecture S is equivalent to the following one.
CONJECTURE I'. Let I~}¢~0 be a non-increasin~ s.e.quence,
~4~%~-0 . Then there exists ~ in B~O A such that ~(S)~ ~,
~0. If the conjecture is true then it would give an analogue of the
well-known Bernstein theorem [5] for polynomial approximation. Note
97
in this connection that Jackson-Bernstein type theorems for rational
approximation in the norm B~0~ were obtained in ~], K7], ~8~.
We are grateful to T.Wolff for valuable discussions.
REPERENCES
I. H • z ~ e p B.B., Xp y~ ~ B C.B. 0nepaTop~raHxe~, ~a~y~e
mpM6xHxeHM2 M CTaUMoHapHMe rayccoBcKMe npoKeccN. - Yc~exH MaTeM.
Hays, 1982, 37, ~ I, 53-124. 2. A ~ aM ~H B.g., A p o B ~.3., Kp e ~ H g.F. BecKoHe~H~e ra~-
KeaeBH maTpM~ H O606~eHH~le 8a~a~ KapaTeo~op~-~e~epa ~ M.mypa. -
~y~K:~. aHa.~. M ero npH~., 1968, 2, ~ 4, 1-17.
3. F o x 6 e p r H.L~., h p e ~ H M.F. BBeAeHHe B TeopMI0 XHHeF~-Ib~X
HecaMoconp~e~ onepaTopoB. M., "HayKa", 1965.
4. H M K o x ~ c x ~ ~ H.K. ~le~ o6 oHepaTope C~B~Pa, ~., "HayKa",
1980. 5. B e p H m T • ~ H C.H. 06 06paTHO~ sa~aue ~eop~H Ha~nnyumero ~p~6-
x ~ m e ~ Henpep~H~x ¢ y ~ . - Co6pa~e c o ~ H . , T.2, I4~-BO AH
CCCP, I954, 292-294. 6. I I e s x e p B.B. 0Hepa~op~ I~a~ex~ ~ a c c a $'~ , ~ Hx npv~omeH~
(pa~MoHax~Ha~ a n n p o x c ~ a ~ , PayccoBcK~e r~po~eccN, npo6xema ~a~o - p a ~ , onepa~opoB).-ga~eM, c 6 o p ~ , I980, I I 3 , ~ 4, 538-58I .
7. P e I I e r V.V. Hankel operators of the Schatten-von Neumann
class %, 0 ~ ~ < ~ . - LO~! Preprints, E-6-82, Leningrad, 1982.
8. S e m m e s S. Trace ideal criteria for Hankel operators, 0~p~.
Preprint, 1982.
VVoo
S. V. HRUSCEV
(C.B.XPm R) V. V. PELLER
(B. B.nF.JL'IEP)
CCCP, 191011, JleHHHrpa~
~OHTaHI<a 27, ~0~
98
3.4. SOME PROBLF~S RELATED TO THE STRONG LAW OE LARGE
N~ERS F~ STATIONARY PROCESSES
Let in the (~E : E~Z) beige pro~ess and in 11='11: IL~(n'~'P)=d2>O. nary wide sense with ._~---- L.o~, ~
Denote the correlation function of the process by
static-
ff, =
and let
be its spectral representation. Here Z(~A) stands for the stochas-
tic spectral measure of the process (~) ; Z(~) is a pro-
cess with orthogonal increments.
It is well-known that the strong law of large numbers (h e r e-
a f t e r a b b r e v i a t e d a s SLLN) holds for all pro-
cesses stationary in the strict sense, that is, the limit of the
means ,.~=~4 ~:~.g~ exists a.e. But there exist processes
stationary in the wide sense such that the means ~ converge in
L~(I~) and diverge a.e. (see[1],[2]). SLLN criteria are given
in [2].
THEOREM (Gaposhkin [~ ). In the above notation
(1)
Thus SLLN holds iff the limit
I Z(d,?,) 111,"'oo -.$ (2)
exists a.e.
99
T~e theorem implies the following: if
R.(~)=o((.to~ bqM,) ) ~ ,_oo (.3)
then SLLN holds provided ~ > 0
hold in general.
In all knov~ counterexamples
, while for 6 =0
E I~,l P=o0 (P>~,)
PROBLEM I. Is the condition
P ]p>2: s,~pIF I~1 <oo
(maz be with the supplementar~ condition
for the SLLN?
PROBLEM 2. Is the condition
it does not
~(~)= 0 ) sufficient
I1~11oo < oo
(may be with the above supplementary c qndition~ sufficient for the
SLI~?
PROBLEM 3. If the answersto problems 1 and 2 ~re negative,
we ma 2 ask: are there stationar2 processes (~) satisfyin~
while SLLN does not hold? 0r .................. condition (~) can be relaxed for oo
L -bouuded processes?
All processes stationary in the strict sense obey SLLN, and so
the Theorem implies the existence of limit (2) as well.
100
PROBLEM 4. W h ,7 does limit (2) exist for stationary (in the
strict sense~ processes?
Analogous problems are of interest not only for unitary opera-
tors determining stationary processes but also for normal operators
in ~(X) (see an ergodic theorem of this kind in [3]). Here is
one of possible problems in this direction.
mo~ 5. ~et T be a no=al o~erato~ ~= ~ (×, 9) ~eln~ a ~-fAnite measure. Su~ose ~TII=~, ~e~(X), ~ IT
Does
exist a.e.?
(~)I ~< C a, e.
REFERENCES
1. B 1 a n c - L a p i e r r e A., T o r t r a t A. Sur la
loi forte des gran~nombres. - C.r.Acad.sci. Paris, 1968, 267 A,
740-743.
2o r a n o m E ~ H B.~. KpHTep~ ycHae~oro 8aKoHa 6oa~mmx ~cea
EaaccoB CTan~oHapHNx B mHpoEoM CMNc~e npo~eccoB ~ o~Hopo~m~x
cay~a~HHX noae~ - Teop.BepOaTH. ~ ee npzM., I977, 22, ~ 2, 295- -319.
3. r a n o m E ~ H B.~. 06 m%E~yaa~Ho~ ~pzD~qecKo~ TeopeMe
HOpMaa~m~X onepaTopoB B I 2 . - ~a~m/.a~aa~8 ~ ero np~., 198I,
I5, ~ I, I8-22.
V. F. GAPOSHKIN
(B.~.rAnommm)
CCCP, 103055, MocEBa
y~.06pasnoBa, 15,
MOCEOBCE~ HHCT~TyT m~zeHepoB zeaesHo~opo~soDo TpaHcnopTa
101
3.5- THE THEORY OP NARKOV PROCESSES FROM THE STANDPOINT
OF THE THEORY OF CONTRACTIONS
Let (X,~) be a Lebesgue space. A contraction P on ~ (X,~) is called a M a r k o v o p e r a t o r if P is order-posi-
tive and preserves the constants. In other words ~ is Narkov if
= , P = and P positive. The integral represen-
tation of such an operator is given by a bistochastic measure
Marker operators form a convex semigroup with a zero (=projection on-
to the constants) and wlth a unit. This is a functional equivalent
of the semigroup of multivalued maps, admitting an invariant measure,
of (X,~) onto itself. A detailed account of an analogous view -
point see in [1].
A ~rkov operator gives rise in a natural way to a stationary
Markov process with the state space X , the initial measure
and the two-dimensional distribution ~ (see above). In the space
2 = ~ X~ ~X~-~) of realizstions of the process a ~rkov
measure M~= ~ appears. The left shift T in (3, ~) generates
a unitary operator UT on ~(~,M) , a unitary dilation of
(non-minimal in general).
The main problem of the theory of Markov processes is the in-
vestigation of metric properties of the shift T in terms of the
~arkov operator ~ . The classical theory virtually used spectral
properties of ~ only. This is insufficient for metric problems,
being nonselfadJoint.
Modern tools of the theory of contractions seem not have been
used for this aim and we want to draw attention to this point (see
also [1] ). The connection between the contractions theory, their di-
lations, the scattering theory on the one hand and N~rkcv processes
theory on the other can be usefully applied in both directions.
I. PROBLEMS ABOUT PAST. It is easy to check that a Markov pro-
cess is forward (back) mixing in the sense of Kolmogorov iff P be-
102
longs to the class ~0. (resp. C.o), see notation in [2]. The oppo-
site class C 4 includes two subcases. The first one is of no inte-
rest and corresponds to an isometric P and to deterministic pro-
cesses. The second one, namely, the case of a completely non-iso-
metric contraction, is very interesting.Its very existence is far
from being obvious (for Markov operators), an example was given by
M.Rosenblatt [3]. An important theorem (see [2]) asserts that the
corresponding process~being non-deterministic~is quasisimilar to a
diterministic one. Our PROBLE~ is as follows: ~e the technique of
the theory of contractions to study mixing criteria of various
k%nds, dete~istic and quasideterministic , the exactness ~1], th e
bernoull!ty etc. A powerful tool for these topics is the characte-
ristic function of a Markov contraction. No adequate metric analo-
gue of this notion seems to be found (e.g. how can one connect this
function with the bistochastic measure ~)
2. NON-LINEAR DILATIONS. The theory of Markov processes imlicit-
ly includes some constructions unfamiliar in the theory of contrac-
already mentioned that a unitary operator UT acting tions. We have
in ~ ) is not the minimal dilation of the Markov operator
. The minimal dilation can be easily described in these terms.
It coincides with the restriction of UT to ~= $~ {~):
~ ~ , ~g(~i) being the subspace of ~,~) consis-
ting of functions depending on the ~ -th coordinate of {~I ~
only. The subspace ~ is the subspace of all linear function~ls of
realizations ("one-particle" subspace). Thus the theory of minimal
dilations corresponds to the linear theory of ~rkov processes whe-
reas the dilation UT has to be interpreted as a "non-linear"
one (clearly OT is a linear operator acting on non-linear func-
tion~ls of realizations).
The investigation of the pair (P, UT) ("a Markov operator
plus a non-linear dilation") is of interest for the theory of con-
tractions connecting it with methods and notions of the metric theo-
ry of processes (mixing, bernoullity etc.) E.g. the problem of the
isomorphism of two Markov processes is analogous with the problem
of existence of the wave operator in scattering theory. The enth-
ropy yields an invariant of the dilation etc. It would be interest-
ing to define the non-linear dilation for an arbitrary (non-positive)
contraction.
103
3. C*-ALGEBRA GENERATED BY MARKOV OPERATORS. Let us mention
a more special problem: to deacribe the C~-envelopeof the set o T
a! ~ Markov ope~tgrs. This algebra does not coincide with the al-
gebra of all operators. (G.Lozanovsky gave a nice (unpublished)
example: the distance between the Fourier transform as an operator
in ~£(~) and the set of all regular operators (= differences
of positive operators) is one). It seems likely that a direct descrip-
tion ef elements of this algebra can be given in terms of the order.
This C* -algebra plays an important r61e in the theory of gruppoids.
REFERENCES
I. B e p m E K A.M. I~orosHa~H~e OTo6pa~eH~ c HHBapMSHTHO~ Mepo~
(nomm~p~s~) ~ ~pEOBCKHe onepaTopH. - 8an.H~.ceMm~.~0~,
1977, 72, 26-61. t
2. Sz.-N a g y B., F o i a ~ C. Analyse h~rmonique des opera-
teurs de l'espace de Hilbert. Budapest, Acad.Kiado ~, 1967.
3. R o s e n b I a t t M~tationary Markov Processes. Berlin, 1971.
A. M. VERSHIK
(A.M.BEF~)
CCCP, 198904, HeTpo~Bope~,
MaTemaT~Eo-Mexasm~e CEm~
~aKyaBTeT JlePmHl?pa~oEoPo
yH~BepO~TeTa
104
3.6. EXISTENCE OF MEASURES WITH GIVEN PROJECTIONS old
Let F denote a measurable subset of the unit cube ~ c [Pv S~ K being the canonical operator of projection onto the k-th
axis, k ~ ~ , ...,~ , and X K the side of ~ situated on the
k-th axis. Consider the linear operator ~ transforming every
finite measure ~ on F into the system (M~ 4 , ~,...,~) of its
margi~als A , ~ A ) , A = X K • I t i s o f t e n
of importance to know whether a ~iven s~stem (~,... ,~) be_ -
longs to the ' image under ~ of a natural class of probability mea-
sures on F • some partial results are known, mostly for ~-~ .
E.g. for a class ~ of subsets F of ~ defined in terms of
measure spaces (Xk, ~K) , k-~ ~, 2 , an existence criterion of a
probability measure on F with marginals ($1~ , m~) is as follows:
the required measure exists iff no subset of ~ x X~ of the form
F rl ( A x B) , where A c X ~ , B = X ~ a~d ~ A + m~ B > 4 , is a union of subsets N~, N2 with ~(~NI) = ~(%~N2) -- 0
(cecil] ~ a smaller class of closed set was considered in [2]). The
class ~ is in particular characterized by the following property:
for every F ~ ~ the set of all measures on F with given margi-
nals is compact in the topology of convergence on sets of the form
A × ~ . A similar condition (where the non-decomposability of
sets FD (Ax B) is replaced by ~6S~ (FQ (Ax ~)) >0)
is a criterion of existence of a probability measure on F with
marginals ( ~4, M$~ ) subordinated to the Lebesgue type (i.e. ab-
solutely continuous with respect to the Lebesgue measure M~65~ ),
[1]. Analogous conditions fail to be sufficient for ~> ~ , and the
corresponding criterion is unknown. For ~--~ it is not known
whether the Lebesgue type can be replaced by any other type in the
last sentence of the previous paragraph. For ~>~ there is no
existence criterion for a positive measure on the cube ~ with
given marginals whose density function with respect to Lebesgue mea-
sure is majorized by the density function of a given probability
measure on ~ (see the discussion in ~I] ).
I. Cy~aEOB
REI~ERENCES
B.H. reoMeTpEqecEEe n p o 6 ~ e ~ Teop~J4 6eCEOHe~Ho--
105
2.
Mepm~xBepoaTHocTH~xpacnpe~eaemm~. - Tpy%M MMAH, 141, M.-Z.,
HayEa, 1976. (Proc. of the Steklov Inst. of Math., 1979,
issue 2).
S t r a s s e n V. Probability measures with given marginals.
- Ann.Math.Stat., 1965, 36, N 2, 423-439.
V.N. SUDAKOV
(B.H.C~0B)
CCCP, 191011, JIeHHHI~8~,
• OHTaHEa 27, .E01v~
COMMENTARY BY THE AUTHOR
Consider a finite or countable family of probability distribu-
tionsI~K,kEK} The answer to the question as to whether there
exists such a familyI~K, KEK I of random variables, each ~K being
distributed according to ~K , that for every pair (~i~ k~) the equa-
lity
holds, depends on existance of a probability measure ~ with marginals
~ on the set
Here ~ is the Kantorovich distance and UKil(2, stands for related
potential function (see e.g. D]):
One o~l show that for such a special type of subsets there always
exists a measure with given marginals ~ WSK, K~K } , so that the fa-
mily { ~K} under discussion does exist.
I am grateful to B.I.Berg for stimulating discussions
3.7. old
o< 1 EI < . The func t ion %E'
indicator of E . Set
106
ON THE FO~T~.R T~L~NSFOP~ OT THE INDICATOR OF A SET IN ~ OP P~ITE ~]~SGEE MEAS~J~
Consider a set E C ~ of finite and non-zero Lebesgue measure:
~(E= 1 ~; X~x@~ ~ is called the
^ ! ~
the ~ourier transform of ~E . We ask whether there is a set E ,
0 "< I EI ~ ~ such that ~E van lxhes on an open non-empty set A ,
A Note that if E is bounded then ~E is analytic in ~$ and
therefore cannot vanish on an open non-empty set. Some other similar
cases are considered in the author's paper [I]. If it turns out that
there exists such a se% ~ , 0 ~ I ~ ~ < co with ~H vanishing on
an open non-empty set, the SECOND PROBLEM will be to describe all
sets E with this property.
These questions are related to the uniqueness problem for a fi-
nite Betel measure ~ (in ~ ) with prescribed values ~(~o+~)
(~e~) , B@ being a given Borel set, 0<JE@J <°° . It follows A
from [1] that if there is no open non-empty ~c~ ~ with~IA~ 0,
then such a ~ is unique. And, conversely, it is not unique provided
does exist.
REFERENCE
1. C a ~ o ~ o B H.A. 0d o~o~ npodxeMe e~CTBeaaocTx ~ ~oHe~-
HHX Mep B e~EyomHx npOCTpaHcTmaX. --3an.aayqB.ce~.~0~, I974,
4I, 3-I3.
N.A. SAPOGOV (H.A. CAIIOIDB)
CCCP, 191011, 2e~I~
~osTaKKa 27, 30~
107
COMMENTARY
The first problem has been solved by Kargayev E2]. Sets E , A
Ec~, with 0<IEJ<oo and ~E vanishing on an interval DO EXIST.
Moreover, it is shown in E~ that given numbers ~,~, 0< @(~<~,
and an even function k: ~-->-EJ,$oo) increasing on ~0,+ @o)
and such that
t o o
0@
there i,, a se~.~no~ { r~ , , , ~'.*~',,,] } ~,--I
ments ~ t i s f y l ~
of disjoint seg-
^
and ~,EI(~,,6)~O, F 4~ U ,a6,E
REPERENCE
2. E a p r a e ~ H.H. Hpec~pasoBan~e Cyp~e xapaavepHcTzqec~ol
~f2~n~ ~oxecTPa, xcqesam~ee ~a ~Tep~e. - MaTeM. C6., 1982,
II7, • 3, 897-4II.
CHAPTER 4
OPRR A TOR THEORY
Due t o f o u n d e r s o f t h e S p e c t r a l O p e r a t o r T h e o r ~ t h e word " o p e -
r a t o r " became a l m o s t i n s e p a r a b l e f r o m t h e word " e e l f a d j o i n t " . T h i s
c o n n e c t i o n was so t i g h t t h a t e v e n now i t i s s t i l l cowmon t o s p e a k on
"NON-selfadjoint operators" as though forgiving general operators
for the absence of appreciable intrinsic struotttre. This kind of in-
feriority complex is being overcome nowadays under pressure from Phy-
sics (recall complex poles of resolvent on "the non-phTsioal sheet"
in the resonance scattering) and under the influence of the increasing
power of Analysis. "Analysis" means here mainly "C~lex A.AlyslS",
and the above tendency may be well illustrated,ln partlonlar, by
operator-theoretic problems of this book (first of all problems in
Chapters 4, 5, 7). Almost all of them are ~elated to the spectral
theory tending to blend with Complex Analysis and, in any case ,to
b o r r o w f r o m i t s o l e t h i n g more s i g n i f i c a n t t ~ Cauch~ f o r ~ l a o r
L i o ~ v i l l e a n d S t o n e - W e i e r s t r a s s t h e o r e m s , t o o l s n o t e x c e e d e d by t h e
c l a s s i c a l a p p r o a c h . T h i s b l e n d i s p r o b a b l y t h e m o s t c h a r a c t e r i s t i c
l e a f y . r e o f t h e p r e s e n t - d a y t h e o r y , o r a t l e a s t o f i t s p a r t s c l o s e t o
t h i s b o o k . ~he f i r s t s t e p s o f t h i s m u t u a l p e n e t r a t i o n w e r e made i n
t h i r t i e s and f o r t i e s ( S t o n e , Wold, P l e s s n e r , I , E r e i n , L i v s h i c ) .
~he new s p e c t r a l t h e o r y b e g i n s w i t h w o r k i n g o u t c o n v e n i e n t f u n c -
t i o n a l models whereas in classical analysis such a model was often
109
t he end o f i n v e s t i g a t i o n . Many problems i n t h i s Chapte r a r e r e l a t e d
t o the m u l t i p l i c a t i o n o p e r a t o r ~ - ~ # whose r e s t r i c t i o n s and compre-
s s i o n s t o s u i t a b l e subspaces y i e l d models we have J u s t ment ioned .
One o f the most p o p u l a r models can be d e s c r i b e d i n t e rms o f t he s o -
c a l l e d c h a r a c t e r i s t i c f u n c t i o n of the o p e r a t o r ( t h e Sz~ 'kefa lv i -Nagy
- Foia~ model and i t s g e n e r a l i z a t i o n s ) . Th is model r e d u c e s s p e c t r a l
problems to the i n v e s t i g a t i o n o f boundary p r o p e r t i e s o f v e c t o r - v a l u e d
f u n c t i o n s o f the Nevanl inua c l a s s . The q u e s t i o n s posed i n Problems
4.8-4.20 exhibit distinctly enough the present state of affairs which
o a n b e suwnar ized as f o l l o w s . P Almost a l l ach ievements o f the H t h e o r y have been e x p l o i t e d
( f r e e i n t e r p o l a t i o n , the d e l i c a t e ~ t l t i p l i c a t i v e s t r u c t u r e o f H P-
f u n c t i o n s , co rona Theorem e t c . ) and a new " o p e r a t o r - v a l u e d " F u n c t i o n
Theory i s needed now. I t s c o n t e n t s a r e e s s e n t i a l l y n o n - s c a l a r though
t h i s f a c t i s o f t e n d i s g u i s e d b y f o r m a l l y d i m e n s i o n - i n v a r i a n t s t a t e -
ments . T h e r e f o r e the new p r o g r e s s r e q u i r e s no t on ly new e f f o r t s i n
HL t h e s p i r i t o f t he s t a n d a r d t h e o r y , bu t r a t h e r c r e a t i n g a k ind
o f s p e c i a l n o n - c o ~ t a t i v e i n t u i t i o n . V iv id examples a r e Problem 4.12
and t he Halmos-Lax theorem d e s c r i b i n g i n v a r i a n t subspaces o f the mul-
t i p l e shift. This theorem is deciphered (for a very special situation)
in Problem 4.14. As to Problem 4.12, its seeming simplicity conceals
many interesting concrete realizations. Namely, Problem 4.12 inclu-
des as partlcular cases the principal question of Problem 4~I0 and
the matrix generalization of the Corona Problem discussed in the Commen-
tary. Many problems of the "vector-valued" function theory admit in-
teresting scalar interpretations (as, for instance, Problems 4,4,
4.9, 4.12). Other problems make sense only when the space of values
is multidimensional.
I% is not easy to classify rapidly growing Operator Theory
with its intertwining ramifications. The same is true even for its
parts presented here. However, we tried to group the problems in
110
accordance with the intrinsic logic of the subject which we understand approximately as follows.
I I I
I I I I I I I l
I I
I
1 I
I !
L
S e l f a d j o i n t S p e c t r a l Problems 1
f orlginatlng in Differential Equatlons~
Schauaer ~ , o r y I s~l ~eorem I ~heory I ~ e o r y I
_ ~ __~_ GENERALIZATIONS i 0ono'e'e I C H A P T E R 5 LModels
II " 1 Integral Opera to r s
BanaOh with Kernels having
Algebra O p e r a t o r - v a l u e d a k ind of Symmetry
Hetho4s : Analytlc (Hankel, T o e p l i t z ,
C*-Algebras, l~tions, Hiener-Hopf, Singular
Representations their £ n t e ~ e l Ope ra to r s )
Commutators, ~ L l t i p l i c a t i v e
C a l c u l i S t r u c t u r e
~ pectral Theor ies module an Ideal,
Systems of a lmoet Commuting Ope ra to r s
Of course, the scheme leaves aside a greet number of links exis-
ting i n Opera to r Theory as w e l l as connec t ions w i th o t h e r b ranches
of Mathemat ics . But the B i n purpose of the scheme i s t o e x p l a i n the
f o l l o w i n g arrangement of Problems i n s i d e the Chapter .
I . S e l f a d j e i n t s p e c t r a l t h e o r y , i n c l u d i n g P e r t u r b a t i o n Theory
111
and Scattering ~S:teozT~ and problems they generate (NN 4.1-4.8, 4.15). This ~romp represents so well-known domains of Operator Theory that
we do not risk commenting It.
II. Punctlonal Models, Characteristic Function and other opera-
tor analytic functions (4.8-4.20, 4.4). Being of quite different
origins, these Problems mostly can be reduced to %he investigation
of the multiplicative structure of operator-f~nctions and they need
certain development of operator (or matrix) analogues of HP-tech-
nlques. Problems 4.12, 4.15, 4.20, 4.21, 4.24 are directly related
to the themes of the next Chapter 5.
IIl. Banach Algebra Hethods are a common feature of Problems
4.22-4.29 and they also play an important role in Problems 4.30-4.36.
We mean here various aspects of "algebraizatlon" as an alternative
approach to generallsatlons of the Spectral Theorem and to interpre-
tations of spectral nature of non-commutlng objects. These aspects
are C*-algebras, calculi, symbols, theories h la Fredholm-Riesz-
Schauder etc.
IV. "Near-normal" (i.e. h~po-, semi-, sub-, quasi-,...nQrmal)
operators as a particular case of the perturbation theory for fami-
lies ( .A t , . . . , A I~) of commuting selfad~oint operators (namely, the case ~=~ ). Such operators are the subject of problems 4.30-4.36
(and 4.37 - from the technical point of view). An analogy with the
classical Perturbation Theory may be drawn as follows: the classical
smallness of "the ima~Insry part" ~ A = (A- f)/3 ~ is replaced here
by the .al~ess of the selfoo=utator [ A f] : A A ~ f A
the aocretlvity condition ~A ..<0 by the ~pono=~llty [A A].<O
etc. The new situation is In a sense "two-dimansional", the operator
A bei~ viewed as a pair CP~ A, ~ A) . This leads to essential
complications in compemlson with the "one-dimensional" (i.e the
selfadOolnt) case. Some quite simple "one-dimensional" problems look
rather difficult in the "%we-dimensional" settle, as for example
112
the stability of continuous spectrum or the solvability of equation
J~A~ for K e~(~ being a given operator ideal). These problems
likely require a hard analysis of operator algebras and finding
out of algebraic obstacles to solve the equation [A,A~E~ in the
form A=N+~ where[NjN~]~-0, ~ . Problems of this direction are rep-
resented rather distinctly by items 4.30-4.38. Problems 4.39, 4.40
deal with interesting concrete questions (circular symmetry of spect-
ra of endomorphisms, quasinilpotency of indefinite integration) which
can hardly be placed at a definite point of our scheme On the other
hand, parts of Operator Theory indicated in the scheme, are included
in Chapters 5 and 7. Let us enlist some close cross-links between pro-
blems of this Chapter and of other Chapters.
Chapter 2 (Banach Algebras): Problem 2.13 could be placed among
items 4.8-4~20; Problems 2.1, 2.4 are related to 4.39. ~inally, Prob-
lems 2.2 and 2-3 deal with a subtle behaviour of the norms ~II~I,
and therefore find a responce in 4.38.
Chapter 9 (Uniqueness). Problem 9.4 admits a clear operator-
theoretic interpretation in the spirit of Problems 4.31-4.36, and it
is recommended to read Problem 9.6 and Problem 4~4 simultaneously.
Numerous problems on multipliers dispersed throughout the volume
are related by their common origins and intrinsic connections (4.21,
4°25, 4.26, 2.6, 10.3, 9.9, I0.8).
Unfortunately (or fortunately - taking into account inevitable
volume restrictions) many parts of Operator Theory are not mentioned
here. Por example, the invariant subspace problem is not presented
explicitly. But it is alluded to, incidentally, in constructions of
items 4.8, 4.13, 4°22 whereas other problems of this "descriptive"
direction of Operator Theory have to wait for another Problem Book.
The same fate (i.e~complete oblivion) is shared by many consumers of
Operator Theory. The reader will find in our Book neither "pseudodif-
ferential operator", nor "integral Fourier operator", nor "operator
-theory" ....
113
4.1. BOUNDEDNESS OF CONTINUUM EIGENFUNCTIONS AND THEIR
RELATION TO SPECTRAL PROBLEMS
We will describe a set of problems for matrices acting on ~) .
There are analogous problems for ~(2 v) and for suitable elliptic
ope~tors on L~CR~ . Let A be a bounded self~djoint o p e ~ t o ~
on ~(Z) whose matrix elements obey 6~ ~- C~ A~j) ~-0
if I $-j I~ ~ . A fundamental result asserts the existence of a mea-
sure ~p(E) , a f~ot ion #(E) taking the values O, t , . . . , c o (infinity allowed)with }~(E)~{ (~)-a.e. E ~nd #(E)=O i f E ~ p and for each E , Y~(E) l i near l y independent sequen-
ces ~(E; ~) ; ~=4~...~(g) (not necessarily in ~ ) so that
(a) I%(E;~)t%c(t+t~l) (b) ~ j ~ ( E ; j ) = E % ( E ; 0 ;
(C) Let = E; ; i.e. functions, ~ , on £ with
~E) having values in CE) (where C °O ~ = ) and let ~ de-
note sequences in ~) of compact support. Define U taking C 0
into ~i by (U~)@ {E)= ~ ~-~ET ~i ~(tr~). Then ~ extends to
a unitary ~p of ~(Z) o~to ~; (d) U(Ao)=ECU~)
These continuum eigenfunction expansions are called BGK expan-
sions in [13 in honor of the work of Berezanskii, Browder, G&rding,
Gel'fand and Kac, who developed them in the context of elliptic ope-
rators. See EI,2,3~ for proofs. These expansions don't really contain
much more information than the spectral theorem. The most significant
additional information concerns the boundedness properties of ~ ;
see E4,5S for applications.
Actually, the general proofs show that (~+l~l) in part (a)
can be replaced bye(4 + If~l) @ for any ~ > ~ . Indeed, one shows
that for any ~ " , one can arrange that for (~)-a.e.
~(') ~ {E~')~% If one could arrange a set, ~ , of good E's where ~ ^ ~ f o r a 1 1 ~ with ~(~\~)=0 , then
on ~ , ~ . Thls leaves open:
QUESTION I. Is it true that for (~)-a.e. E , each I$~(E~')
is bounded?
There is a celebrated counterexamFle of Maslov E6] to the botutd- edness in the one dimensional elliptic case. As explained in Eli,
Maslov's analysis is wrong, and it is not clear whether his example
has bounded u's a,e. We believe the answer to question I (and all
114
other yes/no questions below) is affirmative, but for what we have
to say below, a weaker result would suffice:
QUESTION 2. Is it a t least true that for (~)-a.e. E
~- l~o~(E, ~)1 is ~o~dea? a l Z oD : ~N+~ m,~N
QUESTION 3. Is it true that
~N+~ N-,-oo
exists? T h e ~ w e
Given a subset M , of
I%(E,#)I=- E (a.,E) 1fl,14 N
will denote
{ (E ,~) : E ~I~ , o~< N} b y ~(~,E).
we def ine
I .gE,-)CU ) CE)AoCE) {E:CE,~)~M}
where a suitable limit in meam may need to be taken. Define
M~ ={CE,~): ~CE, ' ) ~ £~}
M~:{CE,~) : ~(~, E) =0 but (E,~) ~ M4}
M3={(E, ~) : i ( ~ , E) ~ 0 } .
Obviously, P(M,) i s the p r o j e c t i o n onto the po in t spectrum o~ A .
QUESTION 4. Is i t t rue tha t P(Mg) ,,is the p,ro,~ection ont,o the
s i m ~ l a r contimuous space of A ~n d P(Ms) th, e p ro j ec t i on onto the
absolutel~ continuous ~pectrum of a ?
Among other things this result would imply that in the Jacobi
case (where the number K of the third sentence in this note is 2),
the singular spectrum is simple.
In higher dimensions, one ca~ see situations where A separa-
tes tie ~( Z~)=~(% v') ® ~ CZar) ~d A:A4®I+ I®A~) where A has a.c. spectrum with eige~functions decaying in ~2 di-
mensions but of plane wave form in the remaiming ~ -dimensions.
115
One can also imagine a.c. spectum from combining singular spectrum
for and In either case for lot. of continuum a.c.
eigenfunctions.
QUESTION 5. Isthere a s ensib!e(i,e ~ not obviously false ) ver-
slon of ~uestion 4 in the multidimensional case?
There are examples [7] of cases where A has only point spect-
rm~ but there is an eigenfunction with ~(~, E)>0 (since it occurs
on a set of ~ -measure zero, it isn't a counterexample to a posi-
tive answer to Question 4). Does the second part of Question 4 have
a positive converse?
QUESTION 6. Is i t true that if A~=~ ~s a bounded ellen-
function with ~ >0 for a set, ~ , of E's of positive Lebesgue
measure r then A has some a.c. spectrum o n ~ ?
QUESTION 7. Wha~ is the ' proper analo~ of ~uestion 6 for singular
continuous spectrum?
REFERENCES
le S i m o n B. Schrodinger semigroups. - Bull.Amer.Math.Soc.,
1982, 7, 447-526.
2. B e p e B a H C ~ ~ ~ D.M. PasxozeHEe no CO6CT~eHH~ SyHEn~M
ca~oconp~eHH~X onepaTopoB. K~eB, HaYEOBa ~D~a, I965 (Transl.Math.
Mono., v. 17, Amer.Math.Soc., Providence, R.I., 1968).
3. K 0 B a~ e H E 0 B.$., C e M ~ H O B DoA. HeEoTopHe BonpocH
pa3~o~eHz~ no 0606~eHH~ C06CTBeHHNM ~yHK~F~K~I onepaTopa mpe~zHrepa
c c~Ho c~zT~L~p~ Howe~dEa~2~. - Ycnex~ M&T.HSyE, 1978, 33,
B~n.4, I07-I40 (Russian ~ath.Surveys,1978,33, 119-157).
4. P a s t u r L. Spectral properties of disordered systems in one-
body approximation. - Comm.I~ath.Phys.,1980, 75, 179.
5. A v r o n J., S i m o n B. Singular continuous spectrum for
a class of almost periodic Jacobi matrices. - Bu~l.Amer.Math.Soc.,
1982, 6, 81-86.
6. ~ a c ~ o B B.H.06 aC~TOTZEe O~06~eHH~X CO6CTBeHHNX~ ypa-
BHeH~pe~Hrepa. - Ycnex~ MaT.HayE,1961,16, ~n.4, 253-254.
7. s i m o n B., S p e n c e r T. unpublished.
BARRY SIMON Departments of ~,~thematics and Physics
California Institute of Technology
Pasadena, California 91125 USA
116
4.2. SCATTERING THEORY ~0R COULO~B TYPE PROBLE~S
1. Let self-adjoint operators A and Ao act in a Hilbert space H and suppose the spectrum of A o is absolutely continuous. Suppose further that there exists a unitary operator function Wo($ )
satisfying the conditions:WoC$)WoC%)=WoC$)WoC~)~ W0(~)Ao=AoWo(t) s - ~ W~(t+~)Wo(t) = E and there exist l~m~ts
S-f/~ ea~P C¢At)ec#C-'~Aot)WoCt)=U+_ CA,Ao) t~-_..+_o~
(1)
s - ~ , Wo(t~)e/Jr, pC~Aof,)ec, p (-~AD P= U_+ (Ao, A) I;--~_+ oo
(2)
where p is the orthogonal projection onto the absolutely conti-
nuous part of ~ . For the generalized wave operators U~ ~-~
the equality AUi(A,A o)= U± (A,Ao) Ao holds. The factor
W0 C$) is not uniquely defined. The factor Wo(~)=WoCt)V~ Ct ~ 0)
can be used too when obvious requirements to V~ are fulfilled. Due to this ambiguity the naturally looking definition
SCA, Ao) =U+CA, Ao)U- CA, Ao) (3)
of the scattering operator becomes senseless since ~(A, Ao) =
= V+* S CA, A~)V- is in fact an arbitrary unitary operator commu-
ting with A0
PROBLEM I. Find physicall2 motivated normalizatio n of W0C$)
when $ =~ , removin~ the non-uniqueness in de~ition of the
scattering operator.
This problem has been solved for the scattering with the Cou-
lomb main part, i.e. when
q---B--@ ~+ L~F¢¢¢+Q ~ *avC~'~J~'T, Ao~ =--~+--~--~ ct~ ~(¢+0 (4)
117
where 0 < ~ <~ , ~ > 0 . For the system (4) the factor W0($)
is of the form ~-~] : Wo(t)= romp [ ~ ( ~ t ) ~ I t l % / { # o ] •
In [2] it is proved that with such Wo(~) the scattering opera-
tor (3) coincides with the results of the stationary scattering theo-
ry. This fact suggests that the normalization of ~o($) is physical-
ly reasonable. A similar problem has been solved for the Dirac equa-
tions with the Coulomb main part [5]. Consider the system
(5)
where ~(OC)~o~±/G),
PROBLEM 2. Find
(5).
When ~+ =-~_
case, when o~+ = ~_
~ ± ~
W0($) and solv,e Problem 1,,,for the sys,t,em
the system (5) can be reduced to the Coulomb
it is considered in [2]. The case %_= 0 ,
~+~0 is of importance in a number of physical problems [6-~.
2. The Coulomb interaction of ~ particles is described by the
system
(6)
where ~K is the radius-vector of the K-th particle, tK,~ =
= I~K - $~I . Taking bound states into account leads to the transi-
tion from Ao~ to the extended operator A o ~8]. The operators
~o and Wo(t ) for the system (6) have been constructed in
[9° The construction is effective when ~=3 . In [9] the existen-
oe of U± (A is proved
PROBLE~g 3. prove the existenc e of U± (A0~ A) for the s2s-
tem (6), i.ue. ~ the completeness of the correspondin~ wave operators ,.
3. We shall consider the non-standard inverse
PROBLE~ 4. Let operator A o and generalized scattering operator
be knewn~ Recover the operator A and the correspondin~
No(t) , find classes where the ~rob!em has on e and only one Golution.
Consider a model example, namely
i% oB (?)
where p(S~) belongs to ~he Holder class and p(s~)>O ,
There is an effective solution of the "direct" problem (the construc-
tion of WoC$) and of $ ) for the system (7) [~:
where ~--- C ~-,) -- ~ ~ C ~)+ and
(9)
Suppose in addition that
g &
Then operator a has no d isc re te spectrum, The formulae (8 ) , (9) give an effective solution of Problem 4 for system (7) by the facto-
rization method, i.e. ~(~) and p(D~) are found by S(G~) and
hence A and W@ ($) . some problems with non-local potentials
and the case (5) with ~+ =o~_ can be reduced to the system (7).
4. Now we come to s t a t i o n a r y i n v e r s e p r o -
b 1 e m s. When ~ > 0 the discrete spectrum of the operator A
(see (4)) is described by the Ritz half-empiric formula
2
its proof is given in [10]. The number ~ is called a quantum de-
fect of the discrete spectrum. The same number ~ serves aS a de-
viation measure of the operator (4) from the case of hydrogen nucleus
-
119
PROBLE~ 5. 1~ind a method to recover the potential ~(~) from
. . . ) .
Here the representation of ~e by solutions of the Schrodin-
ger equation ~10] and the transformation operator ~1 ~ can be useful.
The definition of the quantum defect ~K(K = ±~,t~ ...) is introduc-
ed in ~I~ for the Dirac radial equation too
&
(10)
So problem 5 can be formulated for the case (10) too. Note that the
classic inverse problem for the Dirac equation has not yet been solv-
ed even when the Coulomb member is missing. The systems of the Dirac
type have been thoroughly investigated when (see (10))
~V(~)- m W(~) H( I L
where V($) , W(~) are real functions from t (0, oo) . The pe-
culiarity of the system (10) is defined by the fact that the element
W(~) is known EW(~-K/~ ~ L (0, oo)~ • It is therefore
perhaps not necessary to use the scattering (or spectrum) data over
the whole energy interval--Co <~ < co . We come to a peculiar
half-inverse problem both for the Coulomb and non-Coulomb case.
PROBLEM 6. Let ~,~4~ W($) =-K/~) (K--- ± ~ , ± ~ , o . .)
be known. Reconstruct the element V(~)-----~÷ ~(~) by the scat-
tering (or spectrum) data, be lon~ing to She energy interval
0 ~ < ~ . The model half-inverse problem has been solved in [13].
120
REFERENCES
I. D o i i a r d J. Asymptotic convergence and Coulomb interaction.
-J.~th.Phys.,1964, 5, 729-738.
2. C a x H o B ~ ~ ~.A. 0606~eHHHe BOXHOB~e onepaTopH. - ~laTeM.c6~
I970, 8I, ~ 2, 209-227.
3. C a x H O B E ~ ~.A. 0606mgHHNe BOXHOB~e onepaTop~ ~ peryxapH-
sany~ p~a Teop~ BOSMy~eHm~. -- Teop. E MaTeM.~E3HEa, I970, 2,
I, 80-86.
4. B y c ~ a e B B.C., M a T B e e B B.B. BOa~OB~e onepaTopH
%m~ ypaBHem~ mp~repa c Me~eHHo y6~B~ noTeHn~a~oM. -
Teop. E MaTeM.~EsEEa, 1970, 2, 367-376.
5. C a x H o B H ~ JI.A. llpmn~Lu m~BapHaH~n~OCTE ~ o6o6~eHH~z(
BOJ~HOB~gX ollepaT0pOB. -- ~yHEl~.aHa~H3 E 8tO ITIOE~IO~., 1971, 5, ~ I,
61-68.
6. Tym~ex~H~e ~em~ B T2ep~x Teaax. MMP, 1973.
7. B p o ~ c E ~ ~ A.~., I ~ y p e B ~ ~ A.D. Teop~ s~eETpoH--
HO~ SM~CC;~H ~13 MeTa.EJIOB, 1973.
8. ~ a ~ ~ e e B ~I.~., MaTeMaTE~ecEEe BonpocN EBaHTOBO~ Teop~E
pacce~ ~ CHCTe~ Tpex ~IaCT~I~. - Tp.MaTeM.EH--Ta ~M. B.A.CTeE-
XOBa, 1963, T.69.
9. C a x H o B ~ ~ J~.A. 06 y~eTe BCeX EaHaJIOB pacceH~ B sa~a~e
Te~ C Ky~OHOBCK~M Bsam~o~e~OTBEeM. -- Teop. ~ maTeM.~Es~k~a,
1972, 13, ~ 3, 421-427.
I0. C a x H o B ~ ~ ~.A. 0 ~p~xe P~TSa ~ EBaHTOB~IX ~e~eETax
cneETpa pa~ax~HOrO ypaBHe~ ~pe%~H~epa. - HsB.AH CCCP, cep.Ma-
Tern., I966, 30, .% 6, I297-I310.
II. K o c T e H E o H.M. 06 O~HOM onepaTope npeo6paso2a~m~. - ESB.
BMcm.y~.saB., MaTeMaT~Ea, I977, 9, 43-47.
I2. C a x H 0 B ~ ~ ~.A. 0 CBO~C~BaX ~cEpeTHOrO ~ Henpep~B~o~o
cneE~poB pa~Ea~Horo ypaBHeH~a ~pa~a. - ~oEa.AH CCCP, 1969, I85,
I, 61-64.
13. C a x H o B ~ ~ ~.A. 06 O~H0~ no~yo6paTHO~ sa~a~e. - Ycnex~
MaTeM.HayE, I963, I8, • 3, I99--206.
L.A. SAHNOVICH
(~.A.CAXHOBH~)
CCCP, 270000, 0~ecca,
3xeE~poTexK~e cE~ m~CT~TyT
OB~S~ ~.A.C.HonoBa
121
4.3. A QUESTION OP POLYNOMIAL APPROXIMATION ARISING
old IN CONNECTION WITH THE LACUNAE OF THE SPECTRUM OF
HILL' S EQUATION
Let Q=- ~ ~(×) be a oo ~X ~ + Hill's operator with ~ C~ ,
the class of real infinitely differentiable functions of period I.
The spectrum determined by the periodic and anti-periodic solutions
Q~=~, 0-<x < I comprise a simple (periodic) grotmd state ~o follow-
ed by separated pairs ~_~< i~ ,t$-----~,Z,... of alternately anti-perio-
dic and periodic eigenvalues increasing to + oo , the equality or
inequality signifying the dimensionality (=I or 2) of the eigenspace;
see [1], The intervals [~._i,~ , $$=~... are the I a -
c u n a e of the spectrum- "'~-of ~ in ~,~(~) . Hoohstadt [2] proved
that the infinite differentiability of ~ is reflected in the rapid
vanishing of the lengths ~ of the lac~e [~,-4, ~] as ~+c~.
Trubowitz [3] proved that the real analyticity of ~ is equivalent 4
to ~W<~@6 -~, ~ $ ~ . A comparison between 4~ and-- "~~(~)~-Jg%w~"~(x)~X O
springs to mind. Interest in sharpening these results arises in con-
nection with the following geometrical problem.
Let ~ , ~c C 1 , be the class of functions giving rise to a
fixed periodic and anti-periodic spectrum ~o< ~ ~ ~ < ~ ~<~@ <.. •
an~ let ~ , ~ ~ oo , be the number of pairs of simple eigenvalues
~9,tl,-t < ~ . ~ is a compact ~-dimensional ictus identifiable
as the real part of the Jacobi variety of the hyperelliptic curve of
genus ~ , ~ eo , with branch points over the real spectrum, aug-
mented by the point at oo ; see [4] and [5] for $<oo , and [6]
for ~=oo . ~ admits a family of transitive commuting (iso-spectral)
flows expressible in Hamiltonian form as ~ = X~ with X~--(~%~)',
prime signifying -~ , in which I is~.,~a simple eigenvalue of Q
and grad ~ is the functional gradient dA/d~(X)= the square of
the normalized eigenfunotionLI(x)J ~ - -- . The i 0 c a I flows:
4 0S~ ..... (Korte-
weg-de Vries), e t c. are more familiar. The latter belong to the
span of the former, but can be expressed in an independent fashion
E)~=(#'t~(~ ~i,~)'] via the rule
starting from H_ 1 ~ l ; for example,
122
• i ( .(
o o o
THE GEOMETRICAL QUESTION is to decide if the loca I vecto r fields X~ ,
X% • etc. span the tangent space of ~ at eac h point. This is al-
.ys the case if ~<~ ~ see ES] or E~- MoKean-~bowitz E6J make the question precise for ~-co and prove the following necessary
and sufficient condition. Let ~ be the space of sequences ~(~),
Let ~ be the subspace~(~Z~)~(~) , ~,~,~,... , with
a polynomial. T h e n t h e s p a n n i n g o f t h e
local vector fields takes place if
a n d o n I y i f ~ s p a n s ~ . The condition is met if
is real analytic (~@~-@~, ~ ~ + ~).It is known that ~ , ~ +
+Co÷G~'~C~-4~... (~.~), permitting the application of a result of
Keosis KaS in the case of purely simple spectrum to verify that the
spanning takes place in that circumstance only if E ~ $~
C o n t z ~ z ~ w i s e , t h e s p a ~ n g c a n n e t t a k e p 1 a c e i f
vanishes on an interval; in fact, if the local vector fields
span the tangent space, then the associated gradients ~H/~$
span the normal space of ~ , and the two together (tangent and nor-
real) fill up the whole of the ambient space, taken to be ~(0,I) ,
which is impossible since X~ and ~/~ are universal polynomi-
als in ~ , Sf, $[~ , etc. without constant term and consequently va-
nish on the same interval as $ . I t s e e m s I i k • i y
that the spanning becomes critical
in the vicinity of quasi-analyt ic~ .
The same questions arise for ~ on the line with ~ C~ , the class
of infinitely differentiable functions of rapid decay at ± 0o . The
rate of vanishing of the lacunae is replaced by the rate of decay of
t h e r e f l e c t i o n c o e f f i c i e n t 6,1~(,K) o f F a d d e e v F9], e . g . , $~C~ is
reflected in 61~e C ~ , while the analyticity of ~ in a horizontal strip is reflected in I G~(~)I < @~-~1~1 ~ ~ _ ~ + =o;
see DO]. ~e to~. ~ is n~ replaced by ~-~ - dimensio=l cy-
123
linder specified by fixing I~% I and finite number of bound states
(negative simple eigenvalues) -M= (~= ,., ~) and the vector
fields
from~ K~(~I,=~,.., ~) presumably span the tangent space; see ~1] for preliminary !nformatlon. The local vector fields ~ $ ~ ~/,
r .( f f f ~¢$=3~-~$ operate aS before, and the question is the same as
before: d o t h e y s p a n t h e t a n g e n t s p a c •
o f ~ ? The technical clarification of the question is a necessary
part of any d/scussion.
REPERENCES
I. M a g n u s W., W i n k 1 e r. Hill's Equation, New York,
Interscience-Wiley, 1966.
2. H o c h s t a d t H. Function-theoretic properties of the dis-
crlmi-A-t of Hill' s equation. - Nath. Zeit., 1963, 82, 237-242.
3. T r u b o w i t z E. The inverse problem for periodic poten-
tials. - Comm.Pure Appl.Math.~ 1977, 30, 321-337.
4. ~ Y 6 p o B ~ H B.A., H 0 B ~ ~ o B C.H. Hepzo~ec~a~ sa~a~a yps3ne~ KopTeBera-~e *p~3a ~ IITypMa-~yB~. HX CB~S~ c
a~redpaEecEo~ reoMeTpze~. -~oE~.AH CCCP, 1974, 219, 3, 531- - 534.
5. M c K e a n H.P., P. van M o e r b e k e. The spectrum of
Hill's equation. - Invent.Math., 1975, 30, 217-274.
6. M c K e a n H.P., T r u b o w i t z E. Hill's operator and
hyperelliptic function theory in the presence of infinitely many
branch points. - Comm.Pure Appl.Math.~ 1976, 29, 143-226.
7. L a x P. Periodic solutions of the ~ equation. - Comm.Pure
Appl.Math.~ 1975, 28, 141-188.
8. K o o s i s P. Weighted polynomial approximation on arithmetic
progressions of intervals or points. - Acta Nath.~ 1966, 116,
223-277. 9. ~ a ~ ~ e e B ~.~. CBO~CTBa ~--~Tp~UH o~o~epnoro ypaBneH~
mpe~nrepa. -Tp.MaTeM.~--Tn AH CCCP, I964, 73, 314-336.
IO. D e i f t P., T r u b o w i t z E. Inverse scattering on
the line. - Comm.Pure Appl.~th., 1979, 32, N 2, 121-251.
11. M c K e a n H.P. Theta functions, solutions, and singular
curves. (Proc.Conf., Park City, Utah., 1977, 237-254), Lecture
Notes in Pure and Appl.~th., 48, Dekker, New York, 1979.
H.P.MCEEAN New York University. Courant Institute
of Mathematical Sciences, 25~ Mercer Street,
New York, N.Y. 10012, USA
124
4-4 old
ZERO SETS OF OPERATOR FUNCTIONS WITH A POSITIVE
I~iAGINARY PART
Let E be a separable Hilbert space, ~ be a function analytic
in the unit disc ~ , taking values in the space of bounded opera-
tors on ~ and continuous up to the boundary of D . Suppose also
that
where CC~) is a compact operator on
following properties are satisfied:
A) For a modulus of continuity
. we also assume that the
the inequality
~olds with ~', ~' ~D. B) M has a positive imaginary part in D
i f
A point ~ in ~0~D will be called a r 0 0 t 0 f M
11¢1--4
Since I- MC~) is compact, for any root ~ there exists eeE
such that M (~)e = ~ . It is not hard to verify that the
roots of a function with a positive imaginary part can lie only on
T . Denote the set of all roots of M by A and let ~A~ be the Lebesgue measure of its ~-neighbourhood in
CONJECTURE I. Under hypotheses A. B the inequality m ~Cw(~)
holds for a ~qsitive cQnstant C.
It seems to be natural to weaken hypothesis A and replace it
by the following one taking into account the behaviour of ~ only
near the set A .
A,} I M4C~)I "~ .~a(~,cz , A)).
125
CONJECTURE 2. Under hypotheses ~;,B the inequality
hol,d,s,,,for a positive constant 0 •
Let us note that the validity of any of CONJECTURESS or 2 with
~(~) ~C~ ~ would imply that ~ is finite.
The above CONJECTURES agree with known results of operator theo-
ry and complex analysis. Their proof would permit us to describe the
structure of the singular and discrete spectra of perturbed opera-
tors in terms of "relative smoothness" of perturbation. To indicate
links let us point out a situation of perturbation theory where the
questions of such sort arise.
Let H be a Hilbert space, ~o , V be self-adjoint operators
on H , V~ ,
A=A°+V
E = ~ ~H . It follows immediately from the second resolvent
identity that the following relation between the resolvents of the
original and perturbed operators holds
Here 1{~ :(~-ZIf ~ , 1~° :(A °- zI) -~ by
, Ira, z :,0, (1)
o The function ~ defined
has
relatively
tion ~ of - A °
o
Ncz~ = I E + V ~/~ ~,, V ~1~, N: E -" E
a positive imaginary part. The perturbation V will be called
s m o o t h i f f o r some s p e c t r a l r e p r e s e n t a -
an operator with a smooth kernel corresponds to V . The perturbed
operator A in this representation coincides with the so-called
Friedrichs's model (see for example [1]),
126
The problem of investigation of the singular spectrum ~$ and the
discrete spectrum ~ can be reduced (see [1]) to the investigation
of the zero set of the corresponding operator-valued function ~,
since ~s U ~d C ~ according to (I). Just in this w~y in IS] a theorem
is proved which clain~s that if the kernel function ~has a good be-
haviour at infinity and satisfies the smoothness condition i~@LLp& ,
&~/~ then the discrete spectrum is finite and the singular spect-
rum is empty . The crucial point of the proof of this result is the
fact that ~ has better smoothness properties at the points of
in comparison with I~ and so A') is satisfied for m with
Investigation of the one-dimensional Friedrichs's model with
the kernel of class L~ , & < ~/~ (see[2]) shows that in this
case the same phenomenon takes place.
~mom~ [2 ]. ~ ~'(z,~)=~(o~)c{(~), 0<~,~;<4, S,(o)=c~(~=O, ~ hLp ~ , ~ < 4/~ , then the func,tion M
Mc~)= t-~ ~Li;, I ~ o ,
satisfies A') with ~[~) ~-CO ~l" ~ , the singular and discrete speO- 5
tra of the operator ~ o_~.n E~C-~, ~) defined by
- t
are contained in the set ~ of zeros of m and
It is proved also in [2] that the above theorem is precise (in
a s e n s e ) .
The tool of [ 2] is the scalar analogue of CONJECTURE 2 proved
in [3]. As to CONJECTURE I in the scalar case, apparently it can be
considerably strengthened. See in more details 9.6 of the pre-
sent volume.
We do not venture to formulate so fine conjectures in the multi-
dimensional case.
127
RE~ERENCE S
I. • a ~ ~ e e B ~.2. 0 Moaex~ ~r~xca B Teopz_~ BOSr, grgeam~. -
Tpy~ MaTeM.~ra-.Ta AH CCCP ma.B.A.CTez,,aoBa, 1964, 30, 33-75.
2. II a B a O B S.C., H e T p a c C.B. 0 CKHIV/ITpHOM clleETpe
caa6o BO@MyI~eHHoPo onepaTopa y~mo~eama. - ~/azn.saaa. ~ ero npm~.,
1970, 4, ~ 2, 54-61. 3. II a B ~ 0 B ]3.C. TeopeMa e~]~ICTBeHHOCT~I ~ ~yHEI~ C HOJIO~Te--
~ao~ Mm~o~ ~acT~m. - B Ea. : Hpo6aeM~ MaTeM.~SNI4~, 2LI~, 1970,
II8-I24.
L. D. ~ADDEEV CCCP, 191011, 21eHEHI~a~
$OHTaHEa, 27, JIOM~I
B. S. PAVLOV
(B. C. IIABJIOB)
CCCP, 198904, JleHm~a~
HeTp0~Bopes, SESr~qecEH~ ~8.EyJIBTeT
~eam~pa~cEoro yaazepc~TeTa
COMmeNTARY
The following progress is obtained in [ 4] :
THEOREM. Let C(~) ~ ~ ~ ) and let MC~o) b_~e
invertible ~$ some (and then at any) ~oint ~o E~ . Then
c
where
0 A is a constant dependin~ o n & , 0 < & ~< ~ , and on M(0) only.
This theorem implies that CONJECTURES 1 and 2 hold for
I-~(~) E~I and ~d(~) _~dv.
It is possible to prove an analogous proposition for an arbit-
rary modulus of continuity CO . However, the condition C(~)6 [4
128
seems to be essential [5]. The above Theorem allowed to describe
(see [4]) the structure of the singular spectrum of selfadjoint
Friedrichs model that is discussed after the statement of Conjecture 2
RE FERENCE S
4. H a 6 o E o C.H. TeopeM~ e~CTBeHHOCTH ~X~ onepaTop-~
c nO~O2~Te~BHO~ MRHMO~ ~aCTBm H CEHI~J~H~ cneETp B caMOconp~-
~eHHo~ Mo~e2 ~p~p~xca. -~oF~.AH CCCP, I98S (B HeqaTE).
5. H a 6 o E o C.H. Private communication.
129
4.5. POINT SPECTRUM OF PERTURBATIONS
OF UNITARY OPERATORS
Let ~ be a unitary operator with purely singular spectrum and
let an operator K be of trace class.
QUESTION. Can the point spectrum of the perturbed operator ~+~
be uncountable?
If it is not assumed that the spectrum of ~ is singular the ans-
wer is YES. A necessary and sufficient condition for a subset of T
to be the point spectrum of some trace class perturbation of some (ar-
bitrary) unitary operator was given in [1] : such a subset must be a
countable union of Carleson sets (for the definition see, e.g.,
9.3 of this "Collection")~
A version of the reasoning in [I] allows to reduce our QUESTION
to a question of function theory.
PROPOSITION. Let ~ be a subset of T . The followin~ are equi-
valent.
I) ~ is the point spectrum of some trace class perturbation of
a unitary operator with singular spectrum.
2) There exist two distinct inner functions e4 and $~ such that
~4(~)-~)~_~ }
Note that if one of the inner functions is constant then the lat- ter set is countable since it is the point spectrum of a unitary ope-
rator, namely of a rank one perturbation of a restricted shift (cf
[2]).
REFERENCES
1. M a E a p 0 B H.F. YH~TapH~ To~e~H~ cneKTp nO~TZ yH~TapH~X one- pa~opoB.- 3an.Hay.H.Ce~ZH.20MH, I983, I26, I43-I49.
2~ C 1 a r k D. One dimensional perturbations of restricted shifts~
- J.Analyse Math., 1972, 25, 169-191.
N.G.MAKAROV CCCP, 198904, ~eHZ~I~a~,
(H.F.MAKAPOB) ~eTpo~sope~, JIe~HK~o~ roc~-
;~apcT]3e~ yH~Bepc~TeT
130
4.6. RE-EXPANSION OPERATORS AS OBJECTS O~ SPECTRAL ANALYSIS
I. Notations, ~Z~ and L~@ are subspaces of even and odd functions in [.~(~) ; ~ is the Fourier transform; Z is the multiplication by ,.~ on ~ (~) ; K ~ ~*~ ~ is the Hilbert transform; ~_~L~.= ~*~ ~ ~ . Let ~(~) be the following unitary
mapping from onto :
% and ~& are the E?urier cosine and sine transforms on I. ~ (~+); ~ ¢~ % , ~ 2fL ¢ ~. Let ~ denote the multiplication by
6~ = ~(~+i) on ~(Z) • Integrals with singular kernels are un- derstood in the sense of principal value.
2. Re-expansion operators appear quite often in scattering theory Namely, the wave operators for a pair of self-adjoint operators H,,~
- J~. i~± oO
can be obtained as follows. A given function is expanded with respect to the eigen-functions of He and then the inverse transform using the eigen-functio~ of H is taken. Let for example, Hs sad ~@be the operators -i~S on L~(~+) with the domains ~efined by ~(0)=0 and W(0)--0 respectively. Then ~+(H@,~)=+~ , Indeed, let
Then
I g k oCk) k,
%chc ) e
Simple calculations by the stationary phase method show that ~0(~)~ ~(~) when ~--~+_oo provided ~i=~0.
The re-expansion operator ~ arises in the polar decomposition of A=-{~ on ].~(~+) with the boundary condition ~(0)=0 , na- mely, A = ~ I AI and A = M IA I . Let us verlfy the flrst equall- ty. Since [AI=( A'A)'I/Z= H~, ' ---~14) , we have (using the above notation @4
131
hence JR+
Concre t~ r e - e ~ s i o n o p e r a t o r s a r e a p p a r e n t l y i n t e r e s t i n g f rom
the a n a l y t i c a l p o i n t o f v iew and as the o b j e c t s o f s p e c t r a l ~ n a l y s i s .
In this connection (see Secto5) we propose some problems~ But at
first (in Sect.3,4) we use re-expansion operators as specimens for observations.
The author thanks N.K.Nikol'skii and M.Z.Solomjak, whose remarks
are incorporated into the text.
3, Put 9= ~ ~ ~ , in accordance with the decompositi-
on ~(~)= ~ ~ ~ ~ • It is easy to see that @
hence
The operator
g~= ~, M~ ~,
~- $~ 'S (, M w)($)= ~ (i)
l~+ V defined by
V~(~)= ~/~ ~(~) ~(~+) onto [~ (~) , Clearly, V~coincides w i t h
(2)
(3)
n (res-
: ~) The spectral decomposition of M can be also deduced from [I] (see Oh.IX} but the proof given here is more direct amd simple.
maps isometrically
the convolution operator ~-~ ~ ~ , where
I e sl~ ~(s)- ~ ~ .
Taking the Pourier transform, we obtain *)
M=V*~*ECV, where E is the multiplication by the function
~C~) = co~,h, F~ ' o n L~(~) • I t f o l l o w s from (2) and (3) t h a t the spectrum o f
132
pectively of ~ ) is absolutely continuous and fills out the semi-
circle T ~ { ~ E ~ , ;~ez>zO} (respectively T ). These imply that
is unit ar ily e quival e n t to the
s h i f t o p e r a t o r on ~(~) . Note that
I ¢~-@~1:1 I - r l l : ~ and that the equality ~)@~ : ¢5 ~ is impossible f~ r l ~ e L ~ ( ' ~ \ \ { ~ } , though i t holds f o r ~0(b:~ -II~ and i n fac t ¢~0:
: (~$~,o : I,~ o •
4. A re-expansion operator on [.~(~), A:(-~,~O , with analogous properties appears in connection with the system
I' ] , z ,
(but not with the usual trigonometric system). Let d~:L~(A)-,L~(Z) be the ~ourier transform corresponding to this system. Let &+=(O,~)~
~S and ~ be maps of ~ (A÷) onto ~(~÷) corresponding to the systems
Purther put
The sense of the following notations is clear by analogy with Sect.3 The operator K acts as follows
,I I ~(~)~5 (4) A
Changing variables by the formulae
lc r-g' we red uc@. K to the Hilbert transform- R : ~'K ~ . This implies
that ~ -- ~Z can be written in the form ~ 5 ~ ~ . Further,
decompose ~ into two parts~ven and odd): ~ :~ ~ & . Then
~ 5 = ~ s where ~s--~l~ . S~nce ~$- ¢ we obtain a unitary equivalence of ~ and ~ , namely
where 0+-G I~ (~÷) . Note that M describes the non-trivial part
of the scattering matrix for the diffraction on a semi-~mfimite screen (this is shown in [ 2], where a unitary equivalence of M and of the
133
multiplication by function (5) is presented in an explicit form).
Another (and a more elementary) situa$ion where ~ appears is the
following. Let ~, (resp~ ~ ) be -~ ~ on ~ (/~+)~with the boun- dary condition ~(0)=0J~resp W(~)=0 ~ Then Bo= M I ~ , B~: MI~,|, TO writ t ~ in a matrix form we note that the operator "~ ~ ~'~* (= ~ ~ [ ~* ) on ~(~) has the following biline-
ar form T
5. PROBLEMS. • @
I) equality n 5 implies t at H is boun- ded On LF(~+) , 4< p<~. What is the norm of I- ~ .~ (If ~=~ see (4)). It is not excluded that the answer can be extracted from the results of [I], Ch.IX.
2) Multi-~mensional analogues of the operator ~ can be descri- bed in the following way Let ~9 be unimodular functions on ~ satisfying ~(~)=X(~) , 9($~)-- 9(~) for SF0 Let L,~ be
multiplications by these func, t~on@ on ~,~(~). If ~,~ is the Fourier transform on ~£(~m)then 9 ~ ~ ~ N is a unitary opera-
tor. It would be of interest to investi6ate its spectral properties It might be reasonable to impose some additional conditions on ~ and
(e.g. some symmetry conditions). 3) Let ~ be an even positive function on ~ and {p~} be the or-
thogonal family of polynomials in~~ (-~,~; ~) , 0<~<~o. Then an analogue of the operator ~ appears in ~ ~ (0~) . namely,
the re-expansion operator from the even polynomials {p~'~ to the odd
ones {P~÷4~ • It would be of interest to investigate its spectral properties.
4) Consider the following systems in ~£(A~):
The second system has defect I. Let P be the re-expansion operator . . Z A from sines to cosines . This is a semi-unitary operator on ~ (+) with defect indices (1,0). Is it completely nonunitary? In other
words is the orthogonal system { P~: ~)0} complete in ~(A+) ?
5) The operator ~ is connected with the harmonic conjugation.
What does the theory of invariant subspaces of the shift mean in terms_
134
of ~ ? What is the role of zeros and poles of the function (3) in
this connection?
REFERENCES
I. r o x d e 1o r H.II., Ep y nm ~ K H.H. BBe~eKze B TeoIx~ o~-
Ho~epE~x c~H~11~x HHTerI~JA~HHX onelmTOlOOB, l~m~SeB, mTH~HK~, I973.
2. H ]i L B H E.M. XSpSETelD~0TRI~ ImcCesHms ~. salaam o ~m@Im~sB
S8 F~L~He ~ H8 SEI~He. - 88n~CK~ HBy,.CeMMH.~0MH. 1982. 107. 198- 197.
M. S. Birman
(M.m.z~mH) CCCP, 198904, HSTIDO~80108~I, ~sH~eOl~R~ ~sI~JL7~ Te T
JleHSHI~S~ONRR yHHBeI~TeT
135
4.7. MAXIMAL NON-NEGATIVE INVARIANT SUBSPACES OF
~ -DISSIPATIVE OPERATORS
Let ~ be a ~ -space (Krein space) i.e. the Hilbert space
with an ~_Ter product (~) and indefinite ~ -form[~,~l=(~,~),
~= ~*= (more detailed information see, for instance, in [1]
or [2]). ~ subspace ~ is called n o n - n e g a t i v e if
[~, ~] ~0 for ~ ~ , and m a x i m a 1 n o n - n e g a -
t i v e if it is non-negative and has no proper non-negative exten-
sions.
A linear operator ~ on ~ with a domain ~$ is called
d i s s i p a t i v e ( ~- d i s s i p a t i v e ) if ~(~,~)~O
(~[~,~ ~0) for all O~ ~ . Such an ~ is called m a -
ximal dissipative (maximal ~-dis si-
p a t i v e ) if it has no proper dissipative ( ~ -dissipative)
extensions.
PROBLEM 1. Does there exist a maximal non-negative invariant sub-
space for an~ bounded ~-dissipative o~erator ~ with ~ =~ ?
This problem has a positive solution if ~ is a u n i -
f o r m 1 y ~ -dissipative operator i.e. there is a constant
~Q, ~[~,~] ~ ~llO~l~ ~, In that case ~(~) ~=~ and hence
~e Riesz projection generated by the set 6~(~) ~ C* gives us the
desired subspace. Note that ~-dissipativity of ~ implies the
uniform dissipativity of ~E=~*4~ C8~0) . As ~ posseses
a maximal non-negative invariant subspace, it is natural to use the
"passage to the limit" for ~-*0 . Such a passage - M.G.Krein's
method (see [2]) - leads to a positive solution of Problem I if
(I+ ~)~(I- ~)~ Too ' In the general case Problem I has not yet
been solved and therefore subclasses of operators for which it has a
positive solution are being considered and on the other hand attempts
are being made to construct counterexamples.
THEOREM. If = ~ is a ~ -s~ace, = then
= 0 is a continuous ~ -dissipative operator if and only
if ~ i_~s a continuous d issipat%ve operator in ~ ; Ain tha ~ case
has a maxima! ' non-negative (with respect to the ~-form) inVa-u
riant sub space.
136
PROOF. On~ verifies immediately that a maximal non-negative sub-
space ~ of ~ is invariant under ~ iff it is a graph of an ope- •
rator ~ ) &ere ~ is dissipative in % and ~ =~
(~=V{<~,-~)~£%). Such an operator ~ does exist and is
bounded by the theorem of ~tsaev-Palant[3].
~tSa~v-Palant's result about the square root of dissipative
operator was developed by H.Langer [4]. It was proved there in par-
ticular that each maximal dissipative operator posseses a maximal
dissipative square root. This result allows to omit the requirement
of a continuity of ~ in the above Theorem and replace it by the
maximal dissipativity condition,
where ~ is a continu0us a~-dissipative operator in ~ . Does
there exist a maximal (in ~ ) noD-negative subspace invariant
under the ~ -dissipative operator ~ ?
REFERENCES
I. B o g n a r J. Indefinite inner product spaces. - Springer-
Verl~g, 1974,
2. A s ~ s o B T.H., H o x B H~ O B H.C. ~m~egaHe onepaTOpHB
]IpOOTpaHOTBaX C I~f~e~NB~THO~ MeTpHEo~ E I~X IIpN2Io~eHN~. "MaTeMaT~--
~ecEm~ ssaxHs. ToM 17 (HTOr~ Hay~ ~ TexHzFm)", 1979, MOOEBa,
BMH~TM, 105-207.
3. M as a e B B.H., H aa a H T D.A. 0 cTe~em~x orps~mqe~oro ~cc~naT~m~oro oHepaTopa. - YEp.MaTeM.~pHaa, I962, I4, 829--887.
4. L a n g e r H. Dber die Wurzeln eines maximalen dissipativen
Operators. - Acta Math. 1962, XIII, N 3-4, 415-424,
T.Ya.AZIZOV
(T.H. ASMBOB)
I.S.10HVIDOV
(~.C.H0XB~0B)
CCCP, 394693, BopoHex, YH~Bepc~ Te TOKa2 Ha.l,
BopoHemcEz~ rocy~apcTBeHH~
yHl4 BelOC ~ Te T
137
4o 8° PERTURBATION THEORY AND INVARIANT SUBSPACES
If ~ is a given coefficient Hilbert space, let ~(~) be the
Hilbert space [I] of square summable power series ~(~)~ ~6E~
with coefficients in
2-:.
If B (~) is a power series whose coefficients are operators on ~ and
which represents a function which is bounded by one in the unit disk,
then multiplication by ~(~) is contractive in ~(~) , Consider the
range ~(B) of multiplication by B(~) in ~(~) in the unique norm
such that multiplication by B(~) is a partial isometry of C (~) on-
to 11~(B) o Define ~(B) to be the complementary space to ~(~) in
~(~) . Then the difference-quotient transformation ~(~) into
K~(~)-~(0)]/~ in ~(B) is a canonical model of contractive
transformations in Hilbert space which has been characterized [ 2 S as
a conjugate isometric node with transfer function B(~).
If ~ (E) is a power series whose coefficients are operators on
~and which represents a function with positive real part in the unit
disk, then
is a power series which represents a function which is bounded by one
in the unit disk, Define ~(~ to be the unique Hilbert space of po-
wer series with coefficients in C such that multiplication by
+ B(~) is an isometry of ~(~) onto ~(6) ° Then the differen-
ce-quotient transformation has an isometric adjoint in~(~).
The overlapping space ~ of ~(B) is the set of elements ~(~)
of C (~) such that B(E) ~ (~) belongs to ~(B) in the norm
The overlapping space ~ is isometrically equal to a space ~(0).
A fundamental theorem of perturbation theory [3B states that a
partially isometric transformation exists of ~ (~) into ~ (~) which
commutes with the difference-quotient transformation., The transforma-
tion is a computation of the wave-limit. The wave-limit is isometric
on the square summable elements of ~ (~) and annihilates the or-
thogonal complement of the square sum~able elements of ~(~) , if T
denotes the adjoint in ~(~) of multiplication by B(~) as a trans-
138
formation in C (~) , then the wave limit agrees with ~ ~1-on square
s~unmable elements of ~(~) . A fundamental problem is to determine
the range of the wave-limit in ~(~) . It is known [4] that the ran-
ge can be a proper subspace of ~(~) • The orthogonal complement of
the range of the wave-limit in ~(~) is the overlapping space of a
space ~(C) such that ~(Z) = ~(~)C (E) for a space ~ (~) which
is contained isometrically in ~(~).
CONJECTURE. The range of the wave-limit contains ever 7 element
of ~(@) if the self-a~]oint part of the operator ~(0) is of Ma-
tsaev class.
The llatsaev class seems a reasonable candidate because the existen-
ce of invariant subspaces is known for contractive transformations~-
such that |-T~T is of~atsaev class. Invariant subspaces exist
which cleave the spectrum of the transformation° An integral represen-
tation of the transformation exists in terms of invariant subspaces
[5]. For reasons of quasi-analyticity, such results do not hold for
any larger class of completely continuous operators,
Some recent improvements in the spectral theory of nonunitary
transformations limk the Nmtsaev class to the theory of overlapping
spaoes [6].
REFERENCES
I~ D e B r a n g e s L. Square Summable Power Series, Addison-
Wesley, to appear.
2. D e B r a n g e s L. The model theory for contractive transfor-
mations. - In: Proceedings of the Symposium on the Mathematical
Theory of Networks and Systems in Beersheva, Springer Verlag, to
appear.
3. D e B r a n g e s L., S h u 1 m a n L. Perturbations of uni-
tary transformations. - J.Math.Anal.Appl., 1968, 23, 294-326-
4. D e B r a n g e s L. Perturbation theory, - J~Math Anal.Appl.,
1977, 57, 393-415.
5. r o x 6 e p r H.E., K p e ~ H M.r. Teol~s BO~LTepI~O~X OHe~-
TOI~OB B rM~6elYrOBOM II~OCT~HCTB8 M ee ~ o z e ~ s , M., HSyFm, I967 (Translations of Mathematical Monographs, 24, Amer
Math.Soc., 1970).
6. D • B r a n g • s L. The expansion theorem for Hilbert spaces
of analytic functions, Proceedings of the Workshop on Operator
139
Theory in Rehovot, Birkhauser Verlag, to appear~
L. DE BRANGES Department of ~athematics
Purdue University
West Lafayette, Indiana 47907
USA
140
~-9. old
OPERATORS AND APPROXI~ATION
1. ~nat is a"Blaschke product"? As long as we are concerned
with scalar-valued analytic functions in the unit disc ~ , the an-
swer is well-known: this is a function B satisfying one of the
following equivalent statements. ~k
(i) ~ can be represented as a product ~ = ~ ~(A) of
= ~- ~-~E (here ~ is a functi- elementary factors
on from D to nonnegative integers with ~ ~(~)(I-I~l) < +co)
(ii) ~ is inner (in Beurling's sense) and the part of t~
schift operator ~* on the invariant subspace ~ , ~ = KB a~
~8~ i has a complete (in ~ ) family of root subspaces, Here
~ is the standard Hardy class and • , ~ are operators
(iii) The same for t ra
to ~I~ ( PK stands for the orthogonal projection onto ~ ).
(iv) ~ is inner and
~{4 T The spectral interpretation of (i) and (iv) is of importance for
studying operators in terms of their characteristic functions and the
problems discussed in this section are essentially those of a "cor-
rect" choice of the notion of Blaschke product in the general case,
when operator-valued inner functions are considered (the equality
I~(~)I =~ a.e. is replaced in this case by the requirement that
~(~) be a unitary operator on an auxiliary coefficient space
; ~ is replaced by ~(~) , and so on). Statements (i)-
-(iv), appropriately modified, are still equivalent for operators
~B having a determinant (i.e. when I- ~ B is nuclear). If
this is no longer true, (ii) AND (OR) (iii) prove to be the most
natural definitions of a Blaschke product.
QUESTION 1 !s it true in case of an arbitrar~ operator valued in-
ner function ,that one of the Conditions ,,(ii,~, (iii~ implies the other
one?
The definition under consideration schould presume a metric
criterion for a characteristic function ~ to belong to the class
141
of "Blaschke products" (that is a criterion for T B AND (OR) --~'I'~
to be complete).
QUESTION 2. Do the followin~ conditions ~ive such criteria:
If we restrict ourselves to the case when TB= ~ t K has a sim-
ple spectrum, Question 2 reduces to the following one.
QUESTION 3. How to describe in terms of B the subspaces ~enera-
ted b 2 ,,,ellen-functions of ~ , i . e . the subspac,,es
Here { A~ : ~ ~ } is a family of orthogonal projections on
It seems important to know when the space (V) coincides with H~(F)
i.e.
QUESTION 4. For which families ~AA - ~ ~ } the conditions
SeH~(E) and A~(~)= @, ~D imply ~ 0 ?
If a ~ or i , Question 4 clearly reduces to the scalar uni-
queness theorem ~ (4-I~I)~ ~ . The last condition remains necessa- AyO
ry in general case. Perhaps the answer to Question 4 is the following:
~(~-I~l) IIA~ ~----~ for ~ belonging to a complete family in
As to question 3, in case ~ E ~ ~ the answer can be expressed in
terms of the so called "pseudocontinuation" of functions in (V)
(M.~. DJrbashyan, G.C. Tumarkin, R. Douglas, H. Shapiro, A. Shields and
others). Possibly the same language fits for ~ E > ~.
2. Weak ~enerators of the a!~ebra ~(T@) .In this section ~ is a
scalar inner function and ~(* ) is the weakly closed algebra of opera-
tors generated by the operators, and I .It is known(D.Sarason) that
A~(T@) iff A---~(T@) for some ~ in H@~ (The operator ~(T@) acts in
~@by the rule ~(TG)~--]pK~I ' I~). The description of weak generators
142
of ~(~) = ~ is also known (D. Sarason) and can be expressed
in a geometrical language, in terms of properties of (necessarily
univalent) image ~(~) . Since the algebras ~ (T e ) and ~ H ~ are isometrically isomorphic, it is plausible that the Sarason theo-
rem should admit "projecting":
QUESTION 5. Is it true that ~(~(Te) ) = (le) if and only if ~ + e H~ contains a generator of algebra ~?
QUESTION 6. Which operators ~(~) have simple spectrum?
(I .e . f,o,r which ~ there exists { in K~ with span (PK~{:~)O) =
=Ke?) If ~ is a generator of Hoo then cyclic vectors ~ from
Question 6 do exist and can be easily described. In the particular
~+~ case ~ =~X~ ~_~ Question6 reduces (for some func-
tions ~ at least) to the question whether ~ITO) ~ is unicel lu- la r (or the same question about the operator ~ I'~(5)K(~-$)~5
v~
on ~(0,~) , G.E.Kisilevskii). Related to this matter are a paper
of J.Ginsberg and D.Newman (J.Aprox.T., 1970, 24, N 4) and the prob-
lem 7~19 of this Collection. Other references, historical comments
and more discussion can be found in two papers of N.K.Nikolskii (in
books: ETOI~ HayEH, ~aTeMaT~qecE~ aH~3, T.12, 1974; Teop~ one- paTopoB B ~yH~mOHaX~HHX npooTpaHCTBaX, HOBOO~6~poE, I977).
N.K.NIK0~SKII CCCP, IPIOII, ~eHm~rpa~
~OHTaHEa 27, ~0~
C O~NTARY
B.H.Solomyak has answered QUESTION 5 in the negative (oral ¢ommu-
D/cation). Por the sake of convenience we replace here the unit disc
by the upper half-plane ~----{ ~:I~ ~ 0 } and consider correspon-
ding spaces ~, Hn Let %--~£~ ~--, £~ , , ~>~.
co
Clearly ~ n and it is proved in [I] that ~(~(Te))~(T 0)
143
(This fact is just equivalent to the unicellularity of ~ (~f)(x)~
S(~) %~ in ~(~ I ) ). On the other hand, for any ~n '
~+~(~/~)~+ ~I ; ~(~e~)~n , ~>0 Since
(~/~)~ ~ g is nonunivalent inll~> ~ } and 6 ~ tends to zero
r a p i d l y a s I1,~,~----~+oo , i t can be e a s i l y v e r i f i e d t h a t ~ i s a l s o
nonunivalent. Thus ~ cannot be an ~N -generator. • A n o t h e r c o u n t e r e ~ m p l e f o r ~ - - - ~ , a n i n t e r p o l a t i o n B l a s c l ~ e
product, was constructed by N.G.Makarov.
REPERENCE
I. Frankfurt R., Rovnyak J. Finite convolution operators~
Anal.Appl., 1975, 49, 347-374.
- J.Math.
144
4.10. old
SPECTRAL DECOMPOSITIONS AND THE CARI~SON CONDITION
Completely nonunitary contractions can be included into the
framework of the Sz~kefalvi-Nagy-Foia9 model [I]. Especially simple
is the case when ~ ~T) does not cover ~he unit closed disc 0~ D
and ~ , = & , < C ~ where c{,=dAm,(l-T~T)H, d,, = cl, b ~ , ( l - l T * ) H are the defect nt~nbers of T tions are satisfied then T i . e t o t h e o p e r a t o r
P~Ik
• If S -~4~I,(J)=U and the above condi-
is umitarily equivalent to its model,
K = H~(De e H~(E),
where E is an auxiliary Hilbert space with ~ E=~ , ~ is a boun-
ded a~lytic (E-~E) - operator valued function in O whose boun~-
ry values are unitary almost everywhere on the unit circle T, H2(E) is the Hardy space of E-valued functions, ~ is the multiplication
operator ~ ~-~ , p is the orthogonal projection onto K •
is called the characteristic function of T . it is connected very
closely with the resolvent of T e.g. I ~(X,T)I × (4-IXl) -~x ' l e(X)-~i, X ~ O . In terms of e the operator T can be in- vestigated in details, namely, it is possible to find its spectrt~m,
point spectrum ~p(T) , eigenvectors and root vectors, to calcu-
late the angles between maximal spectral subspases etc. (cf.[1-~).
In particular operator T is complete (i.e. the linear hull ef its
eigenvectors and root vectors is everywhere dense) iff d65 ~ is a
Blaschke product.
A more detailed spectral analysis should include, however, not
only a description of spectral subspaces but also methods of recover-
ing T from its restrictions to spectral subspaces. The strongest
method of recovering yields the unconditionally convergent spectral
decomposition generated by a given decomposition of the spectrum. For
a complete operator T the question is whether its root subspaces
{K A :~ ~ gp (T)} form an unconditional basis. In the case of a
simple point spectrum necessary and sufficient conditions of such
"spectrality" (i.e. in the case under consideration for the opera-
tor to be similar to a normal one) were found in [2],[3]. These con-
ditions are as follows: the v e c % o r i a 1 C a r 1 • s o n
condition
" - e X -t >~ 'Ae :dp<T~>O (I)
145
holds and the following
valid:
imbedding t h e o r e m s are
T_., (HXl)IIA x J~CA)~E<oo , ~ C~-l~i)lt~,~CX)ltE<~, V~{~H(E). (2)
Here ~ is the orthoprojection from E onto the subspace KPJ~,~C~) and A x is the orthoprojection from E onto K ~ e ( X ~ ; e=ex '
X E&X~ x + (I-~)] is the factorization of 0 corresponding to
the eigenspace KX= K~(TIXl) , ~ IXI~ (X-g)(~-~)-i
From geometrica& point of view condition (I) means nothing else
as the so called uniform minimality of the family {Kk: ~ c dpJ,
Moreover,
1 4 exCx) Axl E =sN, C Kx, Kx),
where
cf.[2]. In the case 6 = ~,= ~ L.Carleson proved that (I) implies
(2) (cf.[4])~in the case ~=~,=oo this is no longer true ([3]).
PROBI~M 1. Prove or disprove the implication (I) ;" (2)
in the case, 'I <~ = 4 . < o o The case ~=~.= ~ seems to be an exceptional one, because for
an arbitrary family of subspaces the property to be uniformly mini-
mal is very far (in the general case) from the property to form an
unconditional basis. However for ~ = ~,=~ these conditions coin-
cide not only for eigenspaces but for root subspaces as well and,
moreover, for arbitrary families of spectral subspaces of a contrac-
tion T [5]. The proofs of this equivalence we are aware of (cf,[5],
[4] ) represent some kinds of analytical tricks and depend on the eva-
luation of the angles between pairs of "complementary" spectral sub-
spaces K~ and K ~ ~ K~, corresponding to a divisor ~ of ~ .
Here ~ and ~/ are left divisors of ~ corresponding to a given
~/= e . A divisor pair of subspaces ; if ~ = 61,, = ~ then
is called spectral divisor if ~ is a spectral subspace. So the main
part of the above mentioned trick consists in the following implica-
tion K5S ; let ~E=~ a~d let ~ ~} be an arbitrary family of speot-
146
ral divisors of @ , then the condition
t7 6~E ~D 11611 =~
implies the following one
(3)
' e
d 6~E ~KD
where ~d i s the i nne r f u n c t i o n corresponding to the subspace
proof uses a lower estimate of I le(:)el l E depending on ~g(~)e~ E and II g ' (Oe l l E on ly . However such an es t imate is imposs ib le f o r
E>~ (L.E. Isaev, private communication).
PROBLEM 2. Let ~<~=~,<oo. Prove or disprove the implication
(3) ~ (4) for an arbitrary family {~} of spectral divisors of ~ .
REFERENCES
I. S z ~ k e f a I v i - N a g y B., F o i a @ C. Harmonic ana-
lysis of operators on Hilbert space, North Holland/Akad~miai Kiad~
(Amst erdam/Budapest, 1970 ).
2. H ~ E o a ~ c E n ~ H.K., H a B a O B B.C. Basncw ns COOC~--
BeHHHX BeETOpOB BHO~He HeyH~TapHax C~aT~ ~ xapaETep~cT~ecEa~
~yam/m:. -HsB.AH CCCP, cep.MaTeM., 1970, 34, ~ I, 90--I33.
3. H ~ E O a ~ C E E ~ H.K., H a B a o B B.C. PasaomeHm: no CO6--
CTBeHHRM BeETOpaM HeyHHTS~HHX onepaTopoB ~ xapaETepEcTH~eCE&K
~yHEKm:. --8an.Hay~H.CeM~H.ZONE4, I968, II, I50-203.
4. H ~ E o ~ ~ c E ~ 2 H.K. ~eEKnE o6 onepaTope C~B~ra. H. - 8an.
H~.ceMHH.~0M~, I974, 47, 90-II9.
5. B a c m H Z H B.E. BesycaoBRO cxo~eca cneETpaaRH~e pasaoae-
H~a ~ sa~a~a ~HTepnoamm~. - Tpy~N MaTeM.EH--Ta ~M.B.A.CTeEaoBa
AH CCCP, 1978, 130, 5-49.
N. K. NIKOL ' SKII
(R.K. K0m CK ) B. S. PAVLOV
(B.C. EAB~0B)
V. I. VASYUNIN
(B.M.BACDHMH)
CCCP, 191011, ~eHzHzpaA
@oHTaREa 27, ~0~
CCCP, 198904, HeTpo~Bope~, *~s~eCF~
~Ey~TeT ~eH~HrpaAcEoro yHSBepO~TeTa
CCCP, I9IOII, ZeH~Hrpa~
@OHTa~Ea 27, JlOMI4
4.11 •
old
147
SIMILARITY PROBLE~ AND THE STRUCTURE
OF THE SINGULAR SPECTRUM 0P NON-DISSIPATIVE OPERATORS
The similarity problems under cosideration are to find necessary
and sufficient conditions for a given operator on Hilbert space to be
similar to a selfadjoint (or dissipative) operator. ~or the first pro-
Blem an answer was found in terms of the integral growth of the resol-
.ents [3] (see also [2]): PROPOSITION 1. An opera.tqr
tor if and only if
s~p~, (I II(L-K ~>0 R
~. is similar to a selfad~oint opera-
The second problem is not yet solved. Here we discuss an approach
based on the notion of the characteristic function of an operator KIBo
Fora dissipative operator - - - - - ' - - - " L ( I n ~ L = ( 2 ~ ' ( L - L * ) ~ @ )
there is a criterion of similarity to a selfadjoint operator (due to
B.Sz.-Nagy and C.Foia~) in terms of its characteristic function ~ ,
namely:
s=plS -~(~)I <+°° I~,X>O
The main tool in the proof of this result was the Sz.-Nagy - Poia~
ftuactional model which)yields, a complete spectral description of a
dissipative operator . For a non-dissipative operator L an ana-
logous condition on its characteristic function
(sup IB(~I< +~, ~x~e-'(x)l< +~) is sufficient for L to be I~>O
similar to a selfadjoint operator (L.A.Sahnovich), but not necessary.
It is possible to give counterexamples on finite-dimensional spaces
which show that operators whose characteristic function does not sa-
tisfy the above condition can be similar to selfadjoint operators (cf
[4], where related problems of similarity to a unitary operator and
to a contraction are discussed).
To be more precise, consider the characteristic function
s~i [ + ~t(ivl)'/~(A-~lvi-A)-'(ivl)'~ s(X)E-,-E, I~A>O, i i , u ,
~) Nevertheless &his result can be obtained without using the functional model [3, s]
148
oz ~ a ~ l i a r y dissipative operator A + ~ i vl , where A ~ ~e L , V = ImL, E =c~s Ra~V ~d let V=J-IVl, J= 3 ~ V be the
polar decomposition of V . The latter operator for the sake of sim-
plicity is assumed to be bounded. The characteristic function ~ of
and the function S are connected by a triangular factorization
~)(x) ~=~ I • ~ ;(ivl)~ (2-Z) -' (lvl)W= (X_+ %+S(x))(X++ X_S (x))-' ---- - S()~)X+I ~ J with where Z+_ (I-+~T)/~- , [ 6 ] . Note t ha t IX+ _
I~ > o. Under the additional condition
the above condition of bounded invertibility of ~ is necessary for
L to be similar to a selfadjoint operator and the condition
@~ I ~(~)I < +oo is necessary and sufficient for the similarity
to a aissipative operator [ 7 ] . In this case corresponding selfadjoint
and dissipative operators can be constructed explicitly in terms of
the Sa- Nagy - Foia~ model for A ÷ ~IVI.
In general case (beyond (I)) serious obstacles appear. The reason
is that it is difficult to obtain a complete description of the spect-
ral component of L corresponding to the singular real spectrum The
solution of the LATTER PROBLEM would be of independent interest.
Let us dwell upon this question. The operator L is supposed
to act on the model space K which can be defined as follows~ Let
be the Hilbert space of pairs (~o' ~) of E-valued functions
on tR square sure.able with respect the matrix weight (~ ~),
S (K) ~ S (K + ~0) being the boun~ry values (in the strong
topology) of the analytic operator-valued function S . Then
K = e , ,
where (F) are the Hardy classes of E-valued functio in
the upper and lower half-planese
The absolutely continuous subspace N e in the model representa-
tion of L has the following form [6 ]
Ne = N#,L)- oBs PK (,~ e (,o&~s(X_ U(E), ~÷ C(E)))), (2)
149
P~ I / U( E),-- where i s the orthogonal pro ject ion from onto I%, L: k L , , , , . I . . The s i ~ l a r subspace ,,~ is defined by , -e ,
where Ne ~ Ne (.L* L It is natural to distinguish in" N~ ~ o subspaoes N L and a~ , the f i r s t one corresponding to the point spectrum in the upper half-plane and a part of the real singular con-
tinuous spectrum, the second one to the point spectrum in the lower
half-plane and another (in general) part of the real singular continu-
ous spectrum:
+ ( H i N~(L) a~ N~ (E) e (%_+ S*%+) HI(E),0),
NT~(L) ~{ N[ ~fc~s PK(O,H+(E)e(X++£~_) H + ( ' ' E)). (3)
The subspace ~ i (N~) is analogous to the subspace correspon- din& to the singular spectrum of a dissipative (adjoint of a dissipa-
tive) operator~ Nevertheless if (I) fails, N~ does not necessarily
coincide with 0~$ { N~ + N~ } , In particular the eigen-vectors
and root-vectors of the real isolated spectrum do not belong to
Therefore in general it is necessary to introduce a "complements- 0 (N~c N~) which would permit us to ry spectral component" ~
take into account the real spectrum of k Put
where N~* =~ N +- (L*) •
I - 1 PROBLEM 1. When ~ OBS N~ "+ ~]~, + N~ 0 ---- ? To estimate
~es between NC and N~± ~ N e in terms of the cha.ra.c.teristic g
function @ . To dive aniiiei~plioit description of ~ (similar
+- Me to that of N~ ~ ), ~.£r e~.~.mole as the closure of the projection
onto K some linear manifold in ~ described in termS.of characteris-
tic function.
~OBLEM 2, Find the factor of ~ correspondin~ to ~ . Investi-
gate ..its further factorizati.on. Describe properties of this..., factor .an.d @
..co.nnection of its roots wit. h t.he .spectrum..of m l N~ o How to sepa-
150
of i
O
PROBLEM 3~ Make. clear the spectral structure of N~ , i.e, in
termsof the model ~.pace construct spectral projection onto intervals
of the real singular, spectrum and onto the root-space correspondin~
,to the real point spegtrum.
PROBLE~ 4. Le.t L be similar to a dissipative operator~ Then @
does, e,q,ual to [ N? + + oeo coin-
cide with the..sub.space of K correspondir~ to the. s , i n ~ u l a r continu-
ous .~.pectrum plus point spectrum of the se..lfadjoint par.t of this di..s-
sipative operator?
PROBLEm5. Co=sider concrete examples (Priedrichs mode ~ wit.h
rank one perturbation, SchrBdim~er operator on ~ with a ~owerllke
decreasin ~ potential~ and desqribe No . . . . . . . . . . . . . . . ~ f o r , , , s u c h o p e r a t o r s @
Besides the real discrete spectrum the space N~ apparently can
contain one more spectral component. The elements+of N~ no longer
have the "smoothness" properties as those of N~- (namely for a
dense set of vectors Ue N~ we have (lvl)~(L-~lY'~e H~(E)~ Perhaps, the structure of NI is similar to that of the singular
continuous component of a selfadjoint operator (certainly, one should
take into account the "non-orthogonalily" caused by the non-self-ad-
jointness of L ). It is also important for the similarity problem
to know which factor of the characteristic function ~ corresponds
N ° to ~ . Note also that all difficulties of the problem appear alre-
ady in the case when the imaginary part of V is of finite rank
( ~ E ~). Let us present here one more assertion closely related to the prob-
lem discussed above and especially to the spectral decomposition of L.
PROPOSITION 2o An.operator L on a Hilbert space ~ is similar
to a. dissipative operator L~s~ if and onl~ ~f there exists an
operator M wit.h
~--.- 0
More, over, if such L~i,s s exi,sts then this M can be, chosen satis-
fyi~, ,,the additional inequality "~,a~,k M ~ ~,. ,k ! m Lass.
151
REFERENCES
1. S z . - N a g y B., F o i a ~ C. Harmonic analysis of opera- . t
tors on Hilbert space~ North Holland - Akaddmiai K~ado, Amsterdam
- Budapest, 1970.
2. S z . - N a g y B. On uniformly bounded linear transformations
in Hilbert space.- Acta Sci.Math., 1947, 11, 152-157.
3, H a d o x o C.H. O0 yo,~oBz.qx no~o~mz ca~oconpmte~ ~ ym~Tapm~ ouepaTo~u.-@/~u.a~a~. R ero np~. (B neqaTz).
4o D a v i s Ch., P o i a ~ C. Operators with bounded characte-
ristic functions and their ~-unitary dilation. - Acta Sci,Math
1971, N I-2, 127-139.
5. v a n C a s t e r n J. A problem of Sz.-Nagy. - Acta Sci.
Math., 1980, 42, N I-2, 189-194.
6. H a d 0 E o C.H. A6com~xo Eenpep~ cnezTp Ee~zccznaa~sEoro onel0a1'opa z ~ym~zoma.~Re~ Ho~e.~. If. - :~an.Ha!r~.cemm.~O~, 19T?, V3, 118-135.
7. H a d o x o C.H. 0 c~ryxap~ou onexTpe HecaMooonpe~Imoro onepa- Topa.- 3au.m~m.ce~mn.~0W~, 1981, I13, 149-IV?.
S. N. NABOKO
(C.H.HkBOK0)
CCCP, I98904, ~mrmpa~,
NeTpo~Bopeu, ks=qeo~i @axy~Te~ ~elmH~Ko~ yn~epc,TeTa
152
4.12. A PROBT.~ ON OPERATOR VALUED BOUNDED ANALYTIC FUNCTIONS old
Let ~ , ~* be t w o Hi lbe r t spaces and J~ (~,, ~)*) the space of all bounded linear operators mapping ~ into ~* . The following
was proved in ~].
THEOREM. Suppose O
anal,7%ic inthe unit disc
l.en%:
(a) there exists a bounded
anal,ytic in ~ and satisfyin~
is a bounded ~(~, ~*) -valued function
D . The followin~ assertions are equiva-
~J(~)~ ~)) -valued function
(b) the Kernel function ~8 :
(1)
is positive definite, i~e.
K=i (2)
fo r ~ y finite systems {~4...A.}, {~... ~. }
Condition ( 1 ) obviously implies that
, where ~ e
cIAl< ). (3}
The QUESTION is whether (3) implies ~2} with She same 8 or at
least with some, possibly different, positive constant.
In the special case when dim~ and dim~*< co the equiva-
lence of (1) and (3), and thus the equivalence of (2) and (3),follows
from the Corona Theorem of L.Oarleson, cf.[2]. A proof of the equiva-
lence of (2) and (3) in the general case, and possibly with operator
153
theoretic arguments, would be an important achievement.
REFERENCES
1. S z . - N a g y B., F o ia ~ C. On contractions similar to
isometries and Toeplitz operators. - Ann.Acad.Scient.Pennicae,
Ser.A.I. Nathematica 1976, 2, 553-564.
2. A r v e s o n W. Interpolation problems in nest algebras. -
J.l~anc.Amal., 1975, 20, 208-233.
tl
B. SZOKEFALVI-HAGY Bolyai Inst. of Math.
6720 Szeged Aradi V~rtan~k tere I
Hungary
CO~ENTARY
This interesting question has been considered in several publica-
tioms, but the answers are only partial. Using some refinements of
T.Wolff's corona argument, V.A.Tolokonnikov [3~ (see also [4], p. IO1)
and M.Rosenblum E7] proved (independently) that (3)--7 (I) if
~ ( ~ e ~ ~) ~ ~ . Moreover, Tolokonnikov obtained an esti-
mate of the solution O . This estimate (in somewhat simplified
form) looks as follows:
T)
where
For small values of 8 a better estimate is due to Uchiyama E6~
154
V. I. Vasyunin has shown that
V.l.Vasyum.in [5] has proved that (3) ~ (I) if ~<eo@ (with
the estimate C~(6)~< ~ O~ (6 %) ). In ~5] it is also shown
that 01(~) ~ 4 that C~(%)>i ~ 5 -~*4 ( ~ ~,S, )
and that if the implication (3) > (I) were true for ~, ~
-----~%W% ~----- ~ , then the estimate of ~ wouldn't be better than
6~c~(~6 -~/8 ) (i.e. that ~oo (~)~ ~ ¢*~ (@ 6-~/~ ) ). Tolokon-
nikov has noted also that if (3)~> (I) were always true thence(6)
would be finite for every 6 ~ ( 0~ ) (~npublished).
If ~ C ~* and ~ ~ < + ~ then assertions (a)
and (b) in the Problem are equivalent to the possibility "to enlarge"
to a SQUARE matrix ~ analytic in D and satisfying
~=~I ~ , ~ II~(~)II ~ ~ , A~ D II(~C~))-~II<~ (see
[5]). There exists a connection between the corona theorem (for~=~)
and the left invertibility of the vector Toeplitz operator ~@~ ( ~I],
see also [8]).
REPERENCES
3..T o ~ o K o H H H ~ O B B.A. 0~eHEH B TeopeMe Eap~ecoRa o KOpOHe
H KOHe~4onopo~eHHMe ~ear~ aare6pu H ~ . - ~yHK~.aHa~. H ero
npH~., 1980, 14, ~ 4, 85-86.
4. H H E 0 a ~ C K H ~ H.K. ~e~MH o6 onepaTope CABHra. M., HayKa,
1980.
5. T o a o K o H H H K O B B.A. 0~eHKM B TeopeMe KapaecoHa o ~OpOHe.
H~eam~ aare6p~ H ~ , saAaya C~Ke~a~bBH-Ha~2. - BS/I.HayYH.CeMHH.
~0~4, 1981, I13, 178--198.
6. U c h i y a m a A. Corona theorems for countably many functions
and estimates for their solutions. Preprint, 1981, University of
California at Los Angeles.
7. R o s e n b i u m M. A corona theorem for countably many functions.
-Integral equat, and operator theory, 1980, 3, N I, 125-137.
8. S c h u b e r t C.P. The corona theorem as an operator theorem.
- Proc.Amer.Math.Soc., 1978, 69, N I, 73-76.
4.13.
Let
the characteristic function
acting on the space
155
ON EXISTENCE OF INVARIANT SUBSPACES OF Cm-CONTRACTIONS
T be a compeletely nonunitary Cl0-contraction ~) with
B ~ H~(E E,) so T can be s~pposed
Ke = H~CE,) e @ H'(E)
as follows
where Z is multiplication by
and W8 is the orthogonal projection from
Z (the shift operator) on H~(E~) H~(E,)onto K 0 , i.e.
where ~+ is the Riesz projection from
operator acts as follows
H2 onto . The adjoint
T * f - _ { - f ( o ) , ' ;z eKe (1)
Recall that T~ C,Q iff ~ is inner and ~-outer.
Any C H -contraction is quasi-similar to a unitary operator
and this allows us to prove that the lattice of all T-invariant
subspaces (Lat T ) is non-trivial (see [I]). In our case we have
only a quasi-affine transform intertwining T and the ~-residual
part of its unitary dilation, i.e. multiplication by ~ on
c~sAsL2(E~),where A.=(I-00") Vz. V~iat can we obtain from this for finding a non-trivial invariant subspace?
We can suppose that Ker ~(~)s{p} and Ker ~(~)~-----{~) for every
~I < ~ , because otherwise T or T has an eigenvector. Hence
~_(E~) ~ Ke~ ~*--- { ~} and therefore we have to investigate only
the case ~b~$E=d~wsE, --o° . Indeed, if ~L~E<oo then there exists
an antianalytic solution ~, of the equation ~*~,= ~ , but the
fact that ~ is inner implies that ~#$ ~ ~ ~L~ E
Note that Ke~ ~ ~,La(E,) I~(E~) and
~) A l l used terminology can be found in [ I ] or [2] .
156
P.&,L'(E~)cK,. Her~ P+ intertwines Tandthe ~-resl-
dual part of its unitary dilation, namely IA.< (E~). @* Let SC now be an arbitrary vector in K8 and &~ ~e%
Then
{ T ~ , P ÷ k . ) = ( z " ~ , k ~ ) = ~ < ~ , k w > E g " ~ , T
where ~ is the normalized Lebesgue measure on the c i rc le T . If
then the multiplicity of T is greater than I, i.e. there exists no
cyclic vector. Indeed~for every ~ ~0 we can choose a vector
k~eKe~O* such that <~(~), k~(~)>E =0 a.e. So every
nonzero vector 00 generates a non-trivial invariant subspace.
Further we shall suppose rank A,(~)~4 a.e. Putting ~=
= { ~ T ~i~k ~(~) : ~ ~ , choose a vector ~,~e%~*
such that Uk~(~)~ :~ for a.e. 5e& and k~(~):O fora.e E, ~T\~ . Now we have
Xe~ 8"= A~L'(E.) : { 9k~" 9 eL' }
and
T That is p . ~ k . is cyclic for T * i f f there exists a nonzero vec-
tor ~e K 8 such that < Z ~ . > i s a cyclic function for the multi-
plication by Z on U(~), i . e .
Choosing 0G of special type we can obtain various sufficient
conditions for the existence of a non-trivial invariant subspace.
PROPOSITION. Each of the followin~ conditions implies the exis-
tence o~ a non-trivial invariant subsoace of T •
1) B~eL' such t~t ~ {~e~" < (P.,k.)(~),k~c~)>ET0}*0;
=0}4:0
157
2) ~ L , ~" such that ~,~l<P+~h,,,k,.>l ~L.~; 3) ~[,, ,C~ , I'1¢(~,)~'0 and ~p~.,{h,.G)~,e~,.}÷E.; 4) th,e antianal,ytic function P_ (11P+k~ll~,) admits ,# ps,,eudo-
continuation to the unit disc.
CONJECTURE. For every inner ~-outer function ~ there exists
T , a nonzero npnc,yclic vector for (defined by (I)) of the form
P+ h,~ ,,here k, e Ke'~ g~. If the CONJECTURE is not true a counter example must have a num-
ber of very pathological properties and may be a candidate for an
operator without invariant subspace at all.
REFERENCES
1. Sz.-N a g y B., F o i a ~ C. Harmonic analysis of opera-
tors on Hilbert space, North Holland/Akad~miai Kiad~, Amsterdam
Budapest, 1970.
2. H ~ E o ~ ~ c ~ ~ ~ H.K. ~eE~ od onepaTope c~m~ra, M., HayEa,
I980.
R.TEODORESCU
V.I.VASYUNIN
(B.H.BAC~mS)
Universitatea Bra~ov
Facultatea de ~atematic~
B-dul Gh.Gheorghiu - DeJ 29,
2200 Bra~ov, Romania
CCCP, 191011, ~eH~Hrpa~,
~OHTaHEa 27, ZOMM
158
4.14. TITCH~ARSH'S THEOREM FOR VECTOR FUNCTIONS old
In one version (from which others can be derived) Titchmarsh's
theorem states: i f ~ a n d ~ a r e f u n c t i o n s o f
L~(~+) s u c h t h a t ~ * ~ v a n i s h e s o n (0,~),
a n d i f ~ v a n i s h e s o n n o i n t e r v a 1 (0,6),
t h e n ~ m u s t v a n i s h o n (0,~) . Here is a PROOF.
Fix ~ , and denote by M the set of all ~ such that ~ * ~ va-
rnishes on (0,~) . ~ is a closed subspace invariamt umder shifts to
the right. Beurling's theorem states that M , the space of Fourier
transforms of functions in ~ , is exactly ~Hg , where ~ is inner
in the upper half-plane and H ~ is the Hardy space on the half-planeC
Since ~ contains all functions vanishing on (0,1) ,(~(~))m~p(~%) is
an inner function too. The known structure of inner functions implies
that ~(%)~Xp(~) for some ~ , 0<~G~I . This means that ~ con-
tains all functions that vanish on (0~), and it follows that ~ va-
nishes on (0,~-~) . Hence ~=~ , so ~ must vanish on(034) . @
Suppose ~ and @ are functions in ~(~+) with values in a
Hilbert space ~ , and suppose the expression
What can we
ne~l~, is th
zation of t
that are cl
right shift
tions vanis
By the vectorial
j" G CI; 0
vanishes for 0<~c~<I . (The integrand is the inner product in H .)
s a y a b o u t ~ a n d ~ ? More go-
ere a simple cha ra c t eri-
h e s u b s p a c e s M o f L~(~+)
o s e d , inva riant under
s , and contain all lunc-
h i n g o n (0,~) ?
version of Beurling's theorem (see [I]) the
problem is equivalent to describing the inner functions Q such that
Q(~)-I ~p ~ is also inner. In the vectorial context, an ~nner
function Q is analytic in the upper half-plane, takes values in the
space of operators on H , satisfies ~Q(g)J g I , and has boundary
values Q(~) that are unitary for almost all real ~ . In our case
~(X) has spectrum (the support of its Fourier transform) in[0~]
and so is entire.
We obtain inner functions of this kind in the form e~cp $% A ,
where A is a constant self-adjoint operator satisfying 0~ A ~ I •
The corresponding subspace M is easily described. Let (~) be the
159
spectral resolution of A ; thus H$ = @ for ~ ~ 0 and~=H
for $ > I . ~ is the set of vector functions ~ such that ~(~)
lies in ~$ for almost every $ .
A straightforward extension of Titchmarsh's theorem would as-
sert that the integral above vanishes for 0~ ~ I only if the in-
ner product vanishes identically for such ~ . This is equivalent to
saying the inner function of ~ necessarily has the form ~p $~A.
This is not true, as shov~ by an example of Donald Sarason. His
example leads to a method for constructing such inner functions. Set
~(~)~(¢~p(-~/~)) ~(Z) ; then the unitary function ~(~)
has spect rum i n [ - { , % J . Write ~ = ~ ¢ % T w i t h ~ , T s e t f - a d -
joint. The fact that ~ is unitary means that ~ and ~ commute
at each point, and ~4 T~=I. Suppose ~ is two-dimenqional and ~=~I , 0 ~ • ~ I . Then on
the real axis ~ must be (! ~ where ~ and ~ are entire functi-
ons of exponential type at most -~- , ~ is real on the real axis,
a~d ~ ~q-I~l~=~-'~ ~. ~he choice ~ ' ~ = ~ co~'~, ~ = ~ ~ -~ -~- , g i v e s
Can the s t ruc t ur e of Q be des-
cribed simply in general, or
even when H is two-dimensional?
REFERENCE
I. H e 1 s o n H. Lectures on invariant subspaces. NY-London,
Academic Press, 1964.
HENRY HELSON Department of ~ath.
University of California
Berkeley, California 94720
USA
16o
4.15. SO~ FUNCTION THEORETIC PROBLEMS CONNECTED WITH THE THEORY OF SPECTRA~ ~ASURES OF ISO~rRTRIC OPERATORS
Let V be a completely non-unitary isometric operator in a se- parable Hilbert space H with the defect spaces N and M :
V~HeN -~HeM where it is supposed for defini teness that 0 < ~ N-~M.~®. Let PB denote the orthogonal projection of H onto the subspace L, and let T v -VP~ . The operator V defines in the unit disc D an operator-valued holomorphic function
%vm=z% (l-~Tv f IM which is called the characteristic function of V -
Consider the class B(M~N~ of all operat or-valued contr- active holomorphic functions in ~ taking values in the space of
all bounded operators from M to N . Let ~u ( M, N) =
--{%~B(M,~):~(0)=0], It is known that ; ~ E ' B ° and that for every ~ . there exists an isometric V with given
defect spaces N and M such that ~-- ~ (see [I] -[3]). It is also k11ow~1 t~t in the case ~ N--~ M all unitary
extensions of V not leaving ~ are described by the fo~ula
U 8:7 v +8 Pn where & is a unito~ "pa~meter", & :N - 'M a&*: I I M ). The spectral measure E~
mined (up to a unitary equivalence) by ~=~ of the following formula
of UF_ can be deter- and ~v with a help
V (i)
where ~Q-P,a IN The spectral measures of ,the minimal un i tary extensions of V
leaving H (now the case ~ N~ ~ . ~ M is also pe= i t ted ) can be a l - so determined by .(I ) Where the parameter ~ i s already an a r b i t r a r y function in B ( N , M ) . ~he ~ e c t r a l measure of U8 is absolutely con- tinuous if and only if the measure ~ in (I) is absolutely continu-
161
ous with respect to the Lebes~ue measure on I • (W,N) " N) Consider a subset ~@ of ~ (~, consisting of func-
tions ~ whose measure ~ in the Riesz-Herglotz representation
l
is absolutely continuous for an arbitrary choice of ~ in B(~,M). The inclusion %V ~ B@ (M, N) is clearly esuivalent to the con- dition that all minimal unitary extensions of V have absolutely con- tinuous spectral measures~
PROBLEM. Find criteria for a ~iven % i~ ~°(~,N) to belong
~o B ~ M , N ) • ~oto ~h~ ~or ~ ~ B°~M, N) ~ ~o~.s~on
u)4 b'(T) implies ~ ~@(M,~) being thus a sufficient (but not necessary) condition.
@ Suppose in the sequel that ~ M < +OO and let ~(M,N) (~S~(M,~)) denote the family of all ~ in B(M,~) v~
(~0(M,bN)) with
LE~. G%ven ~ £ ~ (M, N) the follow in~ are equivalent:
I) there exists an isometric operator ~ : N ~r M with the
correspondin ~ measure in (I~ satisfyin~ the S zeg8 condition
2) co.dit%on (~) holds for all isomet~ies ~; ~--~M ;
continuous for almost all ~ (with respect to the invariant measure on the symmetric space of all isometries ~" N -'p M ).
162
We don't know any example of a function in B~ (M, N) not satisfying (2).
B° MN) A more subtle sufficient condition for ~ ~Z ( ~ to be- long to i~n@ (~N) can be deduced4 from results of [4]-[6]. Nszmely, fix % " nd denote by ~ and (~4 ~ (tmique) solu- tions of the factorization problem
in the classes of outer functio~ non-negative at the origin and belonging to BiN, N) and B (M,M)={I~m):k~d/~)~B(M,M),
IZlP~} respectively. Let
It follows that the values of ~.-~. ~ are contra~tions N---I"M a.eo on T • Consi'd%r%%e~Hankel operator 7~ with the
matrix symbol ~0 " The operator F ~aps ~ (N) into ~ (M) and its matrix in the standard basis is (t (-i -k +4))i,~ , where
stands for Fourier coefficients of ~0 . Consider subspaces
No-{e~=t~, o,o,...):'~ ~N}, Mo={~ = oZ, O, o,...): ~, M]
of ~ ( N ) and ~(M) and put
where .~. . . (0,~) , "I~ ~f PNod~-# r'r)-'lNol'~' "and positive square roots
[%od-f rr*)41 MoT ~ - - " . respsctivel, and f inal ly
163
It turns out that ~D~[,'[,|I) provided
0
satisfies (4). then
RE~L~RK. ~or %~B 0 ~M N) condition (4)holds iff the
following formula
[ P+
(4)
establishes a one-to-one correspondence between the set of operator (~---~ a I - ' ' "° valued contractive, functions/.~_. __,~^ _ _ ~ with the same prin-
and all f~c~ions ~ in B C N, MI -
REFERENCES
I. ~ M B ,~ M ~ M.C. 05 OAHOM ~acce aMHe~x onepaTopoB B rHab6epTo-
BOM npocTpaHcTBe. - MaTeM.c6. I946, 19(6I), 236-260. 2. Z H B m ~ U M.C. MsoMeTp~Mec~e onepaTopM c paBH~ ~eSe~TH~H
~Hcaa~H, ~Bas~yHHTapHMe onepaTop~. - MaTeM.c6., I950,26,247-264. 3. Sz.-N a g y B., ~ o i a 9 C. Harmonic analysis of operators in
Hilbert space. Budapest, Akad.K~ado, 1970.
4. A ~ a M 2 H B.M., A p o B A.3., E p • ~ ~ M.£. 5ecHoHe~H~e
~aH~eaeB~ MaTp~U~ ~ O605~eHH~e npod~eMM KapaTeoAop~-~e~epa ~ M.~y-
pa.-~yH~.aHaa.H e~o npHa., 1968,2,B.4, 1-17.
5. A A a M ~ H B.M., A p o B A.3., E p e ~ H M.£. 5ec,oHe~Hue
6aOMHO-~SHEe~eBM MaTpHLIM M CBHSaHH~4e C HMMM npofiaeM~ npoAom~eHHa.- 14sB.AH ApM.CCP, ¢ep.Ma~eM.,1971,6, I81-206.
6. AAaMmH onepaTopos.
V. M. ADAMYAN (B.M.~)
D. Z. AROV
(~.3.APOB) M. G. KR~IN
(M.F.EPE~)
B.M. HeBMpo~AeHHble yHHTapH~e csen~eH~2 No~yyHHTapH~X
-~yH~U.aHaa.~ ero np~., 1973 ,7, BHH.4, I-I6.
CCCP, 270000, 0Aecca, 0AeccEH~
£ooy~&pCTBeHHM~ yHMBepC~TeT
CCCP, 270020, 0~ecca, 0AeccEMB
ne~aror~ecK~ MHCTMTyT CCCP, 270057, 0~ecca,
y~.ApT~Ma 14, ~tB.6
164
4.16. THREE PROBLEMS ABOUT ~-INNER MATRIX-?UNCTIONS
I. Let ~= il ~ 0 ) 0 - I~ . A matrix-function (m.-f.) ~ mero-
morphic in ~ is called ~- i n n e r if
Let ~÷ denote the class of m.-f. with entry functions rep-
resentable as a ratio of an ~-fu_uction and of an outer ~-ftm-
ction. A ~-inner m.-f. ~ is called: I) s i n g u 1 a r if
~4E ~ and 2) r e g u I a r if there exists no nonconstant
singular ~ -inner m.-f. Wo such that WW~ 4 is ~-inner.
THEOREM I. An arbitrary ~-izL~,er m,-f. ~ admits a represe n-
tatiom W = ~ W s , wher e ~% and WS are respectively re~/lar
and singula r J-inner m.-f. ; W~ is.. unique!.y determined by~/uD %o
a constant J-nnitary right fac~Qr
2. The importance of the class of regular I-inner m.-f. is
explained in particular by its connection with the generalized in-
terpolation problem of Schur-Nevanlin
such that
of finding all m.-f.
,
where ~4 , ~£ , ~o (£~) are given m.-f. of order ~ , ~4 and
~ are inner, ~ denotes the set of all m.-f. of order • holomor- phic and contractive in D ; ~ (~< p~< +co') is the class of m.-f.
of order ~ with entries in ~? .
?ix Ith ( k -- 1,2). Whe ra es
over the set of solutions of problem (I), the values S(Ko) fill
a matrix ball. If the right and left half-radii of %his ball are non-
degenerate then problem (I) is called completely indeterminate; this
definition does not depend on K@ •
Let W ~- ~ be an arbitrary ~-iuuer m.-f. It has a
meromorphic quasi-continuation to the exterior D e of the disc ~ .
We denote ~~(~)=~*(~) . We have [2]
165
where ~ and ~ are inner m.-f., ~ and ~ are outer m.-f.,
~4 ~5~ , (~;)-( ~Bm . Singular m.-f. W are characterized by
equalities ~4 = ~ = I~ in (2). The following theorem shows that it is important to establish a criterion of regularity of a J -inner m.f.
TKEOREM 2. Let W =[W~{ be an arbitrai7/ ~ -inner m.-f.
and let ~ , ~Z , ~, b e mo-fo defi=ed in (2~° Then problem (I)
with these data is 90mpletel~ ind,eterminate and the m.-f. ~ ~8 ,
where
(3)
are its solutions, The famil,y { 56 } is the set of all solutions of
problem (I) iff the J -inner m.-f. W is regular. For an~
oompletel,~ indete~na%e problem (I ~ there exists a regular J -in--
W nor m,-f. ~/= [ ~k]~ , fo r which formula (~) establishes a
one-to-one correspondence between the set of all ~ ~B~ , and the
set of all solutions of Rr~lem (1). M.-f. ~ may be chosen so
that m.-f, 6~ and 4~ in (2) be the same as in problem (1)~ in
this case ~ is defined ' bz ...... problem ~S) up to a constant J-tmmi-
tary right factor. @
3. I-inner m.-f. W = [ W4k]4 being arbitrary, let us consi-
der m.-f. ~4 and ~ corresponding~ to it by (2) and define a m.-f.
It takes unitar~j values a.e. on T and
~E H~ • If a m.-f. ~ unitary on
up t o t h e l e f t c o n s t a n t f a c t o r
we w r i t e
- - • +
i s r e p r e s e n t a b l e i n t he
, ~ , b e i n g d e t e r m i n e d by
with ~ Z% 0 , then
~ ~=0 . The following theorem holds (see [I] and ~heorem
166
2).
THEOREM 3. ~ ~-inner ~f .W:[~.K] ~ is regular i f f , for the m,~f. ~I defined in (4) we have ~Ii=0 .
I~ to be regular it COROLLARY. Pot a ~-inner m.-f. W= [W~ is sufficient, that
The proof of Theorems 2 and 3 is based on the results about the prob-
lem of Nehari [3-5] to which problem (S) is reduced by a substitu- tion ~= B~ ~ ~ .
PROBLE~,~ I. Pind a criterion for a ~-inner m~-f. W to be re-
gular without usin~ th e notion of ' the index of a m.-f.
4. It is known [6] that a product of elementary factors of Blaschke-Potapov of the Set, 2hd and 3rd kind with the poles, respec- tively in D , in ~ and on ~ (see~7]) is a J-inner m.-f. We have [1]
THEOREM 4. ~ ~-~ m.-f. ~ is a product of elementary fac-
tors of the let and the 2nd kind iff it is regular and the m.-f. $~
and ~ associated with ~ by (2) are products of (definite) elemen-
tar 2 factors of Blaschke,P0tap0v.
REMARK. Both m.-f. ~ and ~ in (2) are Blaschke-Potapov pro- ducts iff
T T ~& corresponding condition also exists for a product of elementary factors of only the let (2nd) kind. In this case ~= I~ (~-- I~) and instead of (6) we have
T
~,t~ T T
167
T (8)
~4 T T COROLLARY. Suppose condition (5) holds for a J -inner m.-f.
W W~k]~ . The~ W is a product of elementary factors of the
let and 2nd kind (only of the let, only of the 2nd) iff condition
(6) (respectively (7), (8)) is valid.
PROBLEM2. Find a criterion for a l-inne~ m,-f. to be ~ ~T° -
duct of elementary faqtors of the lsto 2nd and 3rd kind.
Theorem 4 gives in fact a criterion of completeness of a simp-
le operator in terms of its characteristic m.-f. ~ in case when
its eigenvalues are not on T . The solution of ~oblem 2 would
give a criterion without this restriction.
3. Find a criterion for a l-inner m.-f. to be a PROBLEM
product of elementary factors of the ~rd kind.
Let us point out that such a product is a singular m.-f. A pro-
duct of elementary factors of the ]st kind arises in the ~angent ~
problem of Nevanlinna-Pick [8] and products of factors of the let
and 2nd kind arise in a"bl-tangent"problem in which '~angent"data for
~(~) and ~*(~) are given in interpolation knots ~ D ) . The
author's attention was drawn to such a'~i-tangen~problem by B.L.Ko-
gan. Products of elementary factors of the 3-d kind arise in the'~an-
gent"problem which has the interpolation knots on T . The defini-
tion and investigation of such problems is much more complicated
[9, ~o].
REFERENCES
I. A p o B ~.3. 05 O~O~ EHTepn~oHHo~ s~a~e ~ EH~e~Ni~2THOM
npoEsBe~eHH~ Bx~Ee-HoTanoBa. Tes~cH AOF~S~OB. Hh~oxa no Teop~
onepaTopoB B ~yHFI~.npocTpa~cTBaX, ~CE, 1982, 14--15.
2. A p 0 B ~.3. Peax~s~ MaTpEI~-~yH~ no ~apln~HPTOHy. -- MSB.
AH CCCP, cep.MaTeM., 1973, 37, ~ 6, 1299-1331.
3. A~ aMH H B.M., Ap o B ~.3., Kp e ~ H M.r. ~ecEo-
Here PaHEe~eB~ Ma~p~n~ ~ o6o6~eHHHe ss~aqz KapaTeo~opH-~e~epa
168
H.N~rpa.-~ym~.aEaxz3 ~ero np~oz., 1968, 2, BWli.4, 1--17.
4. A ~ a M ~ H B.M., A p o B ~.8., E p e ~ H M.P. BeOEoHe~- HHe 6~o~Ho-raHEe~eBNe MaTpHI~ ~i CB~SaH~e C HaME npo6J~eMu npo-
~ a m ~ e ~ . -HsB.AH ApM.CCP, MaTeM. t I 9 7 I , 6 , ~ 2 -3 , 8 7 - I I 2 .
5 . A ~ a M ~ H B.M. HeB~pom~eHHNe y~TapHNe c ~ e r ~ e ~ n ~ T a p - H~X onepaTopoB. - ~.aHa~Hs ~ ero npMo~., 1973, 7, BLIn.4,
1-17.
6. Ap o B ~.8., C HM a Eo B a ~.A. 0 rpavmvgHx sHa~eH~XX
cxo~m~e~ca noc~e~oBaTex~HOCT~ J-cm~a~mx MaTp~-Qy~. - MaTeM.saMeTF~, I976, I9, ~ 4, 491--500.
7. H O T a n o B B.H. Myx~T~mmEaT~BHa~ cTpyETypa J--HepacT~--
l~Ba~m~x Ma~p~-~y~En~. -Tpy~ MOCE.MaTeM,o--~a, I955, 4, I25-- -236 .
8. $ e ~ ~ ~ H a M.II. KacaTe~Ha~ npo6xeMa HeB~-II~Ea C EpaTHMM~ TO~IEaM~. --~oE~.AH ApM.CCP, 1975, 61, ~ 4, 214-218.
9. E p e ~ H M.~. 06m~e ~eope~ o nOS~T~BH~X ~ysEn~oHa~ax. -
B ~H.: Ax~esep H.H., Kpe~H M. 0 HeEoTopux Bonpocax Teop~ MOMeH-- TOB. XapBEOB, 12I-I50. (Ahiezer E.I., Krein M. Some Questions
in the Theory of Moments. Trans.~th.Mon, , AMS, 1962, v.2,
124-153.)
I0. M e x a M y ~ E.H. l~pa~a~ sa~a~a He~-H~Ea ~ l-pa- CT~Ba~X MaTpH~-~. - HsBeCTI~.~ B~IC~I~X y~e6H~x saBe~eHm~, MaTeM., 1984 (B negaTe).
D. Z. AROV CCCP, 270020, 0~ecca,
KOMCOM~CEa~ yX., 26, Q~eccEE~ rocy~apcTBe~
ne~aror~ecE~ HHCTETyT
169
4.17. EXTREMAL MULTIPLICATIVE REPRESENTATIONS
Let ~ be the class of entire functions l.tW of exponential type with values in the space of all bounded operators on a separable Hilbert space and such that
W@ 4, (l X=o3. Fo= e ery
reasing function there exists an operator-valued hermitian non-dec-
E on [0,~] (E(O)--~, V~J~ E=i) satisfying EO,{]
0 (see [1, 2~ . L~t H be the weak derivative of E • Then (I) is e q u i v a -
l e n t to a~ ~
0 The function W determines H uniquely iff I-W(~)~ C~C) and W , d~ W have the same exponential type [3]. To single out a canonical function from the family of all functions H satisfying (2) in the general case the following definition is introduced.
DEFINITION~ Let n H be a weakly measurable function on [ @~] ( 0~ ~, IHI~ ~ [ov,~] ) and suppose that for every S~[~,~] C [ ~ ] the function $
is the greatest divisor (in ~ ) among all divisors of W& (~) of type S-~ • Then H is called an e x t r e -
1 ~ f u n c t i o n o n [~t~]' =E
THEOREM 1. For ever2 W of exponential type ~ in ~ there ~
exists a unique H extremal on [O,~'] and satisf.ying WO,6. --
=Wcx . This result is a special case of a theorem proved in [2] (com-
pare with [4] ).
170
PROBLEM. Find an intrinsic description of functions H extre-
The following theorem shows that the description is very likely
to be of local character.
THEOREM 2 ([5~[6~). Suppose ~ is extremal on [~] . Then it
is extremal on any [~,~] C [~6] . Conversel,y~ if for evel V
S~ [(~,~)] there exists a segment [ ( ~ ] C. [~)~] such
hat S H is,, xtrem l on H
CONJECTURE. Le__~t H b e a continuous (with respect to the norm
t0PolO~V ~ operator -valued function on [@,~] . Then H is extre-
mal iff all values of H are orthogonal projections.
In the particular case A~ H(~)=4 , ~ 6. [~,~] the con-
jecture is true by a theorem of G.E.Kisilevskii [7] (in the form gi- ven in [8]).
Similar questions in case when W(~) is the characteristic function of a so called one-block operator have been considered in
REFERENCES
I. H o T a n 0 B B.H. My~T~a~NaTRBHa~ 0Tpy~Typa --2 epacT~l~Ba- U
MaTp~-~yRE~. --Tpy~ MOON.MaTeM. O6-Ba, 1955, 4, 125--236. 2. F ~ a s 6 y p P D.H. Myx~T~n~mEaT~2HHe npe~cTaBXeH~ ~ ~opaa-
TH ovpaaEenH~x aHaxzT~ecE~x onepaTop-~m~. - Sya~.aHax.
ero np~x., I967, I, ~ 3, 9-23.
3. B p o ~ c ~ E ~ M.C. Tpey20~HHe ~ mop~aHoBH npe~cTaBxe~ x~- n e~Hx onepaTOpOB. -Moc~Ba, "HayEa", 1969.
4. B p o~ c En~ M.C., M c a e B ~.E. Tpeyro~nHe npe~cTasxe- H~S ~CC~naT~B~X onepaTopoB c pesox~BeHTo~ ~Ncno~e~u~ax~o~o ~- Ha. - ~oE~.AH CCCP, 1969, 188, • 5, 971-973.
5. ~ ~ n s 6 y p ~ D.H. 0 ~ex~exsx ~ Mm~OpaHTaX onepaTop-~ ol~a~n~eHHoPo B~Ka. - MaTeM. ~oc~e~oBaR~, E~m~eB, 1967, 2, ~ 4~ 47-72.
6. M o r ~ x e m c ~ a a P.~. HeMOaOTOH~Ne My~ST~NaT~BHMe npe~cTaBxea~ o~paaEen~x asam~TEecm~ onepaTop-~m~. -
MaTeM. ~ccxe~o~a~s, E~m~H~, 4, ~ 4, 1968, 70-81.
171
7. K m c ~ x e B c E ~ ~ r.3. 142BapEaHTH~e Ho~pocTpa~cTBa BOX~-
TeppoB~ ~CCHIIST~BHHX onepaTOpOB c ~AepH~ ~H~m NO~nOHeHTa-
~. -HsBe0T~ AHCCCP, cep.MaTeM., 1968, 32, ~ I, 3-23. 8, r o x 6 e p r H.~., K p e R H M.r. Teop~ BO~TepposI~X
oNepaTopoB B I-~B6epTOBOM IrpOCTpSRcTBe H e~ ~p~ozes~. MoczBa,
"HayKa", I967.
9. C a x s o B ~ q ~.A. 0 /~CClrIaT~BHRX BO~TeppoBr~X onepaTopax.-
fdaTeM.cSopH~, 1968, 76 (I18), ~ 3, 323--343.
yvJ. p. GINZBURG
r rs r) CCCP, 270039, 0~ecca,
0~eccmm~ TexRoxo~ec~m~ HHOT~TyT nm~eBo~ npo~eHHocT~
~.M.B.~oMoaocoBa
172
4.18. FACTORIZATION 0F OPERATORS ON old
I. A bounded operator 5-(, 5+') on L£'(I;;l',~) , is called i o w e r t r i a n g u I a r ( u p p e r
a n g u 1 a r ) if for every ~ (&~<~ ~< ~)
tri-
s: : % 5+P,:% 5+%,
A bounded operator 5 on L ~(&,~) is said to a d m i t
t h e 1 e f t f a c t o r i z a t i o n if 5:5-5+ where
~_ and ~+ are lower and upper triangular bounded operators, with
bounded inverses.
I.C.Gohberg and M.G.Krein [I] have studied the problem of fac-
torization under the assumption
S-I ~ 11"~. <:1)
The operators ~;)4., ~_ have been assumed to be of the form
S+=I+X+, S_:I+X_; X + , X _ ~ .
(~ is the ideal of compact operator)
Factorization method had played an essential role in a number of
problems of the spectral theory. Giving up condition (1) and consi-
dering more general triangular operators would essentially widen the
scope of applications of this method.
EXAMPLE. Consider [21 the operator
%t v.p. I 0
The operator ~ (5~0)j_ clearly does not satisfy (1). Neverthe-
less S~ adraits a factorization S~,,,=W& W&* with ~=~ @%C~ the~lower formula, j and triangular operator W& defined by the
- ( b ( ~ 4 ) ~t.
0
173
The following condition is necessary for an operator ~ to
admit the left factorization:
~ = ~ ~ P is invertible in L~(~,~) for an~ ~ ( ~ ) C*)
PROBLER 1. For what classes of operators condition (*) is suf-
ficient for the existence of t heleftfaqtorization?
In the general case (~) is not sufficient. Indeed, the opera-
tor ~ defined by
S#: ° ~ v.p. I k~ ~,~, ~CZ) + .f~, ~,-i, '
0
0<~<~ (2)
satisfies C~) but does not admit the factorization, [2]. Note that
(*) follows from (~@) defined below:
Operator ~ is bounded positive" a ndhasa bounded inverse. ~,)
An important particular case of problem 1 is the following
PROBLEM 2. Does ~) imply th e existence of a factorization?
If $-I g~ the answer is positive [I]I~ is the Matsaev ideal),
2. It is interesting to study problems I-2 for operators of con-
volution type
0
where
tions of second order make sense:
Wc.~,~)=I_~ ICS)' P~CAo -zl)-~M,~),
$ * CAoS- Ao) ~-: ~ I ~c~) [ Mc¢~) ÷ ~,(b] ~,¢,, (4) 0
A01=* I~{~hl~ . ~f (.~ holds, the fonowi~ matrix-f~c-
T I
174
B (~)=
THEOREM 1. Suppose that the operat0r $ in (~) admits the left
factorization. Then the matrix-functions ~*W(~,Z) and
N~B(~) are absolutely continuous and
~W (5)
where the elements ~ i (''~) of the mat ri x
lhl (~) =l~c~) Flic ~) ,
H (~) satisfy
(6)
and
~('~) I-1%~) +P,~c'~) FI~.c'~) =4. (7)
The functions ~ , ~ can be expressed in terms of ~_, ~+ :
7,~(~)=5[~M, I~(~):ST~ ~. (8)
Every operator ~ satisfying (3) and admitting the factorizati-
on defines (via (6)-(8)) a system of differential equations (5). The procedure of this type in the inverse spectral problem have been de-
veloped by M.G.Krein ~%] provided S ~0 and I- S ~ . Besides,
Theorem I means that the "transfer matrix-function" [5] W(~)
admits the multiplicative representation
W ( ~ , ~ ) = e ~ . (9) g
175
If ~ is positive (4) implies that
teristic matrix-function of the operator
(5), (9) are known ~6, 7]. The equality
(~0, ~) is the charac-
~-~Iz ~o ~I~. Then formulae
=- t , 0 < ac < co, (~o)
is new even in this case.
An immediate consequence of Theorem I is the necessity of the
following condition for the operator in (3) to admit the factoriza-
tion.
Operator $ in (3) satisfies ~) , the matrix-function
B(~) i S absolutel2 continuous and (I0) holds.
Note that all requirements of ~**W) but (10) are satisfied
in example (2).
PROBLEM 3. Does ($~) impl~ the existence of the factoriza-
tion?
THEOREM 2. If the operator ~ satisfies both ~) and (~,) ,
then At admits the factorization.
REFERENCES
I. r o x 0 e p r M.~., K p e ~ H M.r. Teop~ BOJIBTeppoB~X
onepaTopoB B P~B6epTOBOM ~pocTpaHCTBe ~ ee HpE~lO~eH~. M., Hay-
Ea, 1967.
2. C a x H 0 B H ~ ~.A. ~aETOpESaLG~ onepaTopoB B (~) . --
SyHEs.a~ax. ~ ero np~., 1979, 13, B~n.8, 40-45.
3. C a x H o B H ~ SI.A. 06 HHTePpa~BHOM ypaBHeHl~ C ~pOM, 8aBE--
C ~ OT paSHOCTI~ apryMeHTOB. -- MaTeM.ECCJIe~OBaH~, l~z~HeB,
1973, 8, ~ 2, 188--146.
4. E p e ~ H M.r. KOHTEHya~IBH~e s2a~ol~ npe~o~eHE~ o MHOID~e--
Hax, OpTOrOHa~BH~X Ha e~HHH~HO~ oEpy~HOCT~. -~OE~.AH CCCP, 1955,
I06, ~ 4, 687-640.
5. C a x H O B ~ ~ ~.A. 0 ~aETopEsau~H nepe~aTo~o~ onepaTop-
~yHELIH~° -~OE~.AH CCCP, 1976, 226, ~ 4.
6. Jl E B • E ~ M.C. 0HepaTop~, Eo~e6aH~, BO~HN. 0TEpMTNe 0~C-
TEMP. M., Hayza, 1966.
176
. H o T a n o B B.H. Myx~THmmEaT~BHa~ c~pyETypa ~ -EepacT~- l~Ba~x MaTp~11-~yHlg~. -- Tpy~M MOOE.MaTeM.O--Ba, 1955, 4, 125--
--136.
L.A.SAHNOVICH
(~.A. CAXHOBM~)
CCCP, 27002I, 0~ecca
3~e~TpoTex~n~ecE~ ~CTHTyT
C~3~ m~.A.C.HonoBa
177
4.19. EVALUATION OF AN INFINITE PRODUCT OF SPECIAL NATRICES
An important r$1e in studying the integrable models of Field
Theory is played by matrix-functions of complex variable of a spe-
cial form [I]. The simplest example is provided by a rational matrix:
Lo (,z) - z÷p ( i )
where ~ is a matrix of size ~ x~ and ~ is a complex number.
It is natural to call it the matrix Weirstrass factor for the comp-
lex plane C (i.e. a meromorphic function on ~ with one pole and
such that ~(~) = ~ ).
The next interesting example is given by a matrix Weirstrass
factor for a strip. This function LI is meromorphic in the strip
{~EC: 0<~e~ ~ ~ } has only one pole in it and is regular
at infinity, i.e.
where ~+
of L~Cz) are non-degenerate diagonal matrices. The boundary values
satisfy the following relation:
t~(~+~) = A L ~ ( ~ ~ , ~e~, (2)
: "" : "-~-I " One can represent
such a matrix-function as an infinite product of functions (I). For
this purpose introduce the family of matrices
:A" Lo(z÷.OA (3) and their finite product
LN4 Cz) = L, N (z) L Nq (z)-- . L "N÷~ c~} L -N (z). (4)
It is easy to show that the regularized limit
(5)
178
satisfies (2).
Por ~=~
for ~A~
formulae (3)-(5) are nothing but Euler's formulae
, so that
L~ C~)= ~ , ~ (~ +~?
We calculated
A = azo,~ (~,-~) and
S+ -,$~
in ~2]. In this case
so that " '-~Qce(~)= O . The limit in (5) is defined as follows
where
l lI s~ 0 1 DN -- o N ~3 "
The limit matrix
W: l k C 0 ~)'l
L~ C~) has a f o r = .
L~(~) = W -~ L,~C~ W,
I ! O / , ~,(~)= I"(4+1~-z) r ' (4-1t -~) e_~
k, cSD ) t'~ ~' tz ~_ It ~ ) '
I
179
~he~e ~ ~ 5~ ~ 5~ 5_. We pose as a PROBLE~ the explicit calculation of the limit in
(5) for ever~j ~ in terms of known special functions.
REFERENCES
I. F a d d e e v L. Integrable models in I + I dimensional quan-
tum field theory. CEN-SACLAY preprint S.Ph.T./82/76.
2. p e m e T ~ X ~ H H.D., ~ a~ ~ e e B ~.~. raM~TOHOBH
cTpyETyp~ ~ ~HTezp~pyeM~X Mo~exe~ Teop~ nox~. - Teop.MaT.$~8.,
I988, 57, ~ I.
L. D. FADDEEV (~.~.~)
N.Yu. RESHETIHIN
(H.D.P~II~TMXEH)
CCCP, 191011, ~eHEHrps~,
• oHTaHEa 2V, ~0MM
180
4.20. old
~ACTORIZATION OF OPERATOR ~GNCTIONS
(CLASSIFICATION OE HOT OMORPHIC HILBERT SPACE
BODNDLES OVER THE RIEMANNIAN SPHERE)
Let H be a Hilbert space,~,=~(H) the Banach space of boun-
ded linear operators in H , and G~L(H) the group of inver-
tible operators in L . We put ~I~:I~I~I} and~_~I~u~°@} "
~ I~I~<0o} and denote by ~(~,~), ~(~÷, ~), ~(T-, ~) the groups
of holomorphic GL-va!ued functions in a neighborhood of~,T$,~_
respectively. We shall say that two functions ~,T(~,Ta0(T,~I,)) are e q u i v a 1 e n t if~=A_TA+ for someA+_,A±G~(~±,~).
PROBLEM. Classify the functions i n ~(T,@~) with respec t to
thisu notion of equivalence.
REMARK. It is well-known that this problem is equivalent to the
classification problem for hclomorphic Hilbert space bundles over the
Riemannian sphere.
I. What is known about the problem? We shall say that ~ is a
diagonal function if ~(~)=Zt~ P~ , where ~<--- ~ ~ are in- tegers and ~ .... , ~ are mutually disjoint projections in I,(H)
such that p1+...+ ~ '[11 ; the integers ~ are called the
p a r t i a 1 i n d i c e s o f ~ a n d the dimensions
~ ~71~ P~ will b~ called the d i m e n s i o n s of the
p2artial indices ~ It is easily seen that the collection ~, .... ~,
~I""' ~ determines a diagonal function up to equivalence. Por
~ < + ~ it is well-known (see, for exmmple, KIS, ~2~) that every
function in ~(T,~) is equivalent to a diagonal function, a result
that is essentially due to G.D.Birkhoff [3]. ~orS~??~H =@° this is
not true. A first counterexample was given in [4~. We present here
another oounterexample: Let H~-~@ H~ be a decomposition of H and VG~(~I,H~). Then the function defined by the block matrix
(~-' O) V ~ (I)
is equivalent to a diagonal function if and only if the operator V
has a closed image in H~ , as is easily verified. However there are
positive results, too:
HEOREM I [ 5 ] . T et . i f t h e a r e
compact for all ~ , ~V , then A is equivalent to a dia~onal
181
function whose non-zemrO ' partial indices have finite dimensions ,.
For A~(~,~) we denote by W A the Toeplitz operator de-
fined by WA~-----~(A~) , where ~+ is the orthogonal projection from
h~Qq,~) onto the subspace ht(~,M) generated by the holomorphic
functions on ~+ .
THEOREM 2 [4]. A function Ae~(~h) is equivalent to a dia-
~onal function , whose non-zero ' partial indices ~ have finite di-
mension ~ , if and onl.y if ~ is a ~re@olm operator in
h~(~,H) • If the condition is fulfilled,then i%~Wa=~ ~I£
and WA= ~> o
further results see and the re erences in these
papers.
2. A new point of view. In [6] a new simple proof was given for
Theorem I. The idea of this proof can be used to obtain some new re-
sults about general functions in ~(T,~,) , too,
THEOREM 3 (see the proof of Lemma I in [6] ). Every function
from @(~,~b) is equivalent to a rational function of the form
Let A~(T,~[,) . A couple qO=(%0_, ~+) will be called a @5 -
section of A if ~_ , ~ are holomorphic H -valued functions on
T- , T÷ , respectively, and ~_(Z)~ A(~)~. (~) for~T .
Then we put ~(~)~-~+(~) for l~I~ ~ and ~(~3= ~_(~) for ~<I~[~oo .
~or 0=~=~eH and 0~<]ZI<~ co we denote by ~(~,~,~) the
smallest integer ~ such that there exists a ~-section q of A
with ~(~)= ~ . From Theorem 3 it follows immediately that there
are finite numbers ~, ~m~, depending only on A and A -I , such 0~< I%1<~ co and 0=~X~ H • that ~ ~(x,~,A)~ < ~ for all ~ ,
THEOREM 4. For every function ~ , Ae~(T,~) , there exist
unique integers ~<...<~ (the partial indices of A ), uniqu e num-
b grs 14,...,~{4,~,...~} (the dimensions of the partial indices)
and families of (not necessar2 close d~ linear subspaces
182
such that
(~) X~}(~)\ ~ t ( ~ ) i f and only i f ~(x,~,A)--~ (~=~,...,~ ;
0 ~ 1 ~ 1 ~ ). I f ~0 is a ~ j -section of A and (p(z)eM}(zo)\M~_I(Z o)
for some point ~o , then ~0(z)eMj(~) \M~_t(2) for a l l O ~ l ~ l ~ .
I f ~4,...,8~ are l i n e a r l 7 idependent vectors in H and~ f o r some
poin t %. , Cp} are .(~c}.~o,k) -sections of A with (~}(Zo)~---~} ,
then the values ~(~), .... (p~(~) are linearl~ independent for all
@
(ii) The function A is equivalent to a dia~onal functi0n if
and only if the spaces ~(~) (0~I~I~;~=~, ---,~) are close d.
For this it is sufficient that at least for one point ~o the spaces
~ (~o) are closed, Further, it is sufficient that the dimensions
~ are finite with the exception of one of them.
(iii) There are Hilbe~% spaces and ho!omorph~c opera-
tor functions ~;:T~--* ~(H~,H) such that ~(~)=~;(~) forl~ ~ ,
( i v ) f o r a l l (#=1, . . . .
The proof of this theorem uses Theorem 3, the method of the proof
of Lemma 2 in [6] and the open-mapping-theorem. From Theorem 4 we get
a collection of invariants with respect to equivalence, the partial
indices and its dimensions. However this collection does not deter-
mine the equivalence class uniquely~ because, clearly, for every such
collection there is a corresponding diagonal function, whereas not
every function in ~(T~) is equivalent to a diagonal function.
It is easily seen that, for ~£~V=~0} the function (I) has
the partial indices ~4 ~ 0 and ~1(~(~)~H I @ I~ V for all ~ )o
PROBLEM. Are all functions in ~ (~,~) with such partial in-
183
dices equivalent to a function of the form (1)?
PROBLEM. Can we obtain, in ~enera!, a complete classification,
addin~ some special triangular block matrices to the dia~onal func-
tions?
REFERENCES
I. P r o s s d o r f S. Einige Klassen singularer Gleichungen. -
Berlin, S974.
2. F 0 X 6 e p r H.H., ~ e ~ ~ ~ M a R H.A. YpaBBeR~ B cBepTEax
npoe~oRHMe MeTO~H~X pemea~. M., "HayEa', I97I.
3. B i r k h o f f G.D. Math.Ann., 1913, 74, 122-138.
4. r o x 6 e p r H.H., ~ a ~ T e p e p D. 06m~e TeopeM~ o ~a~-
Top~sau~ onepaTop-~R~ oTHoc~Te~no EO~Typa I. roxoMop~R~e
~yHEIMH. - Acta Sci.Math., 1973, 34, 103-120; II. 06o6meH~. -
Acta Sci.Math., 1973, 35, 39-59.
5. r o x 6 e p r H.H. ~a~aya ~a~Top~sa~ onepaTop-~yn~s~. -HsB.
AH CCCP, cep.MaTeM., 1964, 28, • 5, 1055-1082.
6. ~ a ~ • e p e p D. 0 ~a~TOpmSa~ Ma?p~ ~ o~epaTop~J~E~.
Coo~J~.AH rpys.CCP, 1977, 88, ~ 3, 541-544.
J. LE ITERER Akademie der Wissenschaften der DDR Institut f~r Mathematik
DDR, 1030, Berlin
Mohrenstra#e 39
184
4,21. WHEN ARE DIFFERENTIABLE FUNCTIONS DIFPERENTIABLE?
• C * If ~ ~ -~- ~ is continuous and ~ is a -algebra then
there is defined by the usual functional calculus a mapping A {Alin ~-~ { (~) from the linear space of hermitian elements of to
itself.
What is a necessary and sufficient condition on ~ that for all
the function ~A is differentiable everywhere?
Taking A =-~ shows that ~ must be differentiableo In fact:
(1) If ~A is differentiable for all A then ~ C 4 (~).
PROOF, Let A be the algebra of bounded functions on an inter-
val [C~, ~]. The differentiability of ~ A at a function ~ asserts
that for every ~ there is a ~ such that for any function
with II ~ U <
II { ( ~ + k ) - { ( ~ ) - ~ { ~ ) . k II ~ £ .Ukll
#
This shows immediately that ~a (JC)
--({'o ~)~. ~et S.,t e E~, 6 ] and take SC ($) to be the identity function
taut function So - ½o Then
must be the mapping ~--~
satisfy I So- tol ~ and k({} the cons-
~hus I { (~o)- { ( t . ) - changing $@ and to
II kll = I S o - t . I < ~ and ~ (OC+~)- f(OC)-- ~ ( ~ ) ~ is equal at ½= ~. to
~ (So) - ~ ({.) - ~' ( t . ) ( s , - to)
~ ' ( t . ) ( s . - t ) l ~< ~ I s . - t . { , a d ~ , ~nd ~ v i d i ~ by IS, -~. l
• Inter-
give
It is even easier to show that if ~ ~ C 4 and A is commu-
tative than ~ A is differentiable. For general A all I know is this:
(2) If in a nei~hbcurhood of each point of ~ the function ~ i_~s
equal to a function whose derivative has Fourier transform belonging
to ~.~ (~) then ~A is differentiable for all A.
PROOF. Of course "each point" in the assumption on ~ can be rep-
185
laced by "each compact set" and since the differentiability of ~A at ~ depends on the values of ~ in an arbitrary neighbourhood of
the interval [-II~II, II~II] we may assume ~ itself has derivative
whose Fourier transform belongs to L 4 (~) . Let 06 and ~ be hermitian. From the identity
d, e~S(oc~-h,) e4s=c = i.e~SC~c~-h,) ~e4SOc ol, s
we obtain upon integrating with respect to $ over [ 0,~] and right
multiplying by e ~%~
e b { ( ~ b ) __ e ~{* + L I re ~s(~+b) ke~(~-s)~&s . o
Applying the Fourier inversion formula gives
--~ 0 -~ 0
= I÷H,
let us say. The inner integral in i has norm at most Itl" ~Ii and
so (since ~[(%)C ~-~) ) the double integral makes sense and
represents a continuous linear function of ~ o In fact it will de-
fine ~A (~)~ . To show this it suffices to show that the doub-
le integral~ has norm O(U~U) as II~ ~ 0 , But the norm
of the inner integral in ~ is o(~) for each ~ and is at
most 2 ~I II WI~ for all ~ and so the conclusion follows from
the dominated convergence theorem, @
PROBLEM 1. Fill the 6ap between (I) an d (2). In particular, is
~ C I a sufficient condition for the differ anti ability of ~A
for all A ?
Here is a concrete example. Let A be the algebra of bounded ope-
rators on ~2 (~, ~) . If gC is M~ , multiplication by the
identity function, ~ ~ ig the integral operator with kernel
186
(S,~) , then formally
tor with kernel {', (,)k
K (s,b f(4;? S-'i;
is the integral opera-
(*)
(This is easily checked by a direct computation if ~ is a polyno-
mial). Hence we have a concrete analogue of Problem 1:
a necessary and sufficient condition on ~ that PROBLEM 2. Find
~henever ~ ($,~) i A the kernel o~ a bounded operator on ~i(a,~)
then so also is the kernel (*).
HAROLD WIDOM Natural Sciences Diw
University of California
at Santa Cruz,
Santa Cruz, California, 95064
USA
EDITORS' NOTE
Both problems 1 and 2 were extensively investigated by M.~.Birman
and M.Z.Solomyak within the very general scope of their theory of doub-
le operator integralsC[1], [21 and references therein~ see also previ-
ous papers [3], [4]). They obtained a series of sharp sufficient con-
ditions mentioned in Problem 2 and also sharp sufficient conditions
for ~ to be differentiable on the set of all selfadjoint operators
(Birman and Solomyak considered the Ggteaux differentiation but their
techniques actually gives the existence of the Frgchet differential).
Let us cite some results,
Suppose that [~,~]C (0,T) and ~ can be extended from [~,~]
as a ~--~eriodiq function with Fourier series ~ (K) ~ K'~ K=-~
Pu__~t R~(t) = ~ . (K) e -~-" if there exists a sequence I K I ~
f
~,~ ~ of ~ositive numbers w,i,t h ~ ~ < * oO such that
187
then the kernel (*) defines a bounded operator on ~ 2 (@,~)
whenever K (S~#) does, In particular this is the case if
11,= t
< i-oo (I)
Condition (1) is satisfied e.g. if belongs to the H~lder class Aoc with a positive (arbitrary small) O~ or if ~' has
absolutely convergent Fourier series.
If ~ is defined on the whole real line and el [~,~] satis-
fies the above conditions for any G~,~ ~ then ~ is differentiab-
leo n the set of all selfad,~oint operators.
The Birman - Solomyak theory encompasses many other related prob- lems (e.g. for unbounded selfadjoint operators and for the differenti-
ation with respect to an operator ideal), In particular they conside-
red Problem 2 in a more general setting, namely replacing the quoti- ent ~(S) -. ~l%) 6-~ by a function ~ ($, ~) They reformulated
this general problem as follows: for which ~ ($,~) is
~(8) ~(~) ~ (8,~) the kernel of a nuclear operator ~,~ for
any q),tp C L., ~ with ll"[-~,Lp II~d COH~S~ II~UL~ ll~llt~ "~ This equivalence leads (via V,V.Peller's criterion [5] of nuclearity of
Hankel operators) to a NECESSARY CONDITION for ~ to satisfy the re-
quirements of Problems 2 and I. Indeed, putting ~-= ~----~ we
see that ~(~)--~ (~) should be the kernel of a nuclear operator. It
follows from [5], [6] that this is the case iff ~ belongs t,o ithe
Besov class ~44 [ ~, ~ ] for any ~, ~ e ~ . So the condition
~ C ~ is not sufficient in both ..Problems I and 2 •
Let us mention also an earlier paper by Yu.B.Farforovskaya [7]
where explicit examples of selfad,joint operators A~j~m wit h sDectr~
i~ [0,~] and of functions Im are constructed such that IIA~-S~II-~0,
and II F cAD-{ (B )II Note
that the existence of such sequences { AA ' { B~), {~.~ follows also from the above mentioned Peller's results.
RE~ERENCES
I. ~ z p M a ~ M.m., C o a o M ~ X M.3. ~Me~a~ o ~0y~ cne~-
188
paa~Horo O~B~Ta. - 3am~c~ ~ay~H.ce~. ~0MM, 1972, 27, 33-46
2. B ~ p M a H M.N. f~Bo~m~e onelm~opm~e m~erlm~ CT~T~eca m. rlpe-
~ea~m~ nepexo~ no~ sHa~o~ z ~ , r e r p a ~ . - HpodaeJ~ MaT. ~SH~H, ~S~. ~ J , I973 , 6 , 27-53 .
3. E p e ~ ~ M.F. 0 ~e~oTopHx ROBr~X HCC.~0BaH~O:~X no TeopE~ BOSr~- ~ e ~ csMocoup.~z~e~x onepaTopoB. B cd.: "IIelBas . ~ e T ~ maTe~T~- ~ec~. mNoaa" I, l{~eB, 1964, 103-187.
4. ~a ~ e n ~ H ~ D.&, ~p e ~ ~ C.F. M~Terp~poBaeze ~ ~epe~-
n~poBa~e ~p~TOB~X onepaTopoB ~ npz~o~ea~e ~ Teop~z Bos~yme~m~. -
Tpy~ ceM~H, nO ~y~.~saJ~sy, Bopoge~, I956, T.I, 8I-I05.
5. H e ~ ~ e p B.B. 0nepa~op~ Pa~ex~ ~acca ~ ~ ~x np~omee~s
(pan~o~a~a~ annpo~c~aUz~, ~ayccoBc~ze nponecc~, npo6~e~ ~a~o-
p~T~z oNepaTOpOB).-~aTeM. C6OpH~E, I980,I~3, ~ 4, 539-88I.
6. P e 1 1 e r V.V. Vectorial Hankel operators, commutators and
related operators of the Schatten-von Neumann class ~ - Integr.
Equat, and 0per~Theory, ~982, 5, N 2, 244-272.
7. ® ~ ~, o ~ o ~ c ~ ~ ~ ~.~. 0~e~a ~o~ I ~ (~- ~ ~A)I c a ~ o c o n ~ s 2 N x onepaTopoB ~ ~ ~ . -- 8anzc~I ~ iay~H.ce~ .~I0~4 , 1976, 56, I43-162.
189
4,22, old
ARE MULTIPLICATION AND SHIPT
UNIPOR~KLY ALGEBRAICALLY APPROXIMABLE ?
O. NEW DEFINITION. A family ~ ~ I A00 : 006~I } of bounded
operators on Hilbert space H is called u n i f o r m i y a i -
g e b r a i c a i i y a p p r o x i m a b I e or (briefly) a p-
p r o x i m a b i e if for every positive E and for every ~0~/I
there exists an operator A~, 8 such that
ca e ..0.. b) the ~-algebra (i.e. algebra containing ]~* together with
B ) spanned by ~A00,8~ is finite-dimensional *)
In particular, an operator A is called a p p r o x i m a b 1 e
if the family I A } is approximable. In this case ~ ~ A, ~ A
is approximable also. Given an approximable family ~ and ~ ~ 0 let O
~S denote the algebra of the least dimension ~ %j among
algebras satisfying a) and b). The function 6 H(~.,~) ~s'~,~l~,-~ 8 is called t h e e n t r o p y g r o w t h of ~.
I. THE MAIN PROBLEM is to obtain convenient criteria for a family
of operators (in particular, for a single non-selfadjoint operator)
to be approximable, and to develop functional calculus for approximab-
le families. See concrete analytic problems in section 5.
2. KNOWN APPROXIMABLE FAMILIES. The first is I~ } with A~--~*.
Indeed, let ~ ----- ~ ~ (PA~ -- P~_~ ) where ~}~=I forms an
-net for the spectrum of A and ~A} is the spectral measure
of A . In this case ~ (8, A ) coincides with the usual 8-entro-
py of ~p~ ~ considered as a compact subset in ~.
Let ~ I ~,f~..., A~, ~ be a family of commuting selfadjoint
operators. It is clearly approximable with ~$, ~ defined analogous-
ly. The entropy H(~%) is again the 8-entropy of the joint
spectrum in ~.
The same holds for a finite family of commuting normal operators.
Let now % be a finite or compact family of compact operators.
*) We do not require the Ldentity in A 6 to be the identical ope- rator on ~ in order to include compact operators into considera- tion. If the identity of A° is the identical operator I on then ~a ~oes not contain co~pact operators and defines a decompositi- on ~-- ~I ® ~ with ~ ~ <0o and A~,~
~-~ 11~@~, ~ -~,~ ~(H~), /i } . In general it is conve- nient to consider all algebras in the Calkin algebra. (See D~ for definition~of the theory of C*-algebras).
190
Then the operators A~ can be chosen to have finite rank and the- refore ~ is approximable.
Given an approximable family ~! and a finite collection~1,...,~ ~ of compact operators consider the family ~ ~ / .
Then ~ is approximable. In particular, any operator with compact imaginary part is approximable.
Let ~ = I @ ~ and let ~ ~=~ Then ~
~ ~---~ ~[ @ ~ ~X~ : ~ ~--- ~,..., ~,~ is approximable . Consider ~ ~--- {~, P~ ~ ~ being an orthogonal projecti-
on, ~,~ • This is a partial case of the previous example because
there exists a decomposition ~ ~ S ~ ~ ~ such that
C ~ ~ %~ (see ~] for example). X
The unilateral shift U is approximable. If ~ denotes the ~-algebra generated by U then it contains the ideal ~G(~)
of all compact operators and ~/~C(H):C(~) (cf ,e.g., [3])- It follows that~ is approximable in the Calkin algebra
It was actually proved in [4] that any finite family of commuting quasi-nilpotent operators is approximable (see [~).
3. KNOWN NON-APPROXIMABLE FAMILIES THEOREM. If a family ~ = ~ U~, ~=~,..., ~ ~ of .unitar~ operators
is approximable then the C*-algebra generated by I U~ is amenab-
le~ in particulart if~ is a group algebra then the ~roup is amenable
See for example ~6S for the proof.
~=4 ' ~ ~ ~ is a famil 2 of or-
thogonal projections in ~eneral position then ~ is not approximable.
Indeed, set ~ Z~ -- I . Then ~=]J~ = ~ and is a free product of ~ copies of ~ which cannot be amenable for
Therefore a family of two (or more) unitary or selfadjoint opera- tors picked up at random cannot be approximable in general. This imp-
lies that the single haphazardly choosen non-selfadjoint operator is not approximable either. The property of approximability imposes some restrictions on the structure of invariant subspaces (see the foot- note to S e c t i o n 1).
Consider a family ~ [~,...,~ of partial isometries
bound by the relation .~ U£U~ =I , ~ • Then ~ is not approximable [7] although the algebra generated by 4 is amenable [8]
191
Any algebra generated by an approximable famil~ being a subalgeb-
ra of an inductive limit of C~-algebras of type I, is amenable as
a C~-algebra K7~. However, the class of such algebras is narrower
than the class of all amenable algebras. If an approximable family
generates a factor in ~ then it is clearly hyperfinite K6S. All
that gives necessary conditions of approximability.
4.JUSTIFICATION OF THE PROBLEM. Many families of operators arising
in the scope of a single analytic problem turn out to be approximable,
apparently because the operators simultaneously considered in applica-
tions ca~uct be "too much non-commutative" (see ~9~, ~0~, problems
of the perturbation theory, of representations of some non-commutati-
ve groups, etc). Besides, approximable families are the simplest non-
commutative families after the finite-dimensional ones.
On the other hand an approximable family admits a developed func-
tional calculus based on the usual routine of standard matrix theory.
Indeed, functions of non-commutative elements belonging to an appro-
ximable family can be defined as the uniform limits of corresponding
functions of matrices. Therefore it looks plausible that a well-defined
functional calculus as well as symbols, various models and canonical
forms can be defined for such a family. This in turn can be applied
to the study of lattices of invariant subspaces etc. In particular,
if ~ is an approximable non-selfadjoint operator whose spectrum con-
tains at least two points then, apparently, it can be proved that
has a non-trivial invariant subspace.
It is known that the weak approximation, which holds for any fi-
nite family, is not sufficient to develop a substantial functional cal-
culus for non-commuting operators. However, it is possible to consi-
der other intermediate (between the uniform and weak) notions of ap-
proximation (see, for instance, the definition of pseudo-finite fami-
ly in ~).
5. MORE CONCRETE PROBLEMS~ Our topic can be very clearly expressed
by the following questions.
^ a) Let ~ be a locally compact abelian group with the dual group
° Given ~ ~ and ~e ~ consider operators ~(~)
and V~ ~-~ ~ on ~ (Q) ~ For example, for
G=~ let
and for G T
192
and finally for C v = ~,
I,.S the Pair I~.[,~} approximable?
The answer to this question requires a detailed, and useful for
its own sake, investigation of the Hilbert space geometry of spect-
ral subspaces of these operators. One of the approaches reduces the
problem to the following. Consider a partition ofT= .0. ~ by
a finite number of arcs 6~ . Then ~(T) ~-- ~--4 ~ ~{~ "
Let ~-----H~,~,~>0 be the subspace of ~(~) consisting of functi-
ons whose Fourier coefficients may differ from zero only for integers
satisfying ~ ~ j l < 5 • Here { ~ stands for the fracti-
onal part of ~ and ~ is irrational v~at is the mutual position
of subspaces ~@, 6 and ~ ...... ~ in ~ (T) , i.e. what ar..~........their
stationary an~les, t hemutual products of the ortho~onal projectipns
etc? Since ~ and ~ satisfyVUV-~U -I ~ 6~ I (Heisenberg
equation)~ the above question can be reformulated as follows. Is imt '
possible to solve this equation approximately in matrices with any
prescribed accurac~in the norm topology?
The shift U can be replaced by a more general dynamical system
with invariant measure (X,T, ~) . Then UT~(~)= ~(T~) and
, etherI ,V is approximable or not depends essentially qn properties (and not only
spectral ones) of the dynamical system. The author knows no literatu-
re on the subject. Note that numerous approximation procedures exis-
ting in Ergodic Theory are useless here because it can be easily shown
that the restriction of the uniform operator topology to the group of
unitary operators generated by the dynamical system induces the dis-
crete topology on the group.
Note also that if the answer is positive~ some singular integral
operators as well as the operators of Bishop-Halmos type ~I] would
turn out to be approximable which would leadto the direct proofs oftbe
193
existence of invariant subspaces (see sec.4).
b) Let ~ be a contraction on ~ . Are there convenient qrite-
ria for A to be approximable expressed in terms of its unitary di-
lation or characteristicfanction?
c) Let
X
Find approx imab i l i tE c r i t e r i , a,,,,,in terms of K.
Non-negative kerne ls ~ 0 are espec ia l l y i n t e r e s t i n g . d) For what countable solvable groups ~ of rank 2 the regular
unitary representation of ~ in ~(~) generates approximable fa-
milies? Por what general locally compact groups does this hold?
REFERENCES *)
I D i x m i e r J~ Les ~-alg~bres et leurs representations Paris,
Gauthier-Villard, 1969
2. H a 1 m o s P. Two subspaces. - Trans.Amer.Math. Soc., 1969, 144,
381-389.
3- C o b u r n L. G~-algebras, generated by semigroups of isomet-
ties° - Trans.Amer,~ath.Soc., 1969, 137, 211-217.
4. A p o s t o 1 C. On the norm-closure of nilpotents. III. - Rev.
Roum.Math. Pures Appl., 1976, 21, N 2, 143-153.
5. A p o s t o 1 C., F o i a s C., V o i c u 1 e s c u D, On
strongly reductive algebras, - ibid., 1976, 21, N 6, 611-633,
6. B e p m z R A.M. CqeT~e rpy~, d~s~e E NoHe~. - B RH. :
r p z ~ } . N~map~aAT~oe c p e ~ e e ~a ~ono~oz~ecFHx rpynnax. M., ~ p , I973 (Revised English version will be published in "Selecta ~athe-
matica Sovietica", 1983. "Amenability and approximation of infinite
groups"). 7. R o s e n b e r g J, Amenability of cross products of ~ -a±geb-
ras. - Commun.Math.Phys.~1977, 57, N 2, 187-191.
8. Ap s yua ~ ~ B.A., B e p m~ ~ A.M. ,a~Top-npe~cTaB~e~
c~pe~e~soro npo~sBe;~e~ l~OlmlyTaTl~B~lO~ C*- am~edpu ~ nozyrpyn- I~ ee Sa~OMOI~HSm0B. - ~ 0 ~ . AH CCCP, I978, 238, ~ 3, 5II-516.
*) M.I.Zaharevich turned my attention to ~4] and A.A~Lodkin to [7].
194
9~ S z - N a g y B , P o i a @ C. Harmonic analysis of opera-
tors on Hilbert space Amsterdam - Budapest, 1970
I0. r o x d e p r M.~., ~p e ~ R M.F. Teop~Bo~TeppoB~X o~epa-
TOpOB B rE~I~6epTOBO~npocTI~CTBe ~ ee u p ~ o ~ e ~ . ~., HayEa,I967. 11~ D a v i • A. Invariant subspaces for Bishop's operator - Bull
London Math Soc , 1974, N 6, 343-348
A. M o VERSHIK
(A.M.BEP~)
CCCP, 198904, ~eHzHrpa~, UeTpo- ~Bope~, B~6a~oTe~Hs~ina., 2,
MaTeMaT~KO-MexaHM~eOK~ ~KylBTeT
SeHMHrpa~oKOrO y~Bepc~TeTa
COmmENTARY BY THE AUTHOR
During several recent years a considerable progress in the field
discussed in this paper has been made, as well as new problems have
arisen. We list the most important facts.
A C*-algebra will be called an ~-algebra if it is generated
by an inductive limit of finite-dimensional 0~-algebras. C~-sub -
algebras of A~ -algebras will be called API -algebras, A family
of operators generating an ~I -algebra is called approximable.
THE PROBLEM was to find conditions of approximability for a family
of operators or one (non-self-adjoint) operator and to give quantita-
tive characteristics of the corresponding ~ ~-algebras, etc.
I, in ~ a positive answer to QUESTION a), Sec.5 was actually
given. Namely, the approximability problem is solved for the pair of
unitaries ~/~V.
( . V £ ) ( 1 ) - ~ - ; l(r,~), ( V ÷ ) ( I ~ ~ - 1 6 ( I ) ,
~['~(~) , ~, o~E~ This is the simplest of non-trivial cases.
In [12S the authors made use of the fact that these operators are the
only (up to the equivalence) solutions of the Heisenberg equation:
UVU~V -I~ ~I . 2. In [13], [14] the approximability of an arbitrary dynamical system
(i.e. of the pair (~T,~) , where (7JT~)(~)~ ~(T~), ~(X,j~)), T
being an automorphism of (X~), ~ ~ -~- <P~ , %9 ~ ~ ) is proved.
195
The result is based on a new approximation technique developed for
the purposes of ergodic theory ("adic realization" of automorphisms,
Markov compacta). Later in ~5] conditions on a topological dynamical
system (i.e. on a homeomorphism of a compactum)were found under which
the skew product C(~) ~ ~ (~) is an A~ -algebra.
3. These results in turn allowed to describe in some cases an im-
portant algebraic invariant of algebras generated by dynamical systems,
the ~ -functor, cf.~2,16,1~. This makes possible to apply K -theory
in ergodic theory.
Nevertheless, we still have neither general approximability crite-
ria, nor complete information on approximable non-selfadjoint opera-
tors. Since many important families turned out to be approximable, the
questions on construction of functional calculus, estimates of the
norms of powers, resolvents ere are of great importance. Let us men-
tion SOME CONCRETE QUESTIONS.
A. Is the ~roup al~ebr a of a discrete amenable algebra approximab-
le?
B. Is it. true that an arbitrary aDuroximable operat0r ' has a non-tri-
vial invariant subspace?
C. How the ~-functor (as an ordered ~roup~ of an A~I -al__-
~ebra ma~ look like?
D. How are the properties of a dynamical s~stem related to the '
entropy ~r~H~h (i.e. th e ~rowth of dimensions of finite-dimensional
subal~ebras of an ~al~ebra whi.ch ' contains the algebra ~enerated by
the dynamical system)?
REFERENCES
12~ P i m s n e r ~., V o i c u I e s c u D. Imbedding the irra-
tional rotation C*-algebras into an A~-algebra . - J.Oper,
Theory, 1980, 4, 201-210.
13. B e p m ~ K A2~. PaBHOMepHa~ a~re6pazqec~as annpoKcMMa~ onepa- TOpOB C~BMra H yMHO~eHHH. - ~OE~l. AM CCCP, 1981, 259, ~ 3, 526-
529. 14. B e p m H K A.M. TeopeMa o MapKOBCKO~ nepHoA~ecKo~ anHpOECHMa-
L~HM B apro~H~ecEo~ TeopHH. - 3an.Hay~H.ceMHsjl0~, 1982, I15,
72--82.
196
15. P i m s n e r M. Imbedding the compact dymamical system. Pre-
print N 44, INCREST, 1982.
16~ C o n n e s A. An analogue of the Them isomorphism for crossed
products ef a C*-algebra by an action of ~.- Adv. in ~atho,
1981, 39, 31-55.
17. E f f r o s E.G. Dimensions and C~-algebras. C.B.~.S.Region.
Conf.Series, N 46,ANS, Providence, 1981.
4 . 2 3 .
Let T
metry
197
A PROBLEM ON EXTREMAL SIMILARITIES
be a Hilbert space operator which is similar to an iso-
, and
lated to the distortion coefficient; Holbrook [3].
From (I), it follows that, for any polynomial p ,
and ~(T) is re-
and therefore
n P~T)11 ~ ~ ~ II li,,111 llPll-
k(T) (~<T) (2)
Since estimates on ~ ( T ) yield information about functional calculus
and spectral sets, this gives one reason why ~(T) is interesting.
Another reason is the frequent occurence of the quantity~h~IIL-111
(see, fo r e ~ p l e , [2 , p.248J). One way to try to compute ~(T) is to characterize ~ satisfy-
ing (1) and
II ~, II l ~,-'11 = ~cT~.
Suppose L satisfies (I) and
llU~ II = e ~ ~T~I I (4) %-- . oo
(3)
The quantity ~(T)
~ lULl ~,:'H : ~ ~o~o
i s ca l led the k -norm of T
Two interesting quantities associated with T are
k~T,_ ~ . { ~:IlP~T, II ~ - .llP~oo }
where II II® is the su~ over ~ of the polynomial p
AcT) =
198
for all X ; then is L a similarity satisfyin~ (3)?
The answer is "yes" in two (extreme) cases: that in which
II ~ II= ~ and that in which II ~-~ II = ~ . In fact, in the latter
case~4 II=~(T)= ~(T) , by (2) and by
if ~ and ~ are chosen correctly.
If ~ satisfies (I), (4) and Ill U ~
similar and uses the fact that, in this case
, the proof of (3) is
see D] and B ] .
ineq lity Ill, II- 1 holds, in particular, when T is a
contraction. In that case, strict inequality holds in (2).
In D] , I studied ~(T) and ~(T) in connection with some
recent results on similarity of Toeplitz operators. One result was
that, in most cases, the similarity satisfying (1) and (4) could be computed explicitly.
REPERENCES
I. C 1 a r k D.N. Toeplitz operators and ~-spectral sets.--Indla-
na U.Math. J. (to appear).
2. C o w e n M.J. and D o u g 1 a s R.G. Complex geometry
and operator theory.--Acta Nath., 1978, 141, 187-261.
3. H e 1 b r o o k J.A.R. Distortion coefficients for cryptocontrac-
tions.--Linear Algebra Appl.~ 1977, 18, 229-256.
4. S z . - N a g y B. and P o i a ~ C. On contractiOms similar
to isometrics and Toeplitz operators.--Ann.Acad. Sci.Penn. Ser.A.I.
1976, 2, 553-564.
D.N.CLARK University of Georgia
Athens, Georgia 30602
USA
4 . 2 4 .
199
ESTIMATES OF ~UNCTIONS O~ HILBERT SPACE OPERATORS,
SIMILARITY TO A CONTRACTION AND RELATED FUNCTION ALGEBRAS
Pot a given class of operators on Banach spaces we can consider
the problem to estimate norms of functions of these operators. Some-
times, dealing with operators on Hilbert space, we can obtain sharper
estimates than in the case of an arbitrary Banach space. A remarkable
example of such a phenomenon is the following J. von Neumann's inequ-
ality:
for any contraction
any complex polynomial
lynomials).
l~l<- 1
( i .e . (denote by
) on Hilbert space and for
the set of all complex po-
We consider here some other classes ~f operators.
Operators with the ~rowth of powers of orde r ~, ~w 0 . This class
consists of operators satisfying ]T~i~c(~+~)~ ~0 . Clearly, for
any such operator on a Banach space we have
It is easy to see (of, [I]) that the fact that inequality (I) cannot
be improved for Hilbert space operators is equivalent to the fact that
~m) is a n o p e r a t o r a 1 g e b r a (with respect to
the pointwise multiplication), i.e. it is isomorphic to a subalgebra
of the algebra of bounded operators on Hilbert space, It was proved
by N.Th.Varopoulos E2~ that this is the case if ~>~/~ and so for
~ 4[~ (I) cannot be improved. It follows from E2], E3~, E4~ that
~61(~) is not an operator algebra for ~ 4/~ and so (I) can be
improved in this case, Now the PROBLEM is to find sharp estimates of
I~(T) if~or T satisfyingiT~[~o¢4+~) ~ Note that some estimates improving (I) are obtained in [1] o
This g e n e r a l problem a p p a r e n t l y i s v e r y d i f f i c u l t . Let us c o n s i d e r
the most interesting case o~ ~-~-0.
Power bounded operators, We mean operators on Hilbert space satis-
fying [T~ ~ O, ~ 0 It is well-known (see ref. in ~I] )
200
that such operators are not necessarily p o 1 y n o m i a 1 1 y
b o u n d e d (i.e~ I~(T)I~ I I ~ ~ A ). In ~ the follo-
wing estimates of polynomials of power bounded operators are obtai-
ned
(2)
Here
II ll : ~+ k~-~
where ~@ ~ is the injective tensor product;
^ A A
where V I ~ O a ~ N ~(1~) ~ : ~ ( ~ ) ~ - ~ ( l ~ ) , ~ O ,
t
for some ~6C(T)}
is the norm of • in the Besov space B~ . The fact that the inequalities in (2) are precise is equivalent
to the fact that the sets ~, ~0 A ~ ~ , B~I form operator
algebras with respect to the pointwise multiplication. Por ~
this is not the case ~]d For ~; VMO A @ H t the question is
open. It is even unknown whether ~ forms a Bauach al~ebrao
If V~O A @ H I is an operator algebra then the norms ll'I[~
and ]] "~VMO A ~ HI are equivalent. The question of whether this is
the case can be reformulated in the following may.
Let ~HI be the set of all F o u r i e r m u 1 t i p 1 i-
ers of ~I , i.e.
. H
Let V ~ be the set of matrices I ~k}~,k~O such that
N>O
201
where ~@~ ~ is the projective tensor product.
QUESTION I. Is it true that
~,~H ~< > F'~,~V ~ ?
R e c a l ' l t h a t = ( i s t h e H a n k e l m a t r i x o T t i s e a s y
to show (see b] t ~/~ ~H Similarly we can defineM the spaces V rl of tensors {~ZW.,...~..,.,,l}~.~.) 0
a~d the ~el tenor F'~,= { ~ (,,+... + "N ) } o
QUESTION 2. Is it true that
M ll ~ ~H~: ,,, > r~, ~ V
M <. ~ , ~ - . ~ ?
,I If Question 2 has a positive answer then V~0 A H is an ope-
rator algebra (see ~]) and sole, ~and the estimates II %°(T)II ~
~o~U~II~ ~('PIIVI,,IO A ~ i..11 cannot be improved,
QUESTION 3. Is it true that
M VM M
An affirmative answer would imply that ~ is an operator algeb-
ra and the estimate II ~(T)II ~ C0~II 'f IIZ is the best possible
(~]). Moreover in this case the estimates is attained on the Davie's
example (see ~] ) of power bounded non polynomially bounded operator~
Similarit2 to a contraction, Here we touch the well-known problem
(see e.g. ~]) of whether each,,,pol2nomiall 2 bounded operator T o_~n
Hilbert, space%,,s similar to a contraction (i.e. whether there exists
an invertible operator V such that I VTV-II ~ ~ ).
In D] we considered operators ~S on ~@ ~ defined by
where
4 I
is the shift operator on . It was proved in [I] that
is power bounded iff ~ belongs to the Zygmund class A I , i~e~
I~I < I . It was also shown in ~I] that among
202
~ there are many power bounded operators, non p o l y n o m i a l l y bov~ded .
it seems reasonable to try to construct a counterexample to the pro-
blem stated above on the class of the operators ~ . It is very easy
to calculatethe functions of ~. Namely,
THEOREM. If ~/6 BMO A t.hen B~ is polynomially bounded
(Recall that J~M0 A ~ ~ ~ = ~ ~-Cl'l,)~,~': ~(~)~---~,('~,), ~0 , for some
PRO01 ~. By Nehari's theorem (see [6] -
"We have
-'Jr
. To f i ~ s h the p r o o f we use the f a c t t h a t
o
FT] (see ).
It follows that ~( O~l~f~(~C~)I~(~-~)~)~/~O ~ and
QUESTION 4. Is it true that if ~S is polynomially bounded then
~re BMOA ?
This question is related to a question of R.Rochberg KS] concer-
nin6 Hankel operators.
QUESTION 5. Does there exist ~ with ~ ~B~O A such that
~& is similar to no contraction?
Operators with th e ~rowth of resolvents of order a . We consider
here the oper.tors satisfyi~ I ]~A,T)I ~<(ikl_1)- , • It is not difficult to prove that for amy such operator on a Bauach
space we have
203
where t he Besov space B t c o n s i s t s o f t he f u n c t i o n s
(3)
s a t i s f y i n g
being an integer greater than ~ •
Inequality (3) is the best possible on the class of all Banach
spaces • (It is enough to consider multiplication by ~ on B~ ).
The fact that (3) is the best possible for Hilbert space operators
is equivalent to the fact that the algebra BV is an operator algebra.
Consider the following operators on a commutative Banach algebra
A ~, (~,~)~o~ ~ , ~ , ~ A ,
It is proved by A.M. Tonge [9] that if all operators ~ are %-abso-
lutelly summing (see definition in S.1 of this book) then A is an
operator algebra and by P. Charpentier [4] that if ~ is an operator
algebra then all operators ~ can be factored through Hilbert sp&ce.
In the case of ~ the operators~ look as follows
The space ~ i s i so=orphic to (see ~ 0 ] ) . I t f o l l o w s from G r o t h e n d i e k ' s theorem ( see ~11]) t h a t o p e r a t o r s on f a c t o r e d through Hilbert space are ~-absolutely summing.
QUESTION 6. Is it true that for an 2 ~ ~ the opera-
for ~ : ~ ~ B~ ~ is ~-absolutel2 summing?
The answer is positive if and only if ~ is an operator algebra
which is equivalent to the fact that (3) cannot be improved. If the
answer i8 negative, find estimates sharp e r than (3). Even the QUESTI-
ON Of estimatlm ~ I T~I is not ~et solved. It follows from (3)
204
that ~ T~I ~< o(~ + ~)g . This question for ~= ~ was conside-
red in [12]. The best example I know is due to S.N.Naboko (unpublis-
hed), and another one is due to J.A. van Casteren [13]. They construc-
ted for any ~ < I/~ ~ weighted shifts T such that ~(~,T) I ~
REFERENCES
I. P e I i e r V.V. Estimates of functions of power bounded opera-
tors on Hilbert spaces. - J.Oper. Thecry, 1982, 7, N 2, 341-372.
2. V a r o p o u 1 o s N.Th. Some remarks on Q -algebras. - Ann.
Inst,Fourier (Grenoble), 1972, 22, 1-11.
3. V a r o p o u 1 o s N.Th. Sur lea quotiens des alg~bres uniformes.
- C.R.Acad.Sci.Paris, 1972, 274, A 1344-1346
4. C h a r p e n t i e r P. Q-alg~bres et produits tensoriels
tcpologiques. Th~se, Orsay, 1973.
5~ H a 1 m o s P. Ten problems in Hilbert space~ - Bull.Amer Math
Soc., 1970, 76, N 5, 887-933.
6. S a r a s o n D. Function theory on the tlnit circle. Notes for
lectures at Virginia Polytechnic Inst. and St.Univ°, Blacksburg,
1978.
7. F e f f e r m a n Ohm, S t e i n E.M. H P spaces of several
variables. - Acts Math~, 1972, 129, 137-193.
8. R o c h b e r g R. A Hankel type operator arising in deformation
theory. - Proc.Sympos~Pure Math°, 1979, 35, N I, 457-458.
9. T o n g e A.M. Banach algebra and absolutely summing operators. -
Math. Proc.Camb. Phil.Soc., 1976, 80, 465-473
I0. L i n d e n s t r a u s s J., P e ~ c z y n ' s k i A. Con-
tribution to the theory of classical Banach spaces. - J.Funct.
Anal., 1971, 8, 225-249.
11. L i n d e n s t r a u s s J., P e ~ c z y n ' s k i A. Abso-
lutely summing operators in ~p -spaces and their applications
- Studia Math~, 1968, 29, N 3, 275-326,
12° S h i e 1 d s A.L. On~6bius bounded operators. Acts Sci.Msth.,
1978, 40, N 3-4, 371-374~
13. v a n C a s t e r e n J.A, Operators similar to unitary and
self-adjoint ones. - Pacif.J°Math., 1983, I04,N I, 241-255
V.V. PELLER CCCP, I9IOII, ~eHm~rpa~ (B.B.~DD]EP) ~OHTaHEa Z7, ~0~4
4.25. old
205
ESTINATES O~ OPERATOR POLYNO~IALS ON THE SCHATTEN-VON
NEUNANN CLASSES
Precise estimates of functions of operators is an essential
part of general spectral theory of operators. In the case of Hilbert
space one of the best known and most important inequalities of this
type is yon Neumann's inequality (cf. [I~ ) :
-< : 4 }
for any contraction T (i.e. FI ~ 4 ) on a Hilbert space and for
any complex polynomial q (~A is the set of such polynomials). For
a Banaoh space X an explicit calculation of the norm
jjj jjj× I : is a contraction on X ~ ,~e ~At
would be an analogue of yon Neumann's inequality.
In 1966 V.I.Matsaev conjectured that for infinite-dimensional
h P -spaces, ~.J~[pcoincides with the P-multiplier norm= This
means that for any contraction T on 5 P " ,J~(T)Jhp<~J~Jpd¢~ J~(~e P,
where % is the shift operator on ~P : ~(~o,~,,...)=(0,~o,~I,...).
This inequslity was proved in [2] for absolute contractions ~ (i.e.
ITIh4~<I,ITIL~ < ~ ) and in [3], [5] (independently) for operators T having a contractive majorant (for dominated contractions), i.e. for
such ~ that there exists a positive contraction T on hP satis-
fying IT I- TI I for ~hP (See also a survey article [4]).
Let ~=~(~) be the space of compact operators • on an in- r cle~ ~ pIZ "t/P<e o finite dimensioD~l Hilbert space ~ withl[~ =(~= =) ) •
The dual of % , ~< p < co , can be identifiedPwith ~pr with respect
tO the dualitg(~,~)---Sa££~, ~e% , ~ o . ~ r . We are'interested in
the ~p -version of yon Neumann's inequality. Let ~=~(~)
and ~ be the shift operator on ~(~(~o,~1,o..)=($,~o,Xl,...) , X~,
and ~* be the adjoint operator. The operator % on ~ (~) is de-
fined by ~G~-$aG ~ , ~p(~) . This opperator seams to play the
r~le similar to that oi the shift on ~ . Let us introduce the no-
tation J w|T'p . . . . . .
cO~=CT~ I. I 1% . In other words for an[ con-
The conjecture is true for p= 4,Z,~.
206
PROPOSITION 1. Inequalit ~ (1) holds for isometries (not necessa-
rily invert!hie ) on the space ~ .
THE PROOF uses an idea due to A.K.Kitover. Consider an operator
W:~(~) ~ ~(~) defined byW~=(a,~%,~T~,...), where 0 < ~ ' ~ . Then W ~ ( X ) f o r any@~p(~) . P u t ~%~---6*~6 • It is easy to see that ~(~)~(~T), ~ , so
P a,
There fore '
M ~ i n g $ ~ t we obtain ! ~ ( % i~p --~ I ~ l ~ p , . I t is not d i f f i - cu l t to show that I ~ l ~ p = I~1~ , . •
Let u s show that l ~ l z r >- I ~ lp , ~ E ~A . Let Cb =
=.~ ~(.,~C)%J~ ~p(3~) , where ~=(~,@,@,...)~O~. °
• ) l iar s° l~° l~ ~- I ~ l ~ . •
Perhaps it is possible to prove (I) for certain classes of con-
tractions T (or even for any contraction ~ ) applying one of the
following two methods used by the author for hP-spaces. DEFINITION. 1. A net o f ope ra to r s [ ~ } on a Banach space X
is said to converge to an operator T in the ~I~ -topology if
~m(Tj~,,~)=(~:,~) ~ X ~ x ~ ~ o . 2. Let ~ be an isometric imbedding of ~ into ~ , ~ be
the norm one projection from ~ onto ~P (it does exist ~6]) and
T be an operator on ~ . An operator ~ on ~ is called a dila-
tionofT i f PM"J=~T ~, ~0. In view of Proposition 1 an operator T on ~ has to satisfy
(I) if it can be approximated by isometries in the pig-topology or
it has an isometric dilation on T2 . Thus one should describe opera-
tors on T~ , having an isometric dilation on ~ , and the closure
of the set of isometries on Tp in the ~ig-topology. For this aim
207
perhaps it would be useful to apply a known description of the set
of ~ -isometries ~ (cf. K6S). it is known that each contraction
on a Hilbert space has a unitary dilation on a Hilbert space ~] and
can be approximated by unitary operators in the ~-topology (cf.
[4]). The set of operators on ~P~0,1S , ~=~2 , having a unitary di-
lation on an ~P-space coincides with the closure in the p~-topo-
logy of the set of unitary operators and coincides with the set of
operators having a contractive majcrant (cf. [3], ~]). (Earlier for
positive contractions the existence Of unitary dilations was estab-
lished in ETl). QUESTIONS. Is it true that I~ an 2 ~ -contraction has an iso-
metric dilation? 2~ any absolute ~ -contraction (i.e~ a contracti-
on on~ and on ~o@ ) has an isometric dilation? 3) 2 ~ - 6~@~
coincides with the set of all contractions on ~p ? 4) 2~-¢~@~
contains the set of absolute contractions on ~p ?
The affirmative answer to 1) or 3) would imply the validity of
Conjecture I. If Conjecture 2 is also valid then this would imply
the validity of V.I.~tsaev's conjecture because sP can be isometri-
cally imbedded into ~ in such a way that there exists a contrac-
tive projection onto its image. In conclusion let us indicate a class
of operators satisfying (1).
PROPOSITION 2. Let @,~
Then the operator T on ~F
approximated b E isometries.
This follows from the fact that
lation on a Hilbert space and can be
operators on~ .
be contractions on ~ ,TG~@~ ¢ ~
has an isometric dilation and can be
and ~ have a unitary di-
pt~-approximated by unitary
REFERENCES
I. S z . - N a g y B., ~ o i a ~ C. Harmonic analysis of opera-
tors on Hilbert space. North Holland - Akademiai Kiado, Amster-
dam-Budapest, 1970.
2. H e x x e p B.B. ARaxor ~epaBe~cTBa ~x.~o~ He~Maaa ~s npOCT--
paHcTBa ~ . -~o~.AH CCCP, 1976, 231, ~ 3, 539-542.
3. P e i I e r V.V. L'in~galit~ de von Neumann, la dilation iso- • . ° # °
metrlque et l'approximation par isometrles dans . -C.R.Acad.
Sci.Paris, 1978, 287, N 5, A 311-314.
208
4. H e x x e p B.B. ARaxor ~epaBeRcTBa ~x.#oR He,Maria, EBOMeTpE--
~ecEas ~EXaTa~HS CZaTE~ ~ annpoEc~Ma[~Es ~SOMeTp~M~ B npocTpaRcT--
Bax ~sMepEM~X Sys~um~. - Tpy~ M~, 1981, 155, 108-150.
5. c o i f m a n R.R., R o c h b e r g R., W e i s s G.
Applications of transference: The L P version of yon Neumann's
inequality and the Littlewood-Paley-Stein theory. - Proc.Conf.
Math.Res. Inst. Oberwolfach, Intern. ser.Numer.Math., v. 40, 53-63.
Birkhauser, Basel, 1978.
6. A r a z y J., F r i e d m a n J. The isometries of --C~ ~'"~
into Cp . - Isr.J.Math., 1977, 26, N 2, 151-165.
7. A k c o g I u M.A., ~u o he s t o n L. Dilations of posi-
tive contractions o~ ~ spaces. - Canad.Nath.Bull., 1977, 20,
N 3, 285-292.
V. V. PELL~R
(B. B. ~D~EP)
CCCP, IgIOII, ~eR~Rrpa~,
~OSTaREa, 27, ADMH
209
4.26. A QUESTION IN CONNECTION WITH NATSAEV'S CONJECTURE
This problem is closely related to the preceding one [I], where
definitions of all notions used here can be found. I propose to verify
Natsaev's conjecture for the operator ]- on the 2-dimensional space
~$, |'= p < 2 defined by the matrix
the standard basis of
I
2 The contraction ~- is of interest
because it has some resemblance with unitary operators on 2-dimensi-
onal Hilbert space and,as is well-known~the abundance of unitary opera-
tors plays a decisive role in the proof of von Neumann's inequality~
Moreover T has no contractive majorant and so the results of Akcog-
lu and Peller cannot be applied to ito Thus it seems plausible that
the validity of Matsaev's conjecture in general case depends essenti-
ally on the answer to the following
QUESTION. Is Matsaev's conjecture true for 7-- ?
REFERENCES
I. P e i I e r V.V. Estimates of operator polynomials on the Schat-
ten - yon Neumann classes~ - This "Collection", Problem 425
A.K.KITOVER
(A.E.EMTOBEP) CCCP, 191119, ~eHNHl"pa~, ya.EOHCTaHTMHa SaCaOHOBa 14, KB.2.
210
4.27. TO WHAT EXTENT CAN THE SPECTRUM OF AN
OPERATOR BE D!MTNISHED UNDERAN EXTENSION?
Let X be a Banach space, T be a bounded linear operator on X (i.e. T~ J~ (~) ).
QUESTION I. Are there a Banach space Y contain~ X and an
qperat0r ~(~) such thatT~ ~I~ an a th e essential spect-
rum of T is exactl 2 the spectrum.*) of ~ ?
A stronger version of Question I is
QUESTION 2. Given ~,T~ ~ (X), can one find a Banach space
~ ~ X and an isometrical algebra homomorphism ~:JB(~)--~
~(Y) such that ~(~)IX=~ ~(X) and the speQt~ of ~(T) i ss
exactl~the essential spectrum of T ?
B.BOL OB S Department of Pure Mathematics
and Hathematical Statistics
University of Cambridge
16N~ill Lane, Cambridge
CB2 ISB, England
EDITORS' NOTE. Related questions are discussed in S.3.
*) Apparently, here the essential spectrum of
~C such thatI[(~-~I)~ll ~ 0 for a sequence {~} II
i s t h e s e t o f
i n ~ w i t h
211
4.28. THE DECOMPOSITION OF HIESZ OPERATORS old
If X is a Banach space)let B(X) , ~(~) and ~(X) denote the
sets of bounded, compact and quasi-nilpotent operators on X (res-
pectively). T , TaB(X) , is a R i e s z o p e r a t o r
if it has a Riesz spectral theory associatedwith the compact opera-
tors, i.e. the spectrum of T is an at most countable set whose on-
ly possible accumulation point is the origin and all of whose non-
zero points are poles of the resolvent of finite rank. The set of
Riesz operators is denoted by ~(X) •
Ruston D] ~haracterised the Riesz operators as T~ ~(X)<=>
the coset T + K(X) is a quasi-nilpotent element of the Calkin al-
gebra E(X)/R(X) . Clearly ~C X) :~ ~(X)* Q(X) . West [2] proved that if X is a Hilbert space then ~(X)----~cX). Q(X) . This decomposition has been generallsed to a C*-algebra setting by Smyth
~3]. The proof is analogous to the superdlagonalisation of a matrix
which is then written as the sum of a diagonal matrix and a super-
• iagonalmatrixwith a zero diagonal.
Nothln~ is knowa about the decomposition problem in ~eneral
Banach spaogs ~ !t m~ be that the decomposab%lit~ of all Riesz op@-
raters charap,~ri~,~es,,,Hilbert spaces up to equivalence amon~Banach
spaces.
REFERENCES
I. R u s t o n A.F. Operators with a Fredholm theory, - J.London
Nath.Soc.~1954, 29, 318-326.
2. W • s t T.T. The decomposition of Riesz operators. - Proc.Lon-
donMath.Soc.,III Series, 1966, 16, 737-752.
3. S m y t h M.R.F. Riesz theory in Banach algebras. - Nath.Zeit.~
1975, 145, 145-155.
M.R.P.SITTH
T.T.WEST
39 Trinity college
Dublin 2
Ireland
212
4.29. OPERATOR ~GE~RAS IN ~I~ ALL FREDHO~OPERATORS
ARE INVERTIBLE
Let ~(X) be the algebra of all bounded lin@ar operators in
the Banaeh space X . An operator A~X) is called a Fredholm
operator if ~ ~ A <oo and ~×/I~A <oo . It is well known that the operator of multiplication by a func-
tien (in ~P or ~ ) is inver%ible if it is a Predholm operator. The
same is true for multidimensio~l Wiener-Hopf operators in LP<~) ,
for their discrete analogues in ~P(Z~) and for operators in LP(~)
of the form
where ~ is a homeomorphism of the circle T onto itself. This
property is valid for the elements of uniformly closed algebras ~)
generated by the above operators as well (see ~] and the literature
cited therein). The usual scheme of the proof consists of two stages.
First we prove the invertibility of Fredholm operators in the
non-closed algebra ~ generated by the initial operators
(using the linear expansion [2] it is reduced to the same operators
but with matrix coefficients or kernels [3]). Then we have to extend
this statemaut in some way to the uniform closure C~
of the algebra ~ .
QUESTION I. Let eve~ Predholm operator from the algebra
(c ~ (X~ be invertibl e. Is ever~ Fredholm operator in the
algeb~ C~ ~ invertible?
In the examples the passage from ~ to ~0S ~ becomes
easier if A-I~c~ ~ for each invertible operator A ~ .
QUESTION 2. Let ever~ Fredholm operator A ~ be invertible,
~-~C~" I@ ever~ Fre~olm operator in the al~ebra and let
~ invertible~
We point out two eases, when the answer to Question I is posi- tive ~].
~)It is supposed that all algebras under consideration contai~ the identity operator.
213
Algebra ~ is commutative (or (~c~(XT and ~ con- I ° .
sists of operator matrices with elements in some commutative
algebra (~o c ~ &X) )- 2 ° . X is a Hilbert space and ~ is a symmetric algebra.
The answer te Question 2 is positive if one of the following
conditions is satisfied (see [1] ).
3 °. The algebra ~ ~ is semi-simple.
4 0. The system of minimal invariant subspaces of the algebra
is complete in ~ .
5 °. The algebra ~ ~ does not contain nil-ideals consis-
ting of finite-dimensional operators.
We call a non-zero invariant subspace minimal if it doesn't con-
tain other such subspaces. A two-sided ideal is called a nil-ideal if
all its elements are nilpotent.
Either of the conditions 3 °, 4 ° implies condition 5°° ~or 3 ° this
is obvious and for 4 ° this follows from [4] (comp with [5]),
RE~ERENCE S
I. K p y n R ~ E H.~., $ e • B ~ M a H III.A. 06 06paTl~OOTH He-
EOTOpBE( ~pe~IN~XBMOB~X oHepaTOpOB. - ~13B.AH ~CP, cep.~HS.-TeXH.
H MaT.HayE, I882, ~ 2, 8-I4.
2. r o x 6 e p r H.IL, Kp y n H E E H.H. CERzy~pR~e ~HTerpax~-
HHe onepaTopH c EyCO~HO--Henpep~BHMM~ EO3~X~EL~eHTaMH E HX CIgMBOJI~.
--T.~SB.AH CCCP, cep.MaTeM., 1871, 35, ~ 4, Co940--964.
8. Kp y n H E E H.H., ¢ e a ~ ~ M a H H.A. 0 HeBO~MO~U~OCTH
BBe~eHE~ MaTpE~HOr0 C~LMBO~a Ha HeEoTop~x sxPe(Jpax onepaTopOB. -
B F~.: ~e~e onepaTop~ ~ EHTerpa~H~e ypaBHem~. K~m~H~B~
~TEm~a, 1881, 75-85.
4. Jl 0 M O H 0 C 0 B B.H. 06 E~Bap~aHTH~X no~npoc~pa~cTBax ceM~-
CTBa onepaTopoB, EO~TERY}0H~ C BHO~He Henpep~BHm~. - CyHZ~.aHa--
~m8 E ero np~oa,. 1973. 7. BRII.3, 55-56.
5. Map E y c A.C., ¢ e ~ ~ ~ M a H H.A. 06 anre6pax, nopo~-
~eHHRX O~HOCTOpOHHe o6paT~H onepaTop~. - B EH. : ~iccxe~oBa-
no ~epeHn~s~Hm~ ypaBReH~&K~B. ~T~H~a. 1983. 42--46.
N. Ya. KRUPNIK
A. SoMARKUS
I.A.FEL' DMAN
(H.~. m'YSK)
(A.C.m~P~C}
CCCP, 277003, ~ZHS~B, ~zma~'4B- c~z~ rocy~apc~e~ y~BepcXTeT CCCP, 277G~8, K~B. H~C~X~ maTe~aTm~ AH MCCP
214
4.30. ON THE CONNECTION BETWEEN THE INDICES OF AN OPERATOR
MATRIX AND ITS DETERMINANT
Let ~ be a Hilbert space, and ~(~)
linear bounded operators in H . An operator
a ?redholm operator if and
The number ~ A = ~ ~A- ~ ~/I~%
of A •
be the algebra of all
A~ ~(~) is called
H/I A < o o .
is called the index
If
then any operator % 6 ~(~)
A an ope tor m t ix {
~ is the orthogonal sum of ~ copies of the space ~ ,
can be represented in the form
, Let ~ = { Ajk } and suppose
all commutators ~LA~L,~. Aik] to be compact. Define the deter~
mi~aDt ~ in the usual way. The order of the factors Aik in eanh term is of no importance in this connection, since various
possible results differ from each other by compact summand.N.Ja.Krup-
nik showed ([I], see also [2], p.195) that ~ is a Fredholm operator
if and only if det ~ is. On the other hand it is known that under
these conditions the equality
(1)
does not hold in general(see an example below ).
In [3] it was stated that the equality (1) holds if the condi-
tion of compactness of commutators is replaced by the condition of
their nuclearity.
question of preciseness of conditions -*[~ik~ ~irkt]~~ ~ The
arises naturally.
CONJECTURE I. Let ~ be any symmetrically-normed ideal (see
[4]) of the algebra ~(H) , different from ~ ° There exists
a Predholm operator jarv={Aik ~ , such that [Aik,Ai1~,]~
but (I) does not hold.
The weaker conjecture given below is also of some interest.
CONJECTURE 2. For an, Y p>~ there exists a Fredholm opera to r
=[Aik ] such that [-Aik , Aj/k, ] e~'p but (I) does
not hold.
215
Note that in the example below
only for ~ • ~ .
EXAMPLE. Let ~= ~(~) where
sphere. There exist singular integral operators
that
Conjecture 2 is confirmed
S ~ is the two-dimensional
Aik such
is a Eredholm operator and ~ = ~ ([5], Ch.XIV, ~4). As
~ = 0 (~5], Ch.XIII, theorem 3.2) equality (1) does not ^
hold. It can be assumed that the symbols of the operators ~k are
infinitely smooth (for ~ ~ 0 ) and therefore the commutators
[~ik ~ Ai~] map ~(S ~) into W~ (~ ~) ([6], theorem 3).
Hence S~([Aik,Ai~k~] ) = 0(~ ~1~) (see e.g.[7]).
We note in conclusion that some conditions sufficient for the
validity of (I) have been found in ~8-10]. In these papers as well as
in[S],[3] the operators in Banach spaces are considered .
RE~ERENCES
I. K p y n H E E H.H. K BOllpocy 0 HOpMSflIBHO~ paspemzMOCTH E EH--
~e~ce C~ry~spH~X ~HTeI?pS~HRX ypaBHeH~. -- Y~. sa~.K~E~HeBCEOrO
yBL~BepcHTeTa, I965, 82, 3--7.
2. r O X 6 e p r E.~., $ e x ~ ~ M a H M.A. YpaBHeH~ B cBepT-
Eax E npoeEn~oHR~e MeTo~ ~X pemeH~. M., HayEa, I97I.
3. Map E y c A.C., $ex ~ M a H M.A. 06 EH~eEce onepaTop-
HO~ MaTpHn~. --~#H~u.a~a~. E ero np~., I977, II, ~ 2, 83-84.
4. r o x 6 e p r M.H., Kp e ~ H M.r. BBe~eHEe B Teop~0xE--
He~HNX HecaMoconp~eHHRX onepaTopoB B I~B6epTOBOM npooTpaHCTBe.
M., HayEa, I965.
5. M i c h 1 i n S.G. , P r o s s d o r f S. Singulare Integra~-
operatoren. Berlin: Akademie- Verlag, 1980.
6. S e e 1 e y R.T. Singular integrals on compact manifolds. -
Amer.J.Math., 1959, 81, 658-690. 7. H a p a c E a B.M. 0O ac~MnTo~Ee CO60TBeHH~X ~ CEHryJ~HNX
~oe~ Jn~He~z~u~x oHepaTOpOB, H O B ~ P~a~EOCTB. -- MaTeM.cOopH~,
I965, 68 (II0), 623-63I.
216
8, E p y n H ~ E H.H. HeEoTopNe oS~e Bonpoca Teopm~ O~HOMepHI~X
C~I~yJZapH~( onepaTopoB c MaTpE~ EOS~dl~eHTa~m. -- B EH. : He-
caMoconp~eHH~e onepaTopm. K~m~eB, ~TY~Bum, I976, 9I-II2.
9. E p y n H ~ E H.H. YCJIOBH~ cy~eCTBOBaHN21 ~L-C~BO/~a E ~O-
CTaTOqHOrO ~adopa •--MepHHX npe~cTs3xeH~ 0aHSXOBO~ a~re6pH. - B E~.: ~Re~e onepaTop~. I~m~HeB, mT~, 1980, 84--97.
I0. B a c Hx e B C E ~ ~ H.~., Tp y xMX ~ o P, K Teop~
~-onepaTopoB B MaTp~ a~re6pax onepaTopoB. B m~.: ~e~e
ouepaTop~. ~eB, ~IT~m~a, 1980, 3-15.
I.A. l~EL ' DMAN
A. S. ~L~RKUS
(A.C.MAPEYC)
CCCP, 277028, I(~,,~eB, ~HCT~TyT MaTeMaT~EH AH MCCP
217
4.31. SOME PROBLEMS ON COMPACT OPERATORS WITH POWER-LIKE
BEHAVIOUR OP SINGULAR NI~BERS
Classes of compact operators with power-like behaviour of eigen-
values and singular numbers arise quite naturally in studying spect~
ral asymptotics for differential and pseudodifferential operators.
Presented are three problems related to the theory of such classes.
Let ~(~) be the algebra of all bounded operators on a Hil-
bert space H . Given A in the ideal C of all compact opera- tors in ~ define 5~(A), ~=~,~,. . . , the singular numbers of ~ .
~or 0<p<oo let
0 p:{Ae 7.p: s (A) =
See [1-4] for details concerning ~-Tp
While studying spectral asymptotics the main interest is focused
not on the spaces ~, 7.o themselves, but on the quotient spaces P' p
The spaces ~p , O <p<00 (for details see [5]) are complete
and non-separable with respect to the quasi-norm l a, J~ =
~ I ~ / P 5~(A)} , 6L=A+Z~ The natural limit case of
~e -spaces is the Calkin algebra ~ @~=~/C.
The multiplication of operators induces the multiplication of
elements ~E~o , ~E~a , 0< p~ ~< o0 . The product belongs to
the class ~r '[ ~-~=~-~¢~ • Taking adJoints of operators induces
the involution @5 ~ ~ in ~p -spaces. So one can consider commuting
classes, self-adjoint classes, normal classes, etc.
PROBLE~ I. Let ~ , @~6~ --@~* . Is there a normal ope-
rator in the class 8J ?
It is known (see [6]), that the answer is negative if p=oo .
it is due to the fact that in the ~oo -space there is the Index,
i.e. nontrivial homomorphism of the group of invertible elements of
the algebra ~co onto the group ~ , as well as to the fact that
218
the spectrum of an element & ~ can separate the complex plane
. These two circumstances do not occur if p <00 .
An analogous question on self-adjoint classes has the affirmati-
ve answer (and it is trivial): if ~ , @*= ~ , then for an
arbitra~j A~ the operator ~ (A+A is self-adjoint and
belongs to the class ~ .
Closely related to Problem I are the following two problems.
PROBLEM 2. Let ~ , ~ , @~=~
operat.or.s A ~ , Be~ ?
PROBLEM 3. Let ~=~*e dp ,
se l f~d~oin t commut..i.n~..operators
Problem I and Problem 3 in the case
. Are there commuti~
, 0~ = ~ . Are there
?
= p are evidently
equivalent. To the contrary a positive answer to Problem 2 does not
yield automatically the positive answer to Problem ~.
REFERENCES
I. F o x 6 e p r H.IL, E p e ~ a M.F. BBe~ea~e B TeOp~ x~ae~B~x
aecaMoconp~zea~x oHepaTopoB. M., Hay~a, 1968.
2. B E p M a H M.m., C o x o M~ E M.3. 0~eH~ C~ryxSpB~x ~cex
~aTe~0axBa~x oHepaTopoB. -- Ycnex~ MaTeM.Hay~, 1977, XXX~, ~ 1(193)
17-84.
3. s i m o n B. Trace ideals and their applications. - London
Math.Soc.Lect.Note Series, 35, Cambridge Univ.Press, 1979.
4. T r i e b e I H. Interpolation theory. Function spaces. Diffe-
rential operators. Berlin, 1978. 5. B ~pMa ~ M.m., C o ~ o Ma E M.3. Ko~aaKT~e onepaTopH co
cTe~eHHo~ aO~TOTN_WOfi C~HI~pRB~X ~cex. - Ban. aayqn, ce1~H. ~0~,
I983, I26, 21-80.
6 B r o w n L., D o u g 1 a s R , F i i I m o r e P Unitary
equivalence modulo the compact operators and extensions of
C~-algebras, -Lect.Notes in Math , 1973, 345, 58-128
M.S.BIRMA~ CCCP, 198904, ~eHHHrpa~,HeTpo~sope~
(M.[U.BHPMAH) ~XsMqecKM~ ~axy~TeT
~eHm4rpa~cEoro yHHBepCMTeTa
M. Z. SOLO~AK CCCP, 198904, ~eHm~rpa~,He~po~sope~
(~. 3. COJI0~O MaTe~aT~o-~exaH~ec~ Saxyx~TeT
JieH~rpa~c~oro y~BepC~TeTa
219
4.32, PERTURBATION OP SPECTRU~ OF NORNAL
OPERATORS AND OF COMMUTING TUPLES
Recent progress has somewhat clarified the subject of perturba-
tion of spectrum of normal operators and of K-tuples of commuting
self-adjoints. This note is a summary. Only the finite-dimensional
case is treated here. (The infinite-dimensional case is attacked in
[I] .) will be a Hilbert space of Kk dimensions. Th~ spectral re-
solution of a normal operator A will be written A~c~ %41t~ @ .|=4 4 o
here the ~i~ are orthonormal eigenvectors, with eig~nvalues ~
corresponding; and the notation ~C e , for any 3CC~ , denotes
the linear function~ corresponding to ~ , Similarly for normal B
let us write 8= ~=I ~ V~ V~ . As the distance ~ between o(A)
and O (S) let us use
j
the minimum being over all permutations of{ |,2,...,~ ~.
PROBLEM 1. Find the best constant C such that, for all normal
AandB ?
g~ cRA-BII. (2)
It has long been conjectured that C=| (e.g,, [5] ). And
(2) is known with C--4 in some special cases: if A and B
are self-adjoint [7] , if A is self-adjoint and B skew-self-
adjo~nt [ 6 ] , i ~ A and [3 are u n i t a r y [ 3 ] , or i f A - B i s n o r - mal along with A and B [2] . Yet only recently has i% even been
proved that there exists a universal C for which (2) holds in ge-
neral [4] It has been known for many years that if ~ were replaced by
the Hausdorff distance then (2) would h01d with C = | . The follo-
wing stronger assertion is also familiar (e.g., [3] ):
PROPOSITION. If K A is a set of k ei~envalues of A ~d
~Bis a set of ~-k + ~ ei~envalues of B , and if either
(i) the convex hulls of ~A and ~8 are at distance >~
o r
cii) some X C pc R÷ we have gag
220
while ~B ~ { ~[ l~-~l>~p t ~ } , then ~ ~A-BU. PROOF. The spectral subspace for ~ belonging to ~A and the
spectral subspace for B belonging to ~S have dimensionalities
whose sum is > ~ ; hence there is a unit vector ~ in their inter~
section. In case (i), ~-~I~*A~-~*~I ~< IIA-~II . In case (ii),
II(A-X)ocll~ p while II(B-X)~II a+ B(A-D~c-(B-X)~cll ~ II A-BII '& .
p+ SO
COROLLARY (R. Bhatia).
~b~ ~ x l l~il - I ~ [ I ~ IIA-BII. j (Of course, here too we can t ranslate by any ~ C ~ ). Indeed,
the l e f t - h ~ d expression is c lear ly = ~ll~i l- l~l l i f both the O~ i and the ~d are labelled in order of increasing mo-
dulus. Let the maximum be attained for ~--~ , and assume with-
out loss of generality that Ic6Kl < I~KI ~ Then ~A={ o~4,...,c~ k }
and ~B={~k,...~) satisfy hypothesis (ii) of the Proposi-
tion. •
Let ~' denote the greatest ~ obtainable for any .~A and
B~ with and B satisfying (i) or (ii), or the analogue of (ii) A interchanged. Clearly Hausdorff distance ~ ~" ~ ~.
EXAMPLE. I. Set o6~----3,oC~= ~, 063=4 ; and ~ =--c~
for all L i The Hausdorff distance is 2 , ~'----8 (attained
fo r ~A {o~ ,og, } and ~B--- { ~ ' , ~ ' I ), but ~ - - £ ~ . Thus the idea of the Proposition can not be used directly to pro-
ve that the constant C above is
The problem of ~-tuples of commuting self-adjoints may be
more important, but so far seems less tractable, I will use the fol-
lowing notation. If A (q , A~'), , A (K) are self-adjoint and com-
mute, then for orthonormal t~4, • .. , ~ and corresponding real (~) (D ( n ~ .'"
C~j we have A --~ ~, ~; ~= , ~ will let A denote the . j . ~ ~ # o m ,w~ A (1)
operator-ma~ix of one "column whose J-th entry is ~'K , so
that A ---- 2-- O6~t~;* , and we may speak of OC:~ ~ as
the i-th eigenvalue of ~ . (As an operator from 0~ ~o 3~ ,
it doe~ not have eigenvalues in the usual sense° ) Similarly
= ~" ,~iV~ V~W. As above, the distance will be
= ~ ~x II ~-~wll,
the minimum being over all permutations, and the norm being that of ~K.
PROBLEM 2. ~ind the best constant Ok such that~ for all ~ -
221
tuples A and B oKIIA-BII ~ ~ ~
The Proposition and Corollary above have exact analogues for
this situation (with ~ replaced by ~ ~ ~K and so forth);
their proofs are almost as brief, and may be left to the reader
To make precise the relationship between Problem S and Problem
2 (for the case K=~ ), I recall these elementary facts;
FACT 1- For any self-ad~0int H an__~d L on ~ (not neces-
saril~ commuting), ~ II [~]ll ~ II H + bLil.
FACT 2. For a~7 s e l f - a d j o i n t ~ an__~d L , I1[ 111 I I H + J .
"t (H+~ff(H+ZL) ~d • I t f o l l o w s a t once t ha t the cons tan ts C i n Problem 1 and Cl
i n Problem 2 are r e l a t e d by C ~ 02 ~ ~ C . I n p a r t i c u l a r , C~ i s f i n i t e , by v i r t u e of the r e s u l t of [ 4 ] a l ready c i t e d . The e x i s - tence of a finite C K for K •2 has not been proved
CO -3] 0 EXAMPLE 2. Adapt ing Example 1, take ~vt = [ 0 , ~y~vvi=[~] = [ ~ ] ; and ~=-w~v~ for all i " The remarks concerning
Example I apply to this modification, in particular ~ = 2~ . Now
choose the eigensystems
It is not hard to compute that - • For a near-
by example I have found that Ii-- --" "-~-~wl12/~ = d/ t2. The example shows that C~ > ~ However, it remains rea-
sonable to conjecture that C = | , and (as this would imply)
that Ca ~ - I suggest, with little evidence, that in general
CK ,I'R" '
REFERENCES
1. A z o f f E., D a v i s Ch. Perturbation of spectrum of
self-adjoint operators. To appear.
2, B h a t i a Ro Analysis of spectral variation and some inequa-
lities. - Trans.Amer.~athoSoc. 1982, 272, 323-331.
222
3- B h a t i a R~, D a v i s Ch. A bound for the spectral varia-
tion of a unitary operator - Linear ~ultilinear Alg. To appear.
4. B h a t i a R., D a v i s Ch., M c I n t o s h A. Pertur-
bation of spectral subspaces and solution of linear operator equa-
tions. - Linear Alg. and Appl. To apppear.
5. M i r s k y L. Symmetric gauge functions and unitarily invariant
norms. -Quarterly J° Math. Oxford Ser. 2, 1960, 11, 50-59.
6. S u n d e r V.S. Distance between normal operators. - ProcoAmer.
Math.Soc. 1982, 84, 483-484.
7. W e y I H. Das asymptotische Verteilungsgesetz der Eigenwerte
linearer partieller Differentialgleichungeno - Math. Ann., 1912,
71, 441-479,
CHANDLER DAVIS Department of Mathematics
University of Toronto
Toronto M5S IAI
Canada
223
4.33. PERTURBATION 0P CONTINUOUS SPECTRUM AND
NORMAL OPERATORS
Let T be an invertible bounded operator on Hilbert space g °
The cont~muous spectrum (~c (T) of -F is defined as ~ (7-)\ Oo Q~-),
where (~o (T) stands for the set of all isolated points of the spec-
trum O(T) whose spectral subspaces are finite-dimensional. If the
origin lies in the unbounded component of ~ \ •(T) then
0 ¢ ~c( T+ K)for any compact operator K ° on the other hand, if
~(T)separates 0 and co , then for any symmetrically-normed ide-
al~ ~ ~ ~ (~p denotes throughout the Schatten -von Neumann class~
0<p~ ~ ) there exists ~ ~ such that 0~ ~C (T+ K) [1]. The question we are interested in concerns the stability of the continuous
spectrum under "small" perturbations (e.g., finite rank, nuclear, etc~
For rank one perturbations, the problem can be solved easily in terms
of the lattice L~ of invariant subspaces of the operator.
THEOREM I. Suppose 0 does not 'belon ~ to the unbounded compo-
nent of (~ (T) an d 0 ¢ O(T). Then a rank one operator K with
0co (T+K) exi, s,ts if and L tT- ¢ LefT. See [2] , for the proof. Given an operator T denote by g (T )
the weakly closed algebra of operators on ~ generated by T and
the identity I Suppose a is a normal .operator on ~ • Then
M- c N iff , see [3]. herefore Theorem I together with a theorem of Sarason [ 4] imply the following
criterion for the stability of the continuous spectrum under finite
rank one perturbations.Denote~ M the Sarason hull of the spectral
measure of a normal operator ~ , defined in [41.
THEOREM 2. Let M be an invertible ~normal operator The followin~
are equivalent :
1) 0 ¢ (~c (N + K) for every K, ~@tck (K) - ] ,
2) 0¢ C - . - , . In particular, the continuous spectrum of a unitary operator is
stable iff Lebesg~e measure on ~ is absolutely continuous with res-
pect to its spectral ~easure, It is sh~wa in [5] that the continuous
spectrum of a unitary operator is stable under perturbations of ran~
one iff it is stable under nuclear perturbations. This result can be
extended to normal operators with essential spectra on smooth curves.
At the same time it does not hold for arbitrary normal operators ~I~ o
224
QUESTION
valent?
1) O ~
2) 0 ¢ QUESTION
tor N =T
3) o ¢
1. Given an invertible 0perator are the followin~ equi-
O~(T+ K) fo r every K o_~frank one.
dc (T ÷ K) ~or everz K of f i n i t e rank,
2. Are I) and 2) equiv.alen$..f.0ra ~ arbitrar~ normal opera-
? Is it true that they are equivalent to
Oc ( N + K) for every K ~ U! ~ ?
No%e that for ~= ~+~ , where ~ is unitary and K C~ the answer to Question I is affirmative [2] .
QUESTION 3- Is either of the followin ~ implications
0 ¢ O c (T+ K)~ V~, ~@~k (K) < + oo ~ Tc ~ (T -~) true?
The inclusion T~ ~ (T -~) being equivalent to the series of inclusions
L ~ t ~ - ' e e T - ' • " c L ~ : T • ... e T m; j
~= 4,2,... (see [3]), it is natural to ask
QUESTION 4. Are the followin~ statements equivalent for any inte-
ger ~ ?
1) 0 ¢ "~¢(T' I-K) , V K , %,@tsk(K) ~ KI,, .
2) L s t T - ' e e T - ' - . . • -- c L ~ t T e • T . v
By t h e way, we do n o t know t h e a n s w e r e v e n t o t h e f o l l o w i n g q u e s -
t i o n ,
QUESTION 5. Is it true that
T c R (T- ' ) i~f L a t T " . ......... c L~tT
Many interesting problems arise when considering special per- turbations of normal operators. Recall that the problem of stability
of continuous spectra in case of normal operators is reduced to the
calculation of Sarason hulls.
QUESTION 6. Let N and N + K be normal operators. Is it tru. e
225
that
- = C '~ ',,,Q,~,k K < +oo -----~ C. N N÷K •
K ~ : ~ , p<4 "-~" GN
For what normal operators N
== GN +K 9
K ~ F~ ~ GN = GN+K ? (1)
It was noted in [1] that there are normal operators not satisfying
(1). If ~ and ~+ K are unitary then (I) holds with W=N
This is so because the absolutely continuous parts of ~ and ~+
are unitarily equivalent and the Saracen hull of a unitary operator
depends on its absolutely continuous part only~ Therefore Question 6
may be considered as a question of "scattering theory of normal opera-
tors".
Consider a narrower class of perturbations, namely we assume hen-
ceforth that N commutes with N + K Then the symmetric diffe-
rence G N ~ ~N+K consists of the points of the point spectrum of
N or of N + K This reduces the question to the investigation of
metric properties of the harmonic measure, L.Carleson has proved in
[6] that the harmonic measure of any simply connected domain is ab-
solutely conti~uous with respect to Hausdorff measure~ , where
>I/~ is an absolute constant. Using this result, it can be proved
that ~N = G~+~ if ~ and N commutes with N + ~.
QUESTION 7. Let
Is it true that
and N + K be commutin~ normal operators,
REFERENCES
I. M a E a p o B H.r. - ~oE~.AHCCCP (to appear).
2~ M a k a r o v N,G,, ¥ a s j u n i n VoI. A model for noncon-
tractions and stability of the continuous spectrum. Complex Analy-
sis and Spectral Theory, Lecture Notes in Math,, 1981, 864, 365- 412,
226
3. S a r a s o n D. Invariant subspaces and unstarred operator al-
gebras. - Pacific J.Math., 1966, 17, 511-517.
4- S a r a s o n D. Weak-star density of polynomials. - J.reine
und angew.Math., 1972, 252, 1-15.
5. H z E o x ~ c E ~ ~ H.K. 0 BosMYmeHz~x cHeETpa yH~TapHNx onepa-
TopoB, - MaTeM.saMeTEZ, 1969, 5, 341--349.
6. C a r i e s o n L. On the distortion of sets on a Jordan curve
under conformal mapping. - Duke Math.J., 1973, 40, 547-559.
N. G. N~uKAROV
(H.r.~POB) CCCP, I98904, ZeH~Hrpa~, CTap~
HeTepro@, ~eH~Hrpa~cE~ yH~Bepc~TeT,
Ms TeMaTZEO--~exaH~ecK~ ~aEy~IRTeT
N. K. NIKOL ' SKII CCCP, I91011, ZeRz~rpa~,
~o~TaHEa 27, ZON~
227
4 .34 . AI~OST-NOP~LL OPERATORS NOD~O ~'p
O. Notations. H- separable complex Hilbert space of infinite
dimension; ~(~), 11' II) ~ (bounded operators on H , uniform norm); (~p, I' Ip ) = (Schatten- von Neumann p-class, p-norm);
~ ={T~(~) ~ finite rank, 0.<T~I}, ~={P~ ~ , ~= ~,
AN(H) = (almost normal operators on ~)=ITem(H):[T*,7] 6~4~
For T~ ~N[~) we shall denote by PT its Heiton-Howe measure and
by(~. its Pincus G-function (see [6], [10], [5~) so that ~T = _ 4 GT ~X , ~here ~X is ~ebesgue ~easure on ~
I. Basic Analogy. It is known that for V6~N(~) we have:
index (7-~I)=~T(£ ~ for ~eC such that V-~I is Fredholm. In [13] we noticed several instances which suggest that this rela-
tion is part of a far reaching analogy, in which the ~ -function plays the same role for ~ -perturbations of almost normal opera- t6rs as the index for compact perturbations of essentially normal
operators. II. Invariance of PT under ~-per%urbations. This should
correspond %o the invariance of the index under compact perturbati- ons. In [13] we proved that if T,$£~(~)~-~ and T or $ has finite multicyclicity then % = p$ . Our proof in [13] depen- ded on the use of the quasidiagonality relative to ~ . In fact one
needs less. Consider
+ where the liminf is with respect to the natural order on ~4 "
(see [12]) ,
~oPosnION. Ae,~,, T,S~AN(H) and SU!ODOSe k~(T)=O and
~-~6 ~ • Then we have %=%.
PROOF. As in [13] (Prop. 3) the proof reduces to showing that
% [ T * , T ] : % [ S * , B ] . Since k~(T):o there areAB6:~;, A.~I such that I [A~,T: I I~-O as ~ - . and the same holds also for
T replaced by S.Denoting ~ =T- $ we have:
228
i% ( IT ,T* ]- [ ~ ,S* ]1 1 :
=I%([X,T*] + [,~*,S])l--
:~,~ I%(A~,([X,T*] +[ X*,S])lg I,I, ..~oo
~ &~, ~,~p 1%( [ A,,X ,q'~'] + [ A,, X*, S])l ÷ il,.-~ oo
+ ~{,~, ~u~p Cl[ T*, A~]I~ IXI~, +1 [ S, A,,]b.I X* t~): 0. @
We proved in [13] that k~(T)=0 for Te AN(~) with finite
multiplicity.
llI. ~uasitrian~ularity° A refinement of Halmo~ notion of quasi-
triangularity [7] ~vas considered in [11]. The corresponding genera-
lization of Apostol's modulus of quasitriangularity is:
~,p (T)= ~444, ~ I(I P)~PIp
~here the liminf is with respect to the natural order on ~ . We
proved in [13] that for T~ ~N(~)~ ~(~)=0 ~PT'<0 . We conjecture
an analogue of the Apostol-Foia~-Voiculescu theorem on quasitriangu-
lar operators [1].
CONJECTURE 2. i~or T~ AN(H) , we have
1# /
This would imply in particular that PT ( 0 ~(|) ~-0w
Some results for subnormal and cosubnormal operators supporting
the conjecture have been obtained in [13]o
IV. Analogue of the BDP theorem. The following conjecture con-
cerns an analogue of the Brown-Douglas-Fillmore theorem [3] on essen-
tially normal operators.
229
CONJECTURE 3- Let t4,~ ~ ~ A~(~) be such that ~T4 =~T~
Then there is a normal operator ~£~(~) and a unitary
U£~(~s~) such that
This conjecture implies the following
CONJECTURE 4. If T~(~)then there is ~ AN(H) such
that ~8~ = normal + Hilbert-Schmidt.
Note that this last statement corresponds to an important part
in the proof of the BDF theorem, the existence of inverses in Ext or
equivalently the completely positive lifting part in the "Ext is a
group" theorem (see [2]). Even for almost normal weighted shifts,
there is only a quite restricted class for which Conjecture 4 has
been established [8].
Note also that Conjecture 4 implies Conjecture I and that by
analogy with the proof of the Choi-Effros completely positive lift-
ing theorem one should expect the vanishing of k~ to be an essen-
tial ingredient in establishing Conjecture 4.
RE~ERENCES
I. A p o s t o i C., F o i a q C., V o i c u 1 e s c u D.
Some results on non-quasitriangular operators VI. - Rev.Roum.
~th.Pures Appl. 1973, 18, 1473-1494.
2. A r v e s o n W.B. A note on essentially normal operators. -
Proc.Royal Irish Acad. 1974, 74, 143-146.
3. B r o w n L.G., D o u g I a s R.G., F i I I m o r e P.A.
Unitary equivalence modulo the compact operators and extensions .%L
of C~-algebras. Lect.Notes in Math., 1973, 345, 58-128.
4. C a r e y R.W., P i n c u s J.D. Commutators, symbols and
determining functions. - J.Funct.Anal. 1975, 19, 50-80.
5. C 1 a n c e y K. Seminormal operators. Lect.Notes in iv~th.,
1979, 742.
6. H e 1 t o n J.W., H o w e R. Integral operators, commuta-
tor traces, index and homology. Lect.Notes in I~ath. 1973, 345,
141-209 .
230
7. H a I m o s P.R. Quasitriangular operators. - Acta Sci.~ath.
(Szeged), 1968, 29, 283-293.
8. P a s n i c u C. Weighted shifts as direct summands ~ 0 ~
of normal operators. INCREST preprint 1982.
9. P e a r c y C. Some ree~t developments in operator theory.
CBMS, Regional Conference Series in Nathematics no.36, Prodidence,
Amer.~ath.Soc., 1978.
10. P i n c u s J.D. Commutators and systems of integral equations,
I. - Acta T~ath., 1968, 121, 219-249.
11. V o i c u 1 e s c u D. Some extensions of quasitriangularity.
- Rev.Roum.~ath.Pures Appl., 1973, 18, 1303-1320.
12. V o i c u 1 e s c u D. Some results on norm-ideal perturbati-
ons of Hilbert space operators. - J.Operator Theory, 1979, 2,
3-37.
13. V o i c u 1 e s c u D. Remarks on Hilbert-Schmidt perturba-
tions of almost-normal operators. - In: Topics in ~odern Operator
Theory, Birkhauser 1981.
D.VOICULESCU Department of ~thematics
INCREST
Bd.P~cii 220, 79622 Buchareat
RONJLNIA
231
4.35. HYPONORNAL OPERATORS AND SPECTRAL ABSOLUTE CONTINUITY
In the sequel only bounded operators on an infinite dimensional,
separable Hilbert space H will be considered. An operator T on H
is said to be hyponormal if TV-TT Such an operator is
said to be completely hyponormal if, in addition, T has no normal
rt, that is, if there is no subspace ~{01 reducing T on which is normal.
If A is selfadjoint with the spectral family ~E~.~ then the ~ "'2, v. ~ j
set of vectors ~ in H for which II II is an absolutely
continuous function of ~ is a subspace, H@(~) , reducing
(see, e.g., Kato [I], p.516). If H@(A)~{ }0 , then AI~@(A) is
called the absolutely continuous part of A , and if H@(A)=H
then A is said to be absolutely continuous. Similar concepts can be defined for a unitary operator.
If T is completely hyponormal then its real and imaginary
parts are absolutely continuous. In addition, if T has a polar fac- torization
T-Uffl , U ita and
then U is also ~bsolutely continuous.
if l- [T "T (See [23, p.42 and [3],
(i)
p. 193. Incidentally, such a polar factorization (I) exists, ,and is
unique, if and only if O is not in the point spectrum of Y ; see
[4], p.277.) Ingenera~, if7 is completely hyponormal, then its absolute
value ~Y ~= (T* T )i/z need not be absolutely continuous or even have
an abmolutely continuous part. Probably the simplest example is the
simple unilateral shift V for which V*V is the identity.
Of course, V does not have a polar factorization (I), but, never-
theless, there are simple_examples of completely hyponormal T sa-
tisfying (I) for which IV1 has no absolutely continuous part. More- over, it has recently been shown by K.F.Clancey and the author [5]
that if ~ is selfadjoint on H , then there exists a completely
hyponormal T satisfying IT1_ = P and having the polar factori- nation (I) if and only if (i) ~0 and g(P) contains at least
two points, (ii) 0 is not in the point spectrum of P , and
(iii~ neither ~ ~(~) nor min ~(~) is in the point spectrum of r with a finite multiplicity.
Let a nonempty compact set of the complex plane be called radial-
ly symmetric if whenever ~4 is in the set then so is the entire
232
circ:le IZI=Z4 . All examples known to the author of completely hypo-
normal operators T for which IT I does not have an absolutely con-
tinuous part, and whether or not (1) obtains, seem to have radially
symmteric spectra. Por instance, if ~(IT~) has Lebesgue linear mea-
sure zero, then ~(T) is surely radiallE symmetric; see [6],
p.426, also [7]. At the other extreme, if T is completely hyponor-
mal and if there exists some open wedge Wm{E :E~O and
O<~Z<~<~} (or rotated set e&gW ) which does not inter-
sect Q ~(T) then ITI is absolutely continuous. In certain other
instances also one can show at least that H (ffl)÷{0} see [7] and the references cited there. The following conjecture was made in D],
CONJECTURE I. Let T be completel Z hyponormal with a polar fac-
torization (I)~ Suppose that ~(T) is not radiall E s,ymmetric j so
that some circle IZ~=~ intersects both 6(T) and its complement
in nonempt E sets. Then H@(IT~)# ~ 0} •
The following stronger statement was also indicated in [7] and
is set forth here, but with somewhat less conviction than the preced-
ing conjecture, as
CONJECTURE 2. Conjecture 1 remains true without the hypothesis
(i).
REFERENCES
I. K a t o T. Perturbation theory for linear operators, Springer-
Verlag, New York Inc., 1967.
2. P u t n a m C.R. Commutation properties of Hilbert space ope-
rators and related topics, Ergebnisse der Math., 36, Springer-
Verlag, New York Inc., 1967.
3. P u t n a m C.R. A polar area inequality for hyponormal spec-
ra. - J.Operator Theory, 1980, 4, 191-200. 4. P u t n a m C.R. Absolute continuity of polar factors of hypo-
normal operators. - Amer.J.Math., Suppl. 1981, 277-283.
5. C 1 a n c e y K.F., P u t n a m C.R. Nonnegative pertur-
bations of selfadjoint operators. - J.Funct.Anal., 1983, 51 (to
appear).
6. P u t n a m C.R. Spectra of polar factors of hyponormal ope-
rators. - Trans.Amer.Math. Soc., 1974, 188, 419-428.
233
7. P u t n a m C.R. Absolute values of hyponormal operators with
asymmetric spectra. - Mich.~th.Jour., 1983, 30 (to appear).
C.R.PUTNA~ PURDUE UNIVERSITY
Department of Mathematics
West Lafayette, Indiana 47907
USA
234
4.36. OPERATORS, ANALYTIC NEGLIGIBILITY, AND CAPACITIES old
Let ~(T) and ~(~) denote the spectrum and point spectrum of a
bounded operator T on a Hilbert space H . Such an operator is
said to be s u b n o r m a 1 if it has a normal extension on a
Hilbert space K , ~-~ H • For the basic properties of subnormal ope-
raters see [I]. A subnormal T on H is said to be c o m p 1 e -
t e 1 y s u b n o r m a 1 if there is no nontrivial subspace of
H reducing T on which T is normal. If T is completely subnor-
mal then ~p (T) is empty. A necessary and sufficient condition in
order that a compact subset of C be the spectrum of a completely
subnormal operator was given in [2].
If X is a compact subset of ~ , let ~(X) denote the functi-
ons on X uniformly approximable on X by rational functions with
poles off X . A compact subset Q of X is called a p e a k
s e t of ~(X) if there exists a function ~ in~(X ) such that
Q andl l<4 onX\Q ; see p.56. The follow g result was proved in [41.
THEOREM. Let ~ be subnormal on H with the minimal norm~ 1
extension N=I~E ~ o_~n K , K D H . Suppose that Q is a non-tri-
vial proper peak set of ~(~(T)) and that ~(~)=l=0. Then ~(~)H =t= I@}
an,d, H , the spaceE(Q) H reduces T ,TIE(Q)H is subnormaZ, the minimal normal extemsien E(Q)N o_~n E (~)K, and (TI EcOH) G. ~mrt,h,~,,r,, , i f i t , iS,, also assumed that ~ ( ~ ) = O ( Q ) , the___nnTl E(e)H
is norm~l.
Thus, in dealing with reducing subspaces of subnormal operators
, it is of interest to have conditions assuring that a subset of
a compact X(=~(T)) be a peak set of ~(X) .
PROBLEM I. L~% X be a compact subset of C ~nd let C be a
rectifiable simpl 9 closed curve for which Q =clos ((exterior of~)n
X ) is not erupt,7 and C ~ X has Lebes~ae arc length measure O.
Does it follow that Q mus% be a peak set of ~(X) ?
C C In caseCis of class (or piecewise ), the answer is affir-
mative and was first demonstrated by Lautzenheiser [5]. A modified
version of his proof can be found in [6], pp. 194-195. A crucial step
in the argument is an applicatiom of a result of Davie and ~ksendal
[7] which requires that the set C~ X be analytically negligible.
235
(A compact set ~ is said to be analytically negligible if every
continuous function on ~ which is analytic on an open set V can
be approximated uniformly on V U E by functions continuous on
and analytic on ~U E ; see [3], p.234.) The ~ hypothesis is then
used to ensure the analytic negligibility of ~ ~ X as a consequen-
ce of a result of Vitushkin [8]. It may be noted that the collection
of analytically negligible sets has been extended by Vitushkin to in-
clude Liapunov curves (see[9], p.115) and by Davie( DO], section 4)
to include "hypo-Liapunov" curves. Thus, for such curves C , the
answer to PROBLEM I is again yeS. The question as to whether a gene-
ral rectifiable curve, or even one of class C ~ , for instance, is
necessarily analytically negligible, as well as the corresponding
question in PROBLEM 1, apparently remains open however.
As already noted, PROBLEM I is related to questions concerning
subnormal operators. The problem also arose in connection with a
possible generalization of the notion of an "areally disconnected
set" as defined in ~] and with a related rational approximation
question. Problem 2 below deals with some estimates for the norms of
certain operators associated with a bounded operator T on a Hilbert
space and with two capacities of the set @(T) •
Let ~(~) and ~ (E) denote the analytic capacity and the con-
tinuous analytic capacity (or AC capacity) of a set ~ in ~ .
(For definitions and properties see, for example, ~,11,9]. A brief
history of both capacities is contained in K9], pp. 142-143, where it
is also noted that the concept of continuous analytic capacity was
first defined by Dolzhenko ~2].) It is known that for any Berel set
E, (~ ~e~E) ~ ~ ~(E) ~ ~(E)
see 5 q, pp.9, 79. following proved in # 3].
THEOREM. Let ~ be, ' a bo~_ud,ed op,erator on, a, Hil,be~t space and
suppose that
(T ~)(T- ~)*~ ~] ~ ® (i)
holds for some nonnegative operator ~ and for all Z in the un-
bounded component O f the complement of ~(T) . Then ~]4/~(~(T)).
if, in addition, (I~ holds for all ~ i_~n ~ and if, for instance,
~p(T) is contained in the interior of ~(T) (in particular~ if
236
~T) is empty), then also
PROBLEM 2. Does condition (I~, if valid for all 7 i..nn ~ ,but
without an~ rectriction on ~p(T) , alwa[s impl[ that 1~14/~(~(q5) ,
o r possibl, e v e n meas ?
It may be noted that if T* is hyponormal, so thatTT~-T*T~ @ then (1) holds for all ~ in ~ with~-qT*-T~T and, moreover,
meas ( (T) see D4]
REFERENCES
I. H a i m o s P.R. A Hilbert space problem book, van Nostrand
Co., 1967.
2, C 1 a n c e y K.F., P u t n a m C.R. The local spectral be-
havior of completely subnormal operators. - Trans.AmertMath.Soc.,
1972, 163, 239-244.
3. G a m e 1 i n T.W, Uniform algebras, Prentice-Hall, Inc.,1969.
4. P u t n a m C.R. Peak sets and subnormal operators. - Ill.
Jour.Math., 1977, 21, 388-394.
5. L a u t z e n h e i s e r R.G. Spectral sets, reducing sub-
spaces, and function algebras, Thesis, Indiana Univ., 1973.
6. P u t n a m C.R. Rational approximation and Swiss cheeses.
- Mich Math.Jour., 1977, 24, 193-196.
7. D a v i e A.M., ~ k s e n d a i B.K. Rational approximation
on the union of sets. - Prcc.Amer.~th.Soc., 1971, 29, 581-584.
8. B E T y m E H H A.r. AH821~TH~ecE6uq eMEOCT~ ~o~eCTB B ss~a-
~ax TeopzE npESJm~eH~, - YcnexE MaTeM. HayE, 1967, 22, }~ 6,
I4I-I99. 9. Z a i c m a n L. Analytic capacity and rational approximation.
Lecture notes in mathematics, 50, Springer-Verlag, 1968.
10. D a v i e A.M. Analytic capacity and appriximation problems.
- Trans.Amer.Math.Soc., 1972, 171, 409-444.
11. G a r n e t t J. Analytic capacity and measure, Lecture notes
in mathematics, 297, Springer-Verlag, 1972.
12. ~ 0 JI ~ e H E 0 E.H. 0 I~HdJI~eHI4E Ha 3SmEHyTBLY 06~aoTaX H
0 Hy~B--~o~eCTBaX. - ~OF~.AH CCCP, 1962, 143, 9 4, 77i-774. 13. P u t n a m C.R. Spectra and measure inequalities. - Trans.
Amer.~th.Soc., 1977, 231, 519-529.
237
14. P u t n a m C.R~ An inequality for the area of hyponormal
spectra. - ~ath.Zeits., 1970, 116, 323-330.
C.R.PUTNA~ PURDUE UNIVERSITY,
Department of ~th.,
West Lafayette,
Indiana 47907, USA
EDITORS' NOTE. Two works of Valskii (P.8.Ban~c~, ~o~. AH CCCP, 1967, 173, N I, 12-14; C~d~pCE.MaTeM.~., I96V, 8, ~ 6,
1222-1235) contain some results concerning the themes discussed in
this section.
238
4.37. GENERALIZED DERIVATIONS AND SEMIDIAGONALITY
Let A , , A z be bounded l inear operators on a Hi lber t s p a c e
A = A A^ be an operator on the space ~(~) of all bounded ope-
rators on~'#~ defined by ~(X) = A,X-XA , X e If A4 = A=~ A A,,Az is a derivation of ~(~) • That
is why the operators /kA4,A ~ are called sometimes g e n e r a -
i i z • d d e r i v a t i o n s • Put ~ == AA*A ~ . A
question of whether
was raised by various people (see [1], [2] , [3 ] ) . Equality (1) i s true for normal operators A4, Az , whence Fuglede - Putnam theorem
follows (see [4]). This equality means that We~ ~A = Kez&
and so Ke~ ~ = Ke~ a~ = KenYA = Ke% A . zn [3] it is proved that
(1)holds when ~4 =As is a cyclic subnormal operator or a weighted
shift with non-vanishing weights.
Let p e [ { , ~ ] . An operator B in B ( X ) is cal led p - s e m i d i a g o n a 1 if its modulus of p -quasidiagonality
~p(A) = Z ~ { H PA-APII
( ~ being the set of all finite rank projections) is ~inite. Denote
by ~p the class of TP -perturbations of direct sums of p -se-
midiagonal operators. In [ 5] it is proved that (1) is true if one of
~j belongs to ~4 • Though that result covers rather exten-
sive class of generalized derivations ( ~4 contains all nor-
mal operators with one-dlmensional spectrum and their nuclear pertur-
bations, weighted shifts of an arbitrary multiplicity and polynomials
of such shifts, Bishop's operators), it is not applicable to many ge-
neralized derivations with normal coefficients. Namely, a normal
operator belongs in general to ~ , but not to ~4 . It seems
reasonable to try to replace the hypothesis of 1-semidiagonality of &
one of Ai's by 2-semidiagonality of both.
QUESTION 1. Does (1) hold i f A4, A ~ 6 ~ 2 ?
QUESTION 2, Does there exist an operator not in ~ ?
An affirmative answer to the following question would solve QUES- TION 2 (see [5]).
239
QUESTION 3- Do there exist A e ~ ( ~ )
A X -- X A is a, non-zero pro,jection?
, X~ ~2 such that
REFERENCES
I. J o h n s o n B., W i i i i a m s J. The range of normal de-
rivations° - Pacif.J.Math., 1975, 58, 105-122,
2, W i 1 1 i a m s J. Derivation ranges: open problems - In: Top
Modern OperoTheory, 5 Int. Conf.0per. Theory, Timi§oara Birkhauser
1981, 319-328.
3. Y a n g H o . Commutants and derivation ranges, - Tohoku Math J.,
1975, 27, 509-514
4. P u t n a m C, Commutation properties of Hilbert space operators
and related topics. Springer-Verlag, Ergebnisse 36, 1967.
5. m y a ~ M a H B.C. 06 onel)STO~SX ~MHOXeHBS ~ c~e~sx KO~MI~TSTO~OB,
- S81~CER HSyqH.CeM~H.~0~, (to appear).
V. S. SHUL'NAN
(B.C.~E~H) CCCP, 160600, Bogota,
~JI.MS~KOBOK0r0 6,
Ks%e~m .s~e~mT~m
240
4.38. WHAT IS A FINITE OPERATOR?
An operator
is called f i n i t e if the identity operator 1
to the range of the inner derivation ~A induced by
acting on a complex separable Hilbert space is perpendicular
, that is,
w h e r e ~AC~) = A~- ~A and ~ runs over the algebra ~(~)of a l l
(bounded linear) operators acting on ~ .
The notion of finite operator was introduced by J.P. Williams in
~7~, where he proved the following result.
THEOREM 1. These are equivalent conditions on an operato r ~ :
(ii) 0 belongs to the closure of the numerical range of ~A~)
for each X in ~C~)"
(iii) There exists I~(~)*such that I(~)=~--ll~ll an__~d
6- A . The origin of the term f i n i t e is this: if A C - ~ ( ~ )
has a finite dimensional reducing subspace ~ ~ { 0~ ,
{ei}i= 4 (~=~ ~) is an orthonormal basis of ~, and
~(B)=(~/~)~. (~8 i 8~ , then ~ satisfies the conditions of
Theorem 1 (iii) and therefore ~ is a finite operator.
Thus, if ~=~T~ZC~):T has a reducing subspace of dimension ~ ( , = 4 , ~ , . . • ) , then ~= U ~ i s a subset
of the family (Fin) of all finite operators; furthermore, since {~in} is closed ~ ~ (~4) [7], ~-C (~in}.
COnJeCTURE I (J.P.winiams [7] 1. (Fin) = ~-
As we have observed above, if A q~ , then it is possible to
construct ~ a~ in Theorem 1 (iii) such that ~ does not vanish
identically on ~), the ideal of all compact operators. J.H.An-
derson proved in [1] (Theorem 10.10 and its proof) that this fact
can actually be used to characterize ~ : A~ if and only if
there exists ~(~)* such that ~(~)=~ = II~ , and ker ID
241
D~, but ker $~ ~ ( ~ ) ; furthermore, if A ~ ( F i n ) \ ~ ,then
every ~ as in Theoremn ~-~1 (iii) is necessarily a s i n g u 1 a r
functional (i.e., ker ~O~- This is true, in particular, if
Since (Fin) D ~- , it is plain that (Fin) contains all
q u a s i d i a g o n a 1 o p e r a t o r s (in the sense of Hal-
mos; see [4] ). Moreover, (Fin) contains every operator of the form
A ='T' ®B *K, (*)
where T ~(~^~ (for some finite or infinite dimensional subspace
Wo of infinite codimension), ~ = 4 ~ for a suitable uni-
formly bounded sequence ~ ~=~ of operators ac t i ng on f i n i t e
dimensional subspoces (i.e , B is a b i o c k - d i a g o n a i
operator [4] ) and ~ is compact
Let ~0 denote the family of all operators of the form (*). It
is apparent that
and
CONJECTURE 2
then there exits
(D'A'Herrer°)'(Pin)1= ~'0 ; moreover, if A~(~ I
K £~(%) and Q quasidia~ona! such that
A-K=A (Q Q®Q® ,,.'),
Several remarks are relevant here:
(1) J.W.Bunce has obtained several other equivalences, in addi-
tion to the three given by Theorem I (see [2] ).
(2) ~- properly includes U~4(~) [3, p.262], [5 , Example
111.
(3) In [I, Corollary 10.8], J.H.Anderson proved that every uni-
lateral weighted shift is a finite operator, by exhibiting a functio-
nal ~ satisfying the conditions of Theorem I (iii). Recently, D.A.
Herrero proved that all the (unilateral or bilateral) weighted shift
operators belong to ~-[5].
242
(4) Suppose that A Q ~- ; then there exists a sequence
{P~=~ of non-zero finite rank orthogonal projections such
that IIAP~ - P~A II "~0 (~-~oo). Passing, if necessary, to a subse-
quence we can directly assume that ~--~H (weakly, as n-~oo )
for some hermitian operator H , 0 ~ H % ~ . It is easily seen that
commutes with H . If ~ has a non-zero finite rank spectral
projection, then A~ ~ . If either ~=0 or ~ =~ , then it is
not difficult to show that ~ 6 ~0 •
(5) According to a well-knQwn result of D.Voiculescu, A - K ~
Ao~QoQo ) (as in (**~ if and only if the C~-algebra gene-
rated by ~) in the Calkin algebra ~(~)/~(~) admits a
• -representation ~ such that ~-~(~)= ~ ~6]. Suppose that
A ~ ( F ~ ) ' . t~ it posslble to ~e the si~na~ f~c,t!,,onals
provided b~ Theorem 1 (iii) an d a Gelfand-Naimark-Sega ! t,ype const-
ructio n in order to construct a ~ -representation ~ with the
desired properties (i~e., so that ~, ~(~) is quasidia~0nal)?
(6) Is it possible I at l east~ to show that the existence of
such a S~ular functional implies that, for each ~>0~ m ad-
~S, IIII ~ finite rank pert~bat,ion F~ , ~ t h nF~< B , such t~t
A-F ~A A _ _ ~ ~ , where ~ acts on a non-zero finitedimen@io-
nal subspace? (An affirmative answer to this last question implies
that (Fin) ~ = ~ ).
(7) Any partial answer to the above questions will also shed
some light on several interesting problems related to quasidiagonal
operators.
REFERENCES
I. A n d e r s o n J.H. Derivations, commutators and essential
numerical range. Dissertation, Indiana University, 1971.
2. B u n c e J.W. Finite operators and amenable C*-algebras.
- Proc.Amer. Math.Soc.~1976, 56, 145-151.
3. B u n c e J.W., D e d d e n s J.A. C*-algebras generated
by weighted shifts. - Indiana Unlv.Math.J.~1973, 23, 257-271.
4. H a 1 m o s P.R. Ten problems in Hilbert space.-Bull.Amer.
243
Math.Soc.~1970, 76, 887-933.
5. H e r r e r o D.A. On quasidiagonal weighted shifts and appro-
ximation of operators.-Indiana Univ.Math.J. (To appear).
6. V o i c u 1 e s c u D. A non-commutative Weyl-von Neumann
theorem. - Rev.Roum.Math.Pures et Appl.~1976, 21, 97-113.
7. W i 1 1 i a m s J.P. Finite0perators. - Proc.Amer.Math.Soc.~
1970, 26, 129-136.
DON[INGO A.HERRERO Arizona State University
Tempe, Arizona 95287
USA
This research has been partially supported by a Grand of the
National Science ~oundation
244
4.39. THE SPECTRUM OF AN ENDOMORPHISM IN A COMMUTATIVE
BANACH ALGEBRA
The theorem of H.Kamowitz and S.Scheinberg D] establishes that
the spectrum of a ~0n~eriodic automorphism ~ : A-~A of a semi-
simple commutative Banach algebra A (over 6 ) contains the unit
circle ~ . Several rather simple proofs of the theorem have been obtained besides the original one ([2],[3]e.g.) and its various gene-
ralizations found (see e.g.[4],[53,[6]...). It is easy to show that un-
der the conditions of the theorem the spectrum is connected, At the
same t~me all "positive" information is exhausted, apparently, by
these two properties of the spectrum. There are examples (see [7~,
[8]~[9~) demonstrating the absence of any kind of symmetry structure
in the spectrum even if we suppose that the given algebra is regular
in the sense of Shilov.
Let for instance ~ be a compact set in ~ lying in the annu-
lus { : } and containing {~: ~I~ l l Zl and equal to the closure of its interior int K. Denote by A the fami-
ly of all functions continuous on ~ and holomorphic in int K. Clea-
rly, A equipped with the usual sup-norm on ~ and with the point-
wise operations is a Banach algebra. The spectrum of multiplication
by the "independent variable" On A evidently coincides with ~ .
On the other hand the conditions imposed on ~ imply that A is a
Banach algebra (without umit) with respect to the convolution
4 F ~ ~(~) ~(~) ......
corresponding to multiplication of the Laurent coefficients. This al-
gebra being semi-simple, its maximal ideal space can be identified
with the set of all integers. Adjoining a unit to A thus turns it
into a regular algebra. Obviously the above mentioned operator on A
is an automorphism.
1..A.re t.here other necessary conditions, on th.e. spectrum of.~ non-
periodic automorph%sm of % semi-s~mple commut.a.t~.v.~ ' Banach al~ebra
besides the t~0........m.entioned above? In particular, is it obligatory for
the spectrum to have interior points when i% differs from ~ ? It is
known in such cases (see ~7],KB]) that the set of interior points
245
may not be dense in the spectrum and may not be connected either.
Let M A be the maximal ideal space of a commutative and semi-
s~mple Banach algebra A . An automorphism T of A induces an
automorphism of the algebra C ~ C(M A) . The essential meaning of
the Kamowitz - Scheinberg theorem is that ~c(T) c ~A (T) . It is
natural from this point of view to study the inclusion gc (L) c gA ( ~ )
for a more general class of operators L . The case of weighted auto-
morphisms I,@ ~e~ W. T~ with ~ an invertible element of A
has, for example~been studied in [10]. It turns out that the in-
clusion does not hold for this class of operators.
2, Does the spectrum of L=~, cg~tructed for a non-periodic
automorphism T , contain any circle ' centred at %he origin? If it
does then we obtain an instant generalization of the theorem of Ka-
mowitz and Scheinberg.
The spectrum of operators, looking like l. , acting on the al-
gebra of all continuous functions on a compact set has a complete
description[11 . I f A is also a u n i f o = algebra then Thus 6" A (~) ~ ~ ( ], ) provided that I, is a weighted automorphism
of two uniform algebras ~ and ~ having the same maximal ideal
spaceo
3. Let A be a closed subalgebra of a semi-simple commutative
Banach algebra ~ and let ~A--~ MB . Let ~ be a weighted auto-
morphism of A and ~ simultaneously.
IS it true then that 6~ A(~) = ~B ( h ) ?
We OON~aT~R that this question has a negative answer.
The spectrum of an endomorphism apparently does not have any
particular properties even if we suppose that A is a uniform al-
gebrao Given two oompaota ~4 and ~9 it is easy to obtain an endo-
morphism with the spectrum either ~4"U ~ or ~I' ~% . The only
obvious property of spectra is that Iw belongs to the spectrum,
when ~ ranges over its boundary and ~ over the set of non-nega-
tive integers.
4. Let ~ be a compact subset of ~ satisfying ~6 ~ for
~= ~,~,.., and for all points ~ in the boundary of ~ .
Is there an endomorphism of a uniform algebra whose spectrum is
eq~l to ~ ?
Spectra of endomorphisms of uniform algebras (and even those
for weighted endomorphisms) can be described pretty well under the
246
additional assumption that the induced mapping of the maximal ideal
space keeps the Shilov boundary invariant (see D2], where one can
find references to preceding papers of Kamowitz). Roughly speaking,
things, in this~case, are going as well as in the case of Banach al-
gebras of functions continuous on a compact set. The situation
changes dramatically when the boundary, or only a part of it, pene-
trates the interior. In such circumstances it is common to begin
with the consideration of classical examples. Let D be the unit
disc in C , let A(~) be the algebra (disc-algebra) of all func-
tions continuous on the closure of ~ and holomorphic in ~ , and
let H~(~) be the algebra of all functions bounded and holomorphic
in ~ . Both algebras A(~) and H~(~) are equipped with the sup-
nOl~.
5. Every endomorphism of A(O) induces a natural endomorphism
of ~ .
Do the spectra of thes e endomorphisms coincide?
In this connection it is worth-while to note that the answer to
an analogous question concerning the algebra of all continuous func-
tions on a compact set and the algebra of all bounded functions on
the same set is in the affirmative ~3~. The proof of this result
uses, however, a full (though comparatively simple) analysis of the
possible spectral pictures depending on the dynamics generated by the
endomorphism.
The interesting papers ~4~, ~, ~6~ of Kamowitz (see also [6S,
[9~) deal with spectra of the endomorphisms of A (D) whose induced
mappings do not preserve the boundary of ~ . In the non-degenerate
case the spectrum has a tendency to fill out the disc.Discrete and
continuous spirals as well as compacta bounded by such spirals may
nevertheless appear as the spectrum of an endomorphism. (But only
the spirals can appear in the case of Mobius transformations).
6. Is the spectrum of an endomorphism of the disc-algebra a se-
mi-~rouu (with respect to multiplication in ~)? What kind of semi-
~rouDs can arise as spectra.?
7. Is it possible to say something concernin~ the spectra of
9ndomorphisms of natural multi-dimensional generalizations of the
disc algebra?
Note that in the one-dimensional case the theory of Denjoy-
Wolff and the interpolation theorem of Carleson-Newman are often
247
involved in the question.
The problem of describing spectra for weighted automorphisms is
closely related with an analogous one for the so-called "shift-type"
operators which have been studied by A.Lebedev ~7~ and A.Antone-
rich Dsl Let A be a uniform algebra of operators on a Banach space X .
An invertible operator U on X is called a "shift-type" operator
if UAU A . UB lly X a B nach space of f ctio and A
is a subalgebra of the algebra of multipliers for ~ . The transfor-
mation ~ ~ U. ~U ~ determines an automorphism T of ~ which in-
duces the mapping ~: ~A ~ M A . It is assumed that:
I) the set of ~-periodic points is of first category in the
Shilov boundary 8A ;
2) the spectrum of U:X ~ ~ is contained in S A ;
3) each invertible operator @:X--~X , ge~ is invertible as
an element of A ;
4) the topological spaces ~A and ~ have the same stock of
clopen (closed and open) ~ -invarlant subsets.
Then ~(@~)-~-~(~T) for all ~ in ~ [19].
We con,~ecture that Condition $ is superfluous.
If this were true it would be possible (in view of [1 I]) to ob-
tain a complete description of @~ (@~ . It is reasonable to ask
the same question for other algebras A besides the uniform ones.
REFERENCES
I. K a m o w i t z H., S c h e i n b e r g S. The spectrum of
automorphisms of Banach algebras. - J.Punct.An., 1969, 4, N-2,
268-276.
2. J o h n s o n B.E. Automorphisms of commutative Banach algeb-
ras. -Proc.Am.Math.Soc., 1973, 40, N 2, 497-499.
3. ~ e B ~ P.H. HOBOe ~o~asaTe~cTBo TeopeM~ O6 aBTOMOp~zsMax 6a-
EaXOBHX a~re6p. -BeCTH.MFY, cep.MaTeM., Mex., I972, ~4, VI-72. 4. ~ e B ~ P.H. 0d aBTOMOp~SMaX 6aHaXOBRX a~re6p. - ~ym~.aHax~8
E ero np~., 1972, 6, ~ I, 16-18.
5. ~ e B E P.H. 0 COBMeCTHOM cneETpe HeEoTopb~x EOMMyTI~py~E~X oEepa- TOpOB. ~ccepT~, M., 1978.
6. r o p ~ H E.A. EaE BR~JL~ET cHeETp SH~OMOp~EsMa ~cE-a~re~p~? - 3au.HSyqH.ce~m~o~0~, 1983, 126, 55-68.
248
7. S c h e i n b e r g S. The spectrum of an autemorphism. -
Bull.Amer.Math.Soc,, 1972, 78, N 4, 621-623.
8. S c h e i n b e r g S. Automorphisms of commutative Banach
algebras. - Problems in analysis, Princeton Univ.Press., Prin-
ceton 1970, 319-323.
9. r o p~ H E.A. 0 cne~Tpe SH~OMOp~SMOB paBHoMepHHx axre6p. -
B EH. : Tes~cH ~oEx.~oH~ep"TeopeT~xecz~e ~ npm~a~HHe BOnpOC~ Ma--
TeMaT~E~" TapTy, I980, I08--II0.
I0. E ~ T 0 B e p A.E. 0 cne~Tpe aBTO~Op~sMOB C BecoM ~ TeopeMe
EaMoB~um-~s/m6epra. - ~yHE~.aHa~Hs ~ ero np~., 1979, 13,~ I,
70-71.
II. E ~ T o B e p A.E. Cn~Tpax~--e CBOICTBa aBTOMOp~SxOB C Be-
COMB paBHo~eps~x axre6pax. - 3an.H~.CeM~H.~0MH, I979, 92,
288-293.
I2o E E T O B e p A.K. CneETpax~HHe CBO~CTBa rOMOMOp~SMOB C BecoM
B a~re6pax Henpep~Bm~X ~ ~ ~x np~o~eHH~. - 3an.HayqH.ce-
~.~I01~, 1982, 107, 89-103o
13. I{ E T 0 B e p A.E. 06 oiiepaTopax B C (~ , ~H~yI~IpOBaHHIgX l~a~ -
Emm~ OTo6pa~eH~2m. - ~JE~I~.asax~s E ero np~., 1982, 16, ~ 3,
61-62.
14. K a m o w i t z H. The spectra of endomorphlsms of the disk
algebra. - Pacif.J.~ath., 1973, 46, N 2, 433-440.
15. K a m o w i t z H. The spectra of endemorphisms of algebras
of analytic functions. - Pacif. J.Math. 1976, 66, N 2, 433-442.
16. K a m o w i t z H, Compact operators of the form @ C@ . -
Pacif.J.N~th., 1979, 80, N I, 205-211.
I7. ~ e 6 e ~ e B A.B, 06 onepaTopax T~na BSBemeHHOrO C~BEra.
~ccepT~, M~HCE, 1980.
18. A H T O ~ e B ~ ~ A.B. 0nepaTop~ co C~B~rOM, nopom~e~M ~e~c-
TB~eM EOMIIaETHOI l~y211H H~. - CE61~pCE.MaTeM°EypH°I979, 20, ~ 3,
467-478.
19. E ~ • 0 B e p A.E. 0nepa~op~ IIO~CTaHOBE~ C BecoM B 6aHaXOBHX
Mo~y~Ex H8~ paBHOMepH~M~ ax~e6pa~m (B negate). E. A. GORIN CCCP, 117234, ~ocFma ~eH~cK~e rop~
(E.A.IDPHH) MexaHzEo-~aTe~aT~ecEz~ ~ry2~Te T
MOCKOBCE~ rocy~apcTBeRR~ yHzBepc~TeT
A. K. KITOVER (A.K.EETOBEP)
CCCP, 191119, ~eH~Hrpax,
y~ KOHCTaHTNHa 3aC~OHOBa,
~.14, KB.2.
249
4.40. COMPOSITION OF INTEGRATiON AND SUBSTITUTION
Consider a continuous function ~ on [0,1] satisfying~(0)=0,
0~ ~(~) ~ ~ . The function ~ defines a bounded linear operatorI~
On the space C ~o,1~ of all continuous functions on ~0,I~:
0
Recall that a bounded operator T is called quasinilpotent if
£%1I T"lt ¢ = o .
PROBT,~.'~. Describ,~ funct ions ~ correspondin ~ to quas in i lpo tent
operators .
Clearly I~ is quasinilpctent provided
0~< ~ ~ 1 (2 )
Does the inverse., conclusion hold? An analogy with the theory
of matrices provides arguments in favour of the affirmative answer.
Let I@~}~-4 be a nilpotent matrix with non-negative elements.
It follows from the Perron-Frobenius theorem that it can be trans-
formed to a low-triangular form with zero diagonal by a permutation
of the basis. The obtained matrix I ~I defines an operator On ~
with ~(~) < ~ .
Consider now a natural generalization of (I):
0
(3)
where the kernel K >~ 0 is continuous. Orientation preserving ho-
me~nerphisms of EO,1] replace permutations of the basis in the fi-
nite-dimensional case and preserve the inequality ~(X)~ X •
250
Consider a counter-argument to the conjecture. IfI~ is not qua- s i n i l p o t e n t then by Y e n t s c h ' s theorem t h e r e e x i s t ~ ~ 0 and a non- zero continuous function S~ 0 such that
@(x)
o
Suppose q}~C @° ( [ 0 , 1 ~ ) . Then evidently~ C ~ ( K0,1]~and;~(~ ' (O)= O, K~O,~.., ~ because ~(0)~0 . If it were possible to prove that
belongs to a quasianalytic class (under some natural restrictions on
), it would imply clearly that ~ ~ 0 which contradicts
Yentsch's theorem. Are there conditions on ~ no t demanding @(~)~X
but such that a~y solution of ($~ belonss to la lquasianal,ytic Carleman
class? If yes, then there exists ~ such that ~(~o) • $o for a
point to in EO,I] but neve~heless I ~ is q~slnilpotent.
YU. I. LYUBIC
(D.H.~) CCCP, 310077 Xap~EOB
n~.~sepz~HcEozo 4
Xap~EOBCE~ IOCy~apCTBeHmm~ yH~Bepc~TeT
CHAPTER 5
HANKEL AND TOEPLITZ OPERATORS
A quadratic form is called Hankel (resp. Toeplitz) if entries
of its matrix depend on the sum (resp. the difference) of indices
only. These forms appeared as objects and tools in works of Jacobi,
Stleltjes (and then Hilbert, Plemelj, Schur, Szego, Toeplitz ...).
They play a decisive role in a very wide circle of problems (various
kinds of moment problems, interpolation by analytic functions, in-
verse spectral problems, orthogonal polynomials, Prediction Theory,
Wiener-Hopf equations, boundary problems of Function Theory, the ex-
tension theory of symmetric operators, singular integral equations,
models of statistical physics etc.etc.). It wBs understood only later
that the independent development of this apparatus is a prerequisite
for its applications to the above "concrete" fields, and Hankel and
Toeplitz operators were singled out as the object of a separate
branch of Operator Theory. This branch includes:
- techniques of singular integrals ranging from Hilbert,
M.Riesz and Privalov to the Helson-Szego theorem discovered as a fact
of Prediction Theory, and to localization principles of Simonenko
and Douglas;
- algebraic schemes originating from the fundamental concept of
symbol of a singular integral operator (Mihlin), from the semi-mul-
252
tiplicative dependence of Teeplitz operator cn its symbol, (Wiener-
Hopf), and culminating in the operator N-theory;
- methods and techniques of extension theory (Krein), which
have attracted a new interest to metric properties of Hankel opera-
tors and to their numerous connections;
- other important principles and ideas which we have either
forgotten or overlooked cr had no possibility to mention here.
The inverse influence of Hankel and Toeplitz operators is also
considerable. For example, many problems of this chapter fit very
well into the context of other chapters: Banach Algebras (Problem
5.6), best approximation (Problem 5.1), singular integrals (Problem
5.14). Problems 2.11, 3.1, 6.6, 10.2 can hardly be severed from
spectral aspects of Toeplitz operators, and Problems 3.2, 3.3, 4.15~
4.21, 8.13, S.6, from Hankel operators. ~ny problems related to the
Sz.-Nagy-Foia~ model (4.9-4.14)can be translated into the language of
Hankel-Toeplitz (possibly, vectorial) operators, because functions
(~) of the model operator T@ coincide essentially with the
Hankel operators ~G~ ~ , and the proximity of model subspaces ~ _
and K@~ can be expressed in terms of the Toeplitz operator TG*
etc.
Hankel-Toeplitz problems assembled in this book do not exhaust
even the most topical problems of this direction *), but contain many
interesting questions and suggest some general considerations. Many
of the problems are inspired by some other fields and are rooted
there so deeply that it is difficult to separate them from the cor-
responding context. We had to place some Hankel-Toeplitz problems
(not without hesitation and disputes) into other chapters. Examples
can be found in Chapter 3 (3.1, 3.2, 3.3). Moreover, we believe that
*) To our surprise nobody has asked, for instance, whether
every Toeplitz operator has a non-trivial invariant subspace...
253
Problem 3.3 is one of the most characteristic and essential problems
of exactly t h i s Chapter and we hope that the reader looking
through this chapter will turn to Problem 3.3 as well.
Problems 5.1-5.3 deal with metric characteristics of Hankel
operators (compactness, spectra, s-numbers). In connection with
Problems 5.3 and 5.7 concerning operators acting n o t in H~we
should like to mention recent investigations of S.Janson, J.Peetre
and S.Semmes and of V.A.Tolokonnikov (Spring 1983) who have found
(X-~Y) -continuity criteria (in terms of symbols) for Hankel and
Toeplitz operators in many non-Hilbert function spaces X, ~ .
Problems 5.4 and 5.5 treat similarity invariants and some pro-
perties of the calculus for Toeplitz operators.
Problems 5.6, 5.13 are related with localization methods, prob-
lems 5.8-5.10, 5.15 deal with vectorial and multidimensional variants
of Toeplitz operators and with related function-theoretic boundary
problems.
Problems 5.11-5.14 treat "limit distributions of spectra"
(asymptotics of Szego determinants, convergence and other properties
of projection invertibility methods etc).
The theme of Problem 5.16 may be viewed as a non-commutative
analogue of Toeplitz operators arising in the theory of completely
integrable systems.
The field of action and the multitude of connections of Hankel-
Toeplitz operators are so impressive that it became fashionable no-
wadays to find them everywhere - from bases theory to models of
Quantum Physics and ... even where they really do not cccur-~
254
5oio old
APmOX~aTIO~ OF B0~DED F~CTIO~S BY E~TS
OF M~+ C
Every sequence { ~} ~4 of complex numbers defines a Hankel
matrix ~={ ~+k-~ ~ j,K~4 which is considered as an operator
in the Hilbert space ~ . By Nehari's theorem ~ is bounded if and
only if there exists a function ~ in the algebra ~c~ ^of all
bounded and measurable functions on T such that [~=~ (-~) ,
~$= 4, 2,..., ~(~) being the Fourier coefficient [ ~'~n~t~ of
. This function is uniquely determined up to a summand from the
H ~ Hardy algebra . The norm ]~ of ~ coincides with diet (~ ~). Given ~ ~C let ~(~) be the Hankel operator cor-
responding to the sequence {[~>~4 ~ [ = ~(-~) ~ ~= 4,~,,,.
Usual compactness arguments imply that for every ~ ~-~ there
exists ~ E H ~ such that
tial spectrum of ( r*r)~ ~---~ + ~
A criterion of u n i q u e n e s s of the best approxima-
tion ~ as well as a description of all such ~s in the non-uni-
queness case have been obtained in [2 2 . Hartman has shown [3] that
F is compact iff there exists a function ~ in the algebra
of all continuous functions on T such that F=P(~) Moreover,
if r is compact then for every 6>0 there exists ~ C sa-
tisfying r(~6)= r and ~C %Ir~ + 6 •
The results of Nehari and Hartman easily imply the following re- sult discovered independently by Sarason [4] : the algebraic sum
H~+C is a closed subalsebra oZL~Csee ~so [~] .here ~ proper- ties of this subalgebra are disussed), Hartmau's result implies also
the following characterization of ~+C : an element ~ in ~ belongs
t o ~+C if and o n l y i f C(~ ) is a compact o p e r a t o r . Let S~(F), ~=4,~...denote S-numbers of Vcounted with mul-
tiplicities and let S~(F) be the least upper bound of the essen-
(cf°[6], §7). Clearly, S~(F)~ S~(r) as
co THEOREM. Let ~L and ~ = ~(~) . Then
civet (~, H +C) =S~(r). (1)
255
in ~ \ ( H ~ + C ) consider the following
~-numbers
ponding to the Hankel operators ,F'c,~(~) l y Ho f, .~H ) OH1 c . . . ~ H +C
H~+C . Therefore dist (~,H~C)
On the other hand it has been shown in
of finite rank ~< k . Clear-
and ~U~I Fill , is dense in Co
- - ~ ~st C J~ ,H,~).
[2C] that diet (;~ ,H k) = $k (.r) provided SI< (~) >~K÷I(V ) . Hence (I) holds in case 3- In case 2 we
, H~) , which implies (1) have (see [2c], ~5) S~(~)= ~St (~
aV~ also. At last, in case 1 we obviously h e l~ t ( j~ , H )=~st(~, H%C).
aive~ j~eL denote by M~ the set of ~ll geH +C satisfying II ~ .~h ~ ----- Soo(P(~)) " In case I M~ n H ~ ~ ~ and
in case 2 M~{]Hm ~ ~ . A necessary and sufficient condition for
M~ {l H~ to be a one-point set as well as a description of
when it contains more than one element, have been obtained.
As for case 3 it is VNENOWN:
a) Ca._nn M~ be emptE for some ~E Lce and if so how to des-
cribe such ~ ?
b) Can M~ consist of a single element for some ~ i_nn
a@ co L X(.H +C)?
c) I_~f m~ ~ ~ then is it possible to describe at #east a
. . . . . . M~ as a "selected" part of M~ just as i n case ,2, ,,(w'ith N H "selected" part)?
Given a function
possibilities.
CASE I. S~(F)=IFI , i.e. F does not have
greater than ~eo (~)"
CASE 2. There exists only a finite number ~ ,
cf S-numbers greater than Soo ~r) •
CASE 3. The set of S-numbers to the right cf Soo(~s infinlte~
Formula (I) is a simple consequence of theorem 3.1 from [2a]
but for the purpose of this note it is more convenient to connect
it with the investigations of [2c]. co
For any positive integer k let H K denote the set of all
sums ~ = ~+~ , where ~ H c° and ~ is a rational function of
degree % k , having all its poles in the open unit disc ~ and
vanishing at infinity. The set H~ is neither convex nor li-
near. ~lls set coincides with the family of symbols ~ corres-
256
Clearly ~ ~& ~ for #~ [°~=\(H~+C) if and only if there
exists ~C such that for #4 #-~ case ] holds, i.e.
Ir(~)l = S~(~(~4)) ° Question b) remains interesting for cases I and 2
also. The matter is that there are situations when card M~=oo
but card (M~ ~ H ~)=~ (in case I) and card (M~ ~H ~
(in case 2). Indeed, let for example ~ be an inner function with
singularities on an arc ~c T , ~(A) < 4 , and let g be any
function in C satisfying ~(r)--4 on A and l~(r)l<4 for
~ T\ A . Then ~6----geC and setting # ----~/6 we have
S~ (r(#))--~i~t (#, HtC)=~Li~t (#, H5 =] = fig - 1#~. Ho.ever, I1#-~11>~ for every &~H ~ , t 1 ~ > 0 .
A l m o s t e v e r y t h i n g s a i d above can be g e n e r a l i z e d t o t h e case o f
matr~x-valuod fu~otions F-- (#~) .~t~ e~tries ~ belongi~ to , or C • In this case the norm IIF II of F e L~,# ~
should be def~ne~ as ~ ~o I F(~)I where I AI stands for ~e ~ T Hilbert-Schmidt norm of A In connection with these generaliza-
tions we refer to [2d].
REFERENCES
I. N e h a r i A. On bounded bilinear forms. - Ann.~th., 1957,
(2), 65, 153-162.
2. A~ a M ~ B.M., Ap o B ~.B., Ep e ~H M.F. a) BecEo-
He~Hae rsazeaeBH MaTpmm ~ OdO6~eHH~e ssaa~ KapaTeo~op~-~e~epa
$.PHcca. - ~F~m/.aaaa. ~ ero np~a., I968, 2, I, I-I9; b) BecEo-
He,He rss~eaeBH MaTpmm ~ odo6meHHHe sa~a~ EapaTeo~op~-@e~epa n
M.~ypa. - ~/~U.asa~. ~ ero npHa., 1968, 2, 4, 1-17; ~) A~aa~T~-
~ecE~e CBO~CTBa Hap ~Ta raHEeaeBa onepaTopa n odo6me~a~ sa~a-
~a ~ypa-Ta~ar~. - MaTeM.cd., I97I, 85 (I28), ~ I9, 38-78; d) Bec-
EoHe~mae 6ao~Ho-rssEeaeB~ MaTpH~ ~ c~saam~e c ~ npo6~e~ ~po-
~oaxem¢~. - MsB.AH ApMCCP, I97I, YI, ~ 2-3, 87-II2. S.H a r t m a n P. On completely continuous Hankel matrices. -
Proc.Amer.Math.Soc., 1958, 9, 862-866. co
4. S a r a s o n D. Generalized interpolation in H . - Trans.
Amer.Math. Soc., 1967, 127, 179-203.
5. S a r a s o n D. Algebras of functions on the unit circle. -
Bull.Amer.~athoSoc., 1973, 79, N 2, 286-299.
.
257
r o x d e p r Ho~., Kp e ~ H M.r. BBe~eHHe B Teop~no~e~-
HNX HecaMoconlo~eH~x oHepaTopoB. -- M., Hs~Ea, 1965.
V. M. ADAB~AN
D. Z.AROV
( ~. 8.APOB)
M. G. KREIN
( M . r . K P ~
CCCP, 270000, 0~ecca,
0~eccEE~ roc.yHHBepcHTeT;
CCCP, 270020, 0~ecca,
0~eccE~ ne~.EHCTETyT;
270057, 0~ecca,
y~.ApTeMa 14, EB.6
COMmeNTARY BY THE AUTHORS
Soon after the Collection "99 unsolved problems in linear and
complex analysis" LO~&I, vol.81 (1978) was published S.Axler, I.D.Berg,
N.Jewell and A.Schields wrote an important paper on the theory of ap-
proximation of continuous operators in Banach space by compact opera-
tors, where they obtained, in particular, the answers to questions
a) and b) of the Problem.The answers to both questions turned out to
be negative. So for every function ~ L ~ the set M~ is not void and moreover for any ~ co ~ L \(H +C) the set M~ is infinite.
These results were obtained in Eli as consequences of two re-
marquable propositions, which we formulate for Hilbert space opera-
tors only.
THEOREM. Let [~) be a seque.nqe of linear compact operators
on Hilbert space converging in the strong tooolo~y to a bounded opera-
tor T : S-~ ~=T " Supp°sIe, also.that T = s - ~ • Then there
exisSs a sequence ~ J~ of non-negative numbe.rs such that ~=~
and
I T - K I = S=(T)
where K = T % .
258
COROLLARY. Let T a n d {T~} satisf E the conditions of the
theorem. Suppose also that ~ is not compact. Then there exist two
sequenc.,e,s, [6~}. { 8~} oT,,,.,.non-ne~ative~ numbers such that, ~- O,n=
= = 4
I T-Ko, I=IT-K I= s LT)
where ~@=~J~ , ~=[6~T~ , a n d •
The indicated propositions permit also to give answers to ques-
tions of type a)and b)for matrix-functions ~: ~x~(C~×~ + H~l AS we got to know from [~ question a) was raised before us
by D.Sarason.
Question c) concerning the description of the set ~ or
its "selected" part remains open.
REFERENCE
7. A x i e r S., B e r g I.D., J e w e I I N., S h i e I d s
A. Approximation by compact operators and the space H°°+ C . -
Ann.~th., 1979, 109, 601-612.
EDITORS' NOTE
Question a) is solved also in a different way by D.Lueckim~ [8].
Let us mention also a recent reset of O.S~udberg [9] asserting that
the algebra H~+ BUC (BUC is the space of bounded ~niformly continu-
ous functions on ~ ) does not have the best approximation property,
i.e, there exists~ such that there is no ~ in HC~+ BUC saris-
REFERENCES
8. L u e c k i n g D. The compact Hankel operator form an M -ideal
in the space of Hankel operators° - Proc.Amer.Math.Soc., 1980, 79,
222-224.
9. S u n d b e r g C. No°+ BUC does not have the best approximation
property. Preprint, Inst.Nittag-Leffler, 13, 1983
259
5.2. QUASINILPOTENT HANKEL OPERATORS
Hankel operators possess little algebraic structure. This fact
handicaps attempts to elucidate their spectral theory. The following
sample problem is of untested depth and has some interesting function
theoretic end operator theoretic connections.
PROBLEM. Does there exist a non-zero quasinilpotent Hanke 1
operator?
A Hankel operator A on H ~ is one whose representing matrix )~
is of the form (~{+I ~'i'-0 with respect to the standard ortho-
normal basis. A well known theorem of Nehari shows that we may rep-
resent A as ~=$~=~MqlH ~ where P is the orthogonal projec-
tion of ~£ onto HZ;~ is the unitary operator defined by (~)(Z)=~(~),
for ~ in ~£ , and ~ denotes multiplication by a function ~ in
The symbol function ~ and the defining sequence ~ are con- A
nected by ~(~) = ~ , ~= 0,~,~, . The following observation
appears to be new and provides a little evidence against existence.
PROPOSITION. There does not exist a non ' zero nilpotent Hankel
operator,
PROOF. Suppose A~0 and is nilpo±ent. Then ker A is a non
zero invariant subspace for the unilateral shift U, since AU--U*~.
By Beurling's theorem this subspace is of the form ~ for some
non constant inner function ~ . Thus, with the representation
above, we have =0 and hence ~(~) = 0 for ~ . So the
symbol function ~ may be written in the factored form ~=~
for some ~ in H~ , and we may assume (by cancellation) that
and ~ possess no common inner divisors. The operator ~ is a
partial isometry with support space ~= ~e ~ ~ and final space
~= ~ ~ ~ al , where ~)= ~(~) . By the hypothesised nil-
potence of ~ -~ = $ ~ it follows that for some non zero func-
tion ~ in ~0 ~4 belo s to W~ . Hence ~ divides ~ , and . . . . ng . .
~=W~4 with ~ in H . Since ~ belongs to ~T we have
P(~) = 0 . This says that the Toeplitz operator T£,~ has
non t r i v i a l k e r n e l . But k2/£ T * -- k ~ T = k ~ T =
, and we have a contradiction of Coburn's alternative:
Either the kernel or the co-kernel of a non zero Toeplitz operator is
trivial. @
260
Function Theory. The evidence for existence is perhaps stronger.
There are many compact non self-adjoint Hankel operators, so per-
haps a non zero one can be found which has no non zero eigenvalues.
A little manipulation reveals that ~ is an eigenvalue for ~ if
and only if there is a non zero function ~ in ~ (the eigenvec-
tot) and a function ~ in ~£ such that
+
Since continuous functions induce compact Hankel operators it would
be sufficient then to find a continuous function ~ which fails to
be representable in this way for every ~ 0 . Whilst the singular
numbers of a Hankel operator A (the eigenvalues of (~*A) ~l~ )
have been successfully characterized (see for example [3 , Chapter
5] ), less seems to be known about eigenvalues.
O~erator theory. It is natural to examine (I) when the symbol
can be factored as q=~ (cf. the proof above) with ~ an inter- I
polatingBlaschke product. The corresponding Hankel operators and
function theory are tractable in certain senses (see[l] ,[2~Part 2]
and[3,Chapter 4] ), partly because the functions (~-~A~l~)41~(~-~) -~,
where ~,Ai, are the zeros of Wv , form a Riesz basis for ~ O~.
It turns out that ~ is compact if ~(~4), ~CA~), .
is a null sequence. A quasinilpotent compact Hankel operator of
this kind will exist if and only if the following problem for opera-
tors on ~ can be solved.
PROBLEM. Construct an interpolating sequence A~ and a compact
dia~o~l o~erator ~ so that the ' equation ~Xx ~ ~ admits no
proper solutions ~ i_~n ~ whe n ~0 . Here
operator on ~ associated with ~A~
ting matrix
(t. i A,i,l~.) '~1~' (.,I- I ~ ~) '~/~'
is the bounded(!)
determined by represen-
REFERENCES
I. C 1 a r k D.N. On interpolating sequences and the theory of
Hankel and Toeplitz matrices. - J.Functicnal Anal. 1970,5,247-258.
261
2. H r u ~ 6 ~ v S.V., N i ~ o l's k i i N.K., P a v I o v
B.S. Unconditional bases of exponentials and of reproducing ker-
nels. - Lect.Notes Math. 1981,N864, Springer Verlag.
3. P o w e r S.C. Hankel operators on Hilbert space. - Research
Notes in ~athematics. 1982. N 64, Pitman, London.
S.C.POWER Dept. of Mathematics
Michigan State University
E.Lansing, MI 48824
USA
Usual Address:
Dept.of Mathematics
University of Lancaster
Bailrigg, Lancaster LAi 4YW
England
262
5.3. HANKEL OPERATORS ON BERGNAN SPACES
IA
Let ~/~ denote the usual area measure on the open unit disk
17 • ~he B e r ~ ~ ,~ce t . t (1?1 i s the s u b s ~ o e of t } ( l ? ,~A) c o n s i s t i n g of those f u n c t i o n s i n ~ ~D. ~ A) which are analytic on
D , .Let P denote the orthogonal project ion of ~ (D ,~A) onto ~&(D). For ~S~ cD, ~A) , we define the Toeplitz operator
and the Hankel operator
H# : L~ (D) -~ L~(]I), & A) e L~(]D)
by T~I~-P(#I~) and M#l~ =(I-P)(#I~),
# E L ® (D, &A) is the HaZel o p e r . t o r For which functions
compact?
If we were dealing with Hankel operators on the circle T ra-
ther than the disk ~ , the answer would be that the symbol must be
H ~ tC(T) on the dlsk it is ea to s that in the space • " 17 , " sy ee
i f ~ E ~ °O +C(~ ) , then H~ is compact. However, it is not hard to const=ct a n open set 5 ~ D with S n T ~ ~ such that i f is the characteristic function of S , then H, is compact. Thus the
subset of ~(D~ ~ A) which gives compact ~_~nkel t operators is much ~o
bigger ( in a non t r i v ia l way) than H +C (~) , and it is possible that there is no nice answer to the question as asked above.
A mere natural question arises by considering only symbols which are complex conjugates of analytic functions:
For which ~6 i_ss H~ compact?
I% is believable that this question has a nice answer. A good
candidate is that T must be in H®*C(D). The importance of this question stems from the identity
~lid f o r a n ~ ~ H ® . ~hus we are asking which Toeplitz operators
on the disk with analytic symbol have compact self-commutator.
Readers familiar with a paper of Coifman, Rochberg, and Weiss
263
[I] might think that paper answers the question above. Theorem VIII
of [I] seems to determine precisely which conjugate analytic functi-
ons give rise to compact Hankel operators. However, the Hankel opera-
tors used in ~I] are (unitarily e~uivalent to) multiplication follow-
is far bigger than E , the Hankel operators
of [1] are not the s~e as the Hankel operators defined here,
The Hankel operators as defined in [ I ] are more natural when
dealing wlth singular in tegral theory, but the close oo~eo t i on l r i t h
Toeplitz operators is lost, To determine which analytic Toeplitz ope-
rators are essentially normal, the Hankel operators as defined here
are the natural objects to study.
RE~ERENOE
I. C o i f m a n R.R., R o c h b e r g R., W e i s s G. Pac-
torization theorems for Hardy spaces in several variables. - An-
nals of Mathematics 1976, 103, 611-635.
SHELDON AXLER Michigan State University
East Lansing, NI 48824, USA
264
5.4. A SIMILARITY PROBLEM FOR TOEPLITZ OPERATORS old
Co, sider the Toeplitz operator T~ acting on H %~ H~(~) ,
where ~ is a rational function, with ~(~) contained in a simple
closed curve ~ . Let ~ be the ccnformal map from ~ to the in-
terior of ~ , and say that ~ b a c k s u p at 6 ~0 if
arg ~-1 ~(¢t~) is decreasing in some closed interval [@I, ~ , where
0,< 01, and 0 i ~<t~ 01 . L e t ~ t , . . . , ~ be d i s j o i n t a r c s on such tha~ V i s o n e - t o - o n e on each ~ and su c h t h a t U ~K i s t h e
set of all points where ~ backs up. Several recent results suggest
the following
CONJECTURE. Suppose ~(~) has windin~ numbe~ ~ ~ 0 . Then W~
is sizi!a r to
i~ 9>0
i_Xf "~---0
, and t o
M4 ~ - - . • M~ (2)
, where, M k is t h e o ,Rera to r o f multip!ica,tion b~ ~(6¢$) e
One c a s e o f t h e above c o n j e c t u r e goe s back t o Duren ~ , where
it was proved for ~(~)~+~ , I¢I>I;I . In this case ~ and
never backs up, so that MI,..., M~ are not present in (I). Ac-
tually, Duren did not obtain s~milarity, but proved that ~V satis-
fies
LTr ----- D ,
where ~ is some conjugate-linear operator( ~(~,X+ A¢~) -----
=~4~(~) + ~ ( ~ ) ) and ~ is the mapping function for the inte-
rior Of C~03~(T) . I n [2~, the conjecture was proved in case
is $ - to-one in some annulus ~ i~l ~ ~ . Here again ~ never
backs up. In [3~, ~ was assumed to have the form
where ~ and ~ a r e f i n i t e B l a s c h k e p r o d u c t s , ~ h a v i n g o n l y one
zero. In this case ~(~)-----~ and ~ can be taken to be 1.
265
The main tool used in [3] was the Sz.-Nagy-Fola~ characteristic
function of ~p , which we computed explicitly and which, as we
showed, has a left inverse. A theorem of Sz.-Hagy-Foia~, [4], Theo-
rem 1.4, was then used to infer similarity of TF with an isometry.
~oreover, the ,,-!tary part in the Wold Decomposition of the isometry
could be seen to have multiplicity I, and so the proofs of the repre-
sentations (I) and (2) were reduced to spectral theory.
If ~ is of the form (3) where ~ and ~ are finite Blaschke
products, ~ having m o r e t h a n o n e zero, the compu-
tation of the characteristic function of T~ is no longer easy.
However, left invertibility can sometimes be proved without explicit
computation. This is the case if ~ and ~ have the same number of
zeros; i.e., when T~ is similar to a unitary operator. This and
some other results related to the conjecture are given in [5]. The
Sz.-Xagy-Fola~ theory may also be helpful in attempts to formulate
and prove a version of the conjecture when (3) holds with ~ and
arbitrary inner functions. For example, it follows from ~3~ that if
is inner and ~ is a Blaschke factor, then ~F is similar to an
isumetry.
For the case in which ~ is net the unit circle, the only
successful %ech-lques so far are those of [2], which do not use mo-
del theory. We have not been successful in extending them beyond the
case of ~ satisfying the "annulus hypothesis" described above.
There is a model theory which applies to domains other than ~ [6],
but to our knowledge no results on similarity are a part of this
theory. ~ore seriously, to apply the theory, one would NEED TO KNOW
that the spectrum of r~ W is a spectral set for Tp ; a result which
does not seem to be known for rational ~ at this time.
Pinally, I% seems hardly necessary to give reasons why the con-
Jecture would be a desirable one to prove. Certainly detailed infor-
mation on invariant subspaces, commutant, cyclic vectors and functio-
nal calculus would follow from this type of result.
REFERENCES
1. D u r e n P.L. Extension of a result of Beurling on invariant
subspaces. - Trans.Amer.Nath.Soc. 1961, 99, 320-324.
2. C 1 a r k D.N., M o r r e 1 J.H. On Toeplitz operators and
similarity. - Amer.J.Math., 1978, 100, N 5, 973-986.
3. C 1 a r k D.N., Sz.-Nagy-Foia~ theory and similarity for a
class of Toeplitz operators. - Banach Center Publiaations,v 8,
266
Spectral Theory, 1982, 221-229 4. S z. - I~ a g y B., F o i a 9 C. On the structure of in-
tertwining operators. - Acta Sci.Math. 1973, 35, 225-254.
5. C 1 a r k D.N. Similarity properties of rational Toeplitz
operators. In preparation.
6. S a r a s o n D. On spectral sets having connected comple-
ment. -Acta Sci.Math. 1965, 26, 289-299.
DOUGLAS N. CLARK The University of Georgia, Athens,
Georgia 30601 USA
COmmENTARY BY THE AUTHOR
Since my first note on the similarity problem, the following
results have been obtained.
T ~ O ~ 1 ( [8 ] ) , I__~ F is a ra t iona l funct ion. ~ p ~ T ~ : o
a simple closed curve P , which is anal,ytic in a neighborhood o,f
FCT ) ; i_~f V>0 , where V is the windin~ number of F(T)
about the points interior to F ; and if F(>-)@F , where ~- i_~s
the set (o~ T ) where F backs up; then TF is similar to
T (V) • V where T (y) , ....... _,~ is the sum of ~ copies of the ana-
lytic Toeplitz operator ' as s0ciated with the mapping function
from D to the interior of F and where V is a normal operator
whose spectrum is F(~) • V is absolutely continuous and the
spectral multiplicity of a 90int ~ in th e spectrum of V is ,equal
to the number of points ~$ where F backs up and F(e~)=~ .
THEOREM 2. ([9], [10] ). If F is a rational function~ if
F~T) divides the plane ̀ into disjoint re6ions~ from which the ones
in which the index of T F -~I is negative (resp,positive) are
labeled ~ (rasp. ~ ); if the closures of any two of these
(~ , ~) inters,act St onl~ finitely many points (called the mul-
tiple points of F ); if the boundary of each ~ , ~ is an ana-
267
l~tic curve except at th e multiple points= where it is piecewise
smooth with inner angle ~0 ; if no multiple point is the image I
under F of a point ~o~:T where ~(~o) = 0 ; and if
never backs up at a multiple point; then ~ is similar to h
where ~ (resp. T ) is the maopin~ function of ~ o_~n ~ (resp.
~ ), each summand is included with multi p!icit E equal to the
absolute value of the index of TF-~I for ~ (resp. ~ ),
and V is as described in Theorem I.
THEOREM 3. (Wang, ~I~ ). I_~f F~C4~T) and if ~Ce ~) is the
restriction t° T of a function F analytic in
some ~<~ , i_~f F(e ~$) i_~s 1-te-1 , ~t(e~$)
and $~F~e '~$) is orientation preserving, then ~F
, where ~ is the mappin~ function from 0
F ( T ) •
<I ~I < 4 fo__~r
never vanishes
is similar to
to the interior
Theorem I, of course, PROVES THE CONJECTURE posed in my first
note on the similarity problem, EXCEPT WHEN the curve FCT ) has sin-
gularities. The case of a singularity with nonzero inner angle can be
settled using the methods of ~0], but the case of zero inner angle
remains open, owing to our lack of understanding of the behavior of
and ~i near such a point.
Theorem 2 also excludes zero interior angles in the ~ and~ .
The hypothesis that Fk~o)~0 at the inverse images of the mul-
tiple points can be somewhat weakened ~I0] but not removed. In fact,
~0] contains an example of a TF satisfying Theorem 2 (with such a weakening) and a rational, orientation preserving, homeomorphism
of T such that t F and TFo ~ are not similar. The case of intersecting loops of F~T) was discussed in ~ .
As indicated there, nothing is known unless F~) is the image of~
under a function analytic is ~ . Unfortunately, even the proof of
Theorem 5 of ~ ~ is incomplete without fur~er hypotheses, as point- ed out by Stephenson ~I~.
Theorem 3 is the first attempt at a systematic similarity theory
268
for non-rational F , and one hopes that it may be generalized to
the point of a non-rational version of Theorems I and 2. Other examp-
les of similarity for non-rational T F may be obtained using Cowen's
Equivalence Theorem ~ , which implies that TFo ~ is unitarily
equivalent to a direct sum of countably many copies of T F , when
is a non-rational inner function.
An obstruction to similarity between T F and a "reasonably
nice" operator can occur when T F has a boundary eigenvalue, see
Clancey [7, ~4.~, In fact, as Clancey has pointed out to me, the
spectrum of T F is not a k -spectral set, for the example given
in ~ [7]-
REFERENCES
7. C I a n c e y K~F. Toeplitz models for operators with one di-
mensional self-commutators (to appear).
8. C 1 a r k D.N. On a similarity theory for rational Toeplitz
operators.- J.Reine Angew.Math. 1980, 320, 6-31.
9. C 1 a r k D.N. On Toeplitz operators with loops. - J.Operator
Theory, 1980, 4, 37-54.
10. C 1 a r k D.N. On Toeplitz operators with loops, II. - J.Ope-
rator Theory 1982, 7, 109-123.
11. C 1 a r k D.N. On the structure of rational Toeplitz opera-
tors. - In:Contributions to Analysis and Geometry, supplement to
Amer. J.Math. 1981, 63-72.
12. C o w e n C.C. On equivalence of Toeplitz operators. - J.Ope-
rator Theory 1982, 7, 167-172.
13~ S t e p h • n s o n K. Analytic functions of finite valence,
with applications to Toeplitz operators (to appear).
14. W a n g D. Similarity and boundary eigenvalues for a class of
smooth Toeplitz operators (to appear).
269
5.5. ITERATES OP TOEPLITZ OPERATORS WITH UNIMODULAR SYMBOLS
Each invertible Toeplitz operator ~ on H ~ can be represented
as T~ ~T~ T@ where ~ is an outer function with modulus I~I and
~=~/~ is a unimodular function. The operator ~ , being invertible
analytic Toeplitz operator, had simple spectral behaviour. Therefore
the Toeplitz operators with unimodular symbols play an especial role
(see [I] ).
I would like to propose the following questions concerning these
operators.
Suppose X is one of the function classes H°°~O~ Q C ~¢---~
= I-I°'% C, n FI°°÷ C, C,, C k, C, ~ . QUESTION I , Le t ~ be a unimodular function in X such that
Ko~T~ ~0] . Is it true that there exists ~ i~ ~ with
II 0 z ~4>0
If the answer is positive, it is reasonable to ask whether the
following stronger conclusion can be done.
QUESTION 2. Is it true that under the h~potheses of Question I
II T s II 0
for every non_zero ~ i_~n ~ ?
It follows from Clark's results ~2~ that the answer to Question 2
is positive for rational functions ~ (see also Commentary to 5-4).
In view mf T.Wolff's factorization theorem ~3~ (asserting that each
unimodular function t~ in H°°+ G can be represented as ~ 0 with
~r~C and 0 inner) it seems plausAble that if Question I (or 2) has
a positive answer for X =~c then so is for X~°° C . Note that
for general unimodular functions ~ with ~c~T~ ~ I@} it may happen
that ~011T~II=0 for any ~ in ~ . For example, if ~ is a
measurable subset of T , 0 < ~4ea~ E < ~ and ~=~ on ~ and-~
on~ then it follows from M.Rosenblum's results ~4~ that T~ is
a selfadjoint operator with absolutely continuous spectrum on E-~, ¢~.
An affirmative answer to Question I for X~-~ C would imply the
existence of a non-trivial invariant subspaces for Toeplitz operators
with unimodular symbols in ~ C (see some results on invariant subspa-
ces of Toeplitz operators in [5~ ). Indeed, either one of the kernels
~£~T~, ~T~ is non-trivial or T~ and T~ satisfy the hypothesis
at Question I and so both subspaces
270
are invariant under T~ and ~ ==I=H~ ~{~) of these sub@paces is non-trivial or ~={ ~},
would be a 0~ contraction (see [6]). But each elf
non-trivial invariant subspace [6].
. Therefore either one
~ = H~ i~e, T~ contraction has a
REFERENCES
I. S a r a s o n D. Function theory on the unit circle. - Notes for
Lect.at a conference at Virginia Polytechnic Inst. a~d State Univ.,
1978.
2. C 1 a r k D.N. On a similarity theory for rational Toeplitz ope-
rators. - J.Reine Angew.Math., 1980, 320, 6-31.
3. W o 1 f f T. Two algebras of bounded functions~ - Duke Math J.,
1982, 49, N 2, 321-328.
4. R o s e n b 1 u m M. The absolute continuity of Toeplitz's matri-
ces. - Pacif.J.Math., 1960, 10, N 3, 987-996.
5o P e 1 1 e r V.V. Invariant subspaces for Toeplitz operators. -
LOMI Preprints, E-7-82, Leningrad, 1982.
6. S z . - N a g y B., P o i a ~ C. Harmonic analysis of operators
on Hilbert space, North Holland, Amsterdam, 1970.
V. V. PELLER
(B.B.~P)
COOP, 191011, ~eHs~m~,
#OH~aHKS 27 , ~0MH
271
5.6. LOCALIZATION OF TOEPLITZ OPERATORS old
Let H ~ and H °@ denote the Hardy subspaces of ~,~(~) and ~°°(T) respectively, consisting of the functions with zero negative Fourier
coefficients and let ~ be the orthogonal projection from h~(~)
onto H~ . For ~ in ~,~(T) the Toeplitz operator with symbol
is defined on H% byT9~=P(~) . ~[uch of the interest in Toeplitz
operators has been directed toward their spectral characteristics
either singly or in terms of the algebras of operators which they generate. In particular, one seeks conceptual determinations of why
an operator is or is not invertible and more generally Fredholm. One
fact which one seeks to explain is the result due to Widom [I] that
the spectrum 6~(T~) of an arbitrary Toeplitz operator is a connected
subset of C and even [2] the essential spectrum 6~e(Tq) is connec-
ted. The latter result implies the former in view of Coburn's Lemma.
An important tool introduced in [2], [3] is the algebraic notion
of localization. Let ~ denote the closed algebra generated by all
Toeplitz operators and ~ C be the subalgebra
(H°% C) n (H°°+ C,)
of ~ , where C denotes the algebra of continuous functions out .
Each ~ in the maximal ideal space ~QC of ~C determines a
closed subset ~E of M~ and one can show that the closed ideal%
in S generated by
is proper and that the local Toepli%~ operator%+ % in~=~/%
depends only on ~I X~ ° Moreover, since ~. ~. equals the ideal~
of compact operators on ~ , p rope r t i es ~ l c h are t rue modulo
can be established "locally". For example,T~ is FTedholm if and on-
ly if--'J~ ~ is invertible for each ~ in MQC . These localiza- tion results5 are established [4] by identifying Q C as the center of
~/~ . One unanswered problem concerning local Toeplitz operators
is:
CONJECTURE I. The spectrum of a local Toeplitz operator is
connected.
In ~] it was shown that many of the results known for Toeplitz
operators have analogues valid for local Toeplitz operators. Unfortu-
272
nately a proof of the connecteduess would seem to require more re-
fined knowledge of the behavior of ~ functions on ~ H @@ than ava-
ilable and the result would imply the connectedness of ~,(T?) •
A mere refined localization h~s been obtained by Axler replacing
X~ by the subsets of ~L ~ of maximal antisymmetry for H@@+
using the fact that the local algebras S~ have nontrivial centers
and iterating this transfinitely.
There is evidence to believe that the ultimate localization
should be to the closed support X~ in~L ~ for the representing
measure ~ for a point ~ in ~H ~ . In particular, one would
like to show that if H~(#V) denotes the closure in ~(~$) of the
functions ~I X~ for ~ in M ~ , P~ the orthogonal projection
from ~(~) onto H~(~N) , then the map
extends to the corresponding algebras, where the local Toeplitz ope-
rator is defined by
Zor ~ in . ~f N i s a point i n M L ~ , then H~(~) = C and it is a special case of the result ~2] that ~ modulo its commu-
tator ideal is isometrically isomorphic to ~ , that the map extends
to a character in this case. A generalized spectral inclusion theorem
also provides evidence for the existence of this mapping in all cases.
One approach to establishing the existence of this map is to try
to exhibit the state on ~ which this "representation" would deter-
mine. One property that such a state would have is that it would be
multiplicative on the Toeplitz operators with symbols in H~ . Call
such states a n a 1 y t i c a 1 1 y m u 1 t i p 1 i c a t i v e .
Two problems connected with such states seem interesting.
CONJECTURE 2 (Generalized Gleason-Whitney). I_~f ~ and 6~ ar__~e
analytic all.v multiolicative states on S which a~ree on ~@@ and
such that the kernels of t h e t.wo representations defined b Z ~ an d
are equa!, then the representations are equi~lent.
CONJECTURE 3 (Generalized Corona). In the collection Q fana!y-
ticallymultiplicative states the ones whichcorrespond to pqints °f
273
One consequence of a localization to XV when ~ is an analytic
disk ~ ,would be the following. It is possible for ~ in L ~ that
its harmonic extension ~I~ agrees with the harmonic extension
of a function continuous on ~ . (Note that this is not the same as ^
saying that ~ is continuous on the boundary of ~ as a subset of
H~ which is of course always the case.) In that case the inverti-
bility of the local Toeplitz operator would depend on a "winding num-
ber" which should yield a subtle necessary condition for T~ to be
Predholm. Ultimately it may be that there are enough analytic disks A
in ~ H~ on which the harmonic extension ~ is "nice" to determine
whether or not ~ is Fredholm but that would require knowing a lot
more about ~H~ than we do now.
REFERENCE S
1. W i d o m H. On the spectrum of a Toeplitz operator. - Pacif.
J.Math.~ 1964, 14, 365-375.
2. D o u g 1 a s R.G. Banach algebra techniques in operator theo-
ry. New York, Academic Press, 1972.
3. D o u g 1 a s R.G. Banach algebra techniques in the theory of
Toeplitz operators. CBMS Regional Confer. no.15, Amer.Math.Soc.,
Providence, R.I., 1973.
4. D o u g 1 a s R.G. Local Toeplitz operators. - Proc.London
Math.Soc., 1978, 3, 36.
R.G.DOUGLAS State University of New York
Department of Math.
Stony Brook, N.Y. 11794,
USA
274
5.7. TOEPLITZ OPERATORS ON THE BERGMAN SPACE
Let ~ denote the Bergman space of analytic functions in
~ (0) , and let P be the orthogonal projection of L I(0) .9
OntO ~" : Por~. ~. U--(D) we define the Toeplitz operator with - p
rators may be quite different from that of the Toeplitz operators on @
the Hardy .space H • However i% is shown in [I] that Toeplitz opera-
%ors on ~ with h a r m o n i c symbols behave quite similarly to "" o U% . • nse on I1 , ana one can prove analogues for this class of many
results about Toeplitz operators on H ~ .
An important result about Toeplitz operators on H t is Widom's
Theorem, which states that the spectrum of such an operator is connec-
ted ([2]). This suggests our problem. @
CONJECTURE. A Toeplitz operator on A ~ with harmonic sxmbol ' has
a 9o~ected spectrum.
In support of this conjecture we mention the following cases for
harmonic ~ in which the spectra can be explicitly computed.
I) If ~ is analytic then ~(T~)=~(D).
2) I f ~ is real-valued then ~ ) = [ ~ q , ~ ] ,
3) If ~ has piecewise continuous boundary values then ~(T~)
consists of the path formed from the boundary values of ~by Joining
the one-sided limits at discontinuities by straight line segments,
together with certain components of the complement of this path.
For proofs of these see [1].
In connection with our conjecture it should bementioned that
there are easy examples of Toeplitz operators on ~ with disconnect-
ed spectra - e.g. izl - ) is disconnected since T4.1zl: is positive and compact. The proof of connectedness in the case of H
breaks down almost immediately in the ~ case. I would expect a
solution to the present problem to shed light on Toeplitz operators
in general, and perhaps %o lead %o a different proof and a better
understanding of Widom's Theorem.
REFERENCES
I. M c D o n a 1 d G., S u n d b e r g C. Toeplitz operators
on the disc. - Indiana Univ.Math.J.~ 1979, 28, 595-611.
275
2. W i d o m H. On the spectrum of Toeplitz operators. - Pacific
J.Nath. 1964, 14, 365-375.
CARL SUNDBERG University of Tennessee
Dept. of ~ath.
Knoxville, TN 37916
USA
and
Institut Mittag-Leffler
Auravagen 17
S-182 62 Djursholm, Sweden
276
5.8. VECTORIAL TOEPLITZ OPERATORS ON HARDY SPACES
Let ~ be a separable Hilbert space (dim ~ may be finite), be the algebra oZ all bonded liner ope~tors on ~ ~d L~(q)
be the Banach space of weakly measurable ~ -valued functions
~: ~ ~ ~ with the norm
T
We denote by ~r(~) the Hardy space of functions in L,~('~) with zero negatively indexed Fourier coefficients and by ~ the
Riesz projection onto HP(~) (4<p<~), Let l.~( ~ ) be the space of all essentially bounded ~ -va-
lued ftmctions and a~(~) be the Hardy space corresponding to
then the operator
is called a v • c t o r i a 1 T o e p 1 i t z o p • r a t o r.
The following criterion of the invertibility of ~ on ~(~)
has been obtained by Rabindranathau [I ]°
~ o ~ ~. ~,et ~ C c~) • T~ is,,invertible ,,~ H~c~) if, ,a,nd only if ~ = ~ I~, ~. w,h,,ere
2) ~ is a unitary-valued f unc t i on i n L~(~0 " ) ;
3) there exists an operator-valued function ~ with ~+-46
e H~°C~) such t~t
II '~,- ,.~ I/ < 4. (1) L®c~}
See also [2-3] C ~ , ~ = ~ ) ~d [4] (~,~<~). A sufficient condition for the invertibility of To$ on
H ~ (~) has been given by the authors [5] (the case ~ ~ = had been considered earlier by Simonenko [6] ).
THEOREM 2. ~et p ~. ({ , oo) , ~ = ~*~£~ and suppo,~e that
al~ conditions of Theorem 1 for ~)~4)~ hold except for (1)
which has to be ~eplaced b,y
277
I
Then ~ is inver~, ible ,,O n HP( ) (and o n HPc ) ).
PROBLEM 1. Are the conditions of ' Thgorem 2 necessary for the
HPc I HPc I? It is shown in [7] that the answer is affirmative if
Let us note that the class of operator-valued functions in Theo-
rem 2 admits an equi~mlent description [8]: & = ~ Z , G ~o(.~),
~( , ~4 ~ Heo(~ ) and the numerical range ~/C~(~)) lies in
a fixed angle with the vertex at the origin and wlth the size less
than ~/~vP~C (p, p') a.e. on T •
It is well-known that the problem of inver%ibility of T~ in
~P(~) can be reduced by means of faotorization [2-4,73 to the
problem of boundedness of ~ in weighted C -spaces. In the case
~ ~ = ~ a criterion for boundedness of ~ was given in [9].
tP What are the conditions for ~p~'~ to be bounded on (~) ?
We don't know the solution of Problems I and 2 even in the case
of matrix-valued Toeplitz operators (~ ~ < oo) .
REFERENCE S
I. R a b i n d r a n a t h a n M. On the inversion of Toeplitz
operators.- J.N~th.Mech., 1969/70, 19, 195-206.
2. H e 1 s o n H., S z e g B G. A problem in prediction theo-
ry. -Ann.Mat.Pure Appl., 1960, 51, 107-138.
3. D e v i n a t z A. Toeplitz operators on H Z space. - Trans.
Amer.Nath.Soc. 1964, 112, N 2, 304-317.
4. P o u s s o n H°R. Systems of Toeplitz operators on H Z . -
Trans.Amer.Math.Soc., 1968, 133, N 2, 527-536. 5. B e p d E n ~ H ~ M.3., K p y n H H E H.H. TO~HHe EOHCTaHTH
B TeopeMax od O~SHH~e~HOCTH cm~Pyaapm~x onepaTopoB B npocTpaHCT-
Bax c BecoM. --B ~. : ~HHe~e onepaTopM. Km~ZHeB, mT~B, I980,
21-35.
6. C ~ M O H e H ~ o ~I.S. KpaeBa~ s~a~a Pm~aHa~ nap ~YH~d~ C
EsMepmva~m EOS~J~4eHTa~S~ ~ ee npm~eHeHEe E Ecc~e~oBaH~0 C~HPyJLKp-
m~x ~HTerpaaoB B npocTpsacTBaX c Becalm. - M3B.AH CCCP, cep.MaT.
1964, 28, 277-306.
278
7. K p y n H E E H.& HeEOTOp~e c~e~CTBH~ ES TeopeM~ Xa~Ta--~EeH--
xaynTa-B~eHa. - B EH. : 0nepaTop~ B 6aHaXOBLZ( IIpOCTpaHCTBaX. I~--
~HeB, IHT~H~a, 1978, 64--70.
8. C H ~ T E 0 B C E ~ ~ H.M. 0 ~aETopHsa~ MaTp~-~yHF~, xayc-
~op~oBO M~ozecTBO EOTOpHX pacnoaozeHo BHyTp~ yraa. - Coo6m.
AH Pp.CCP, I977, 86, c.561-564. 9. H u n t R., M u c k e n h o u p t B., W h e e d e n R.
Weighted norm inequalities for conjugate function and Hilbert
transform.- Trans.Amer.Math.Soc., 1973, 176, 227-251.
N.Ya.ER~D~K
. .VERBI Sr
CCCP, 277000, IG~I~HeB,
~eBcE~ l~Cy~apOTBeHH~
yHKBepc~TeT
CCCP, 277000, EI~HeB,
MHCT~TyT reo~sHE~ H
reoaor~AHMCCP
279
5.9. FACTORIZATION PROBLEM FOR ALMOST PERIODIC MATRIX-E3NCTIONS
AND FREDHOLM THEORY OF TOEPLITZ OPERATORS WITH
SEMI-ALMOST PERIODIC MATRIX SYMBOLS
I. We consider (~ x ~) -matrices G defined on ~ with ele-
ments from the usual algebra ~ ~ of almost-periodic functions and
Toeplitz operators TG = ~ G I I~ ~ generated by these matrices.
Here ~ is the Riesz projection onto the Hardy class ~ in the
lower half-plane, ~ < p < co .
It is well known (this fact holds for an arbitrary G ~ ~ )
that condition G ~ ~ ~ is necessary for T8 to be semi-Predholm.
In the case ~=~ the converse is true. Moreover, ~ is left-
invertible if the almost periodic (a.p.) index ~ of the function
is non-positive and TGis right-invertible if $ ~ 0 . There
exists a certain parallel between Fredholm Toeplitz operators and
the factorization problem of their symbols, in accordance with which
the formula
(1)
is valid in the case ~=~ , G +-~ ~ AP . Here ~(~)= ~,
the functions (~ ÷~)-~ ~+~ belong to Hardy class H ~ in the
upper half-plane and the operator G+ ~_ G+~ is bounded in all ~,
Formula (~) with G ~ ~ ~.possessing~ ~ ~ t h e above properties (with the natural change of e ~ by ~@~e~V4t, o. ,a~Y~tJ,
V ~ ) will be called a P-factorization of G . It is easy
to °check that the partial a.p. indices $~, ,~ are umiquely de-
fined by G provided G admits P-factorization; and it is not
difficult to describe the freedom of choice of G+ . However in the
case ~ > ~ not each matrix ~ invertible in ~ ~ admits P-fac-
torizations. In this connection the following problems appear.
PROBLEM 1. Obtain a criterion (or at least more or less ~eneral
sufficient conditions~ of existence of P-factori~ation.
PROBLEM 2. Find out whether the existence of P-factorization of
is a necessar~ condition for T~ to be (semi-)Fredholm. If
not~ then is it possible to chan~e the definition of P-factorizatipn
280
in such a wa~, as to ~et an equivalent of the semi-Predholm ~roDsrty
Zo~ T~ Note that if ~ admits a P-factorization then 7~ is left-
(right-) Predholm iff @i ~0(>0) , i=~, ,~. Consequently an
affirmative answer to Problem 2 would mean that "Fredholm character"
of --T~ is the same in all spaces H~ and its fredholmness implies
the invertibility. We do not know whether these weaker statements are
true if ~ ~ .
2. The class SAP (of semi-almost-periodic functions) is a
natural extension of AP . This class has been introduced by
D.Sarason [ I] and may be defined, for example, as {~=(0,5+~)~+
The a°p. components I' ~ (of ~ ) are uniquely determined by
~0 - A criterion for TG ( G~ 5~P , ~=~ ) to be semi-Fredholm
in the space ~ was obtained in ~] amd was generalized in ~2] to
the case of an arbitrary ~ , ~ ~ (~,oo) . The case ~>~ is con-
sidered in [3,4], where the fredholmness and semi-~redholmness crite-
riaha~been established. These results, however, were obtained under
the a priori assumption of existence of P -factorization of a.p. com- u +
ponents F , H of G . The latter means that the factors ~; ,
(~*)±( from the P-factorization F = ~+ A Ff belong to the class AP + of those matrices from ~P whose ~ourier exponents
are all non-negative; the same holds for ~ . The following problems
arise in connection with the question of removing these a priori as-
sumptions.
PROBLEM 3. L e~ G be an (~x~)-matrix from ~ ~>~, F
and ~ are its aop% components. Is it true that the semi-fredholm-
hess of ~G implies the semi-fredholmness of ~F and 7 H ?
PROBLEM 4. I sth e set of matricesadmitting ~ Po-factorization
dense in the set of all matrices admittin~ a P-factoriz~tion? What
would the situation be like if we restricted ourselves
to matrices wit h a fixed (non-zero~set of partial a.p~ indices?
The positive answer to Problems 3, 4 would allow to extend the
criterion for ~r. to be (semi-)Fredholm [3, 4] to the case of ar-
bitrary matrices ~ G ~ 5A~ .
281
3. Let us consider a triangular matrix ~ £ AP of the second
order. Under some additional assumptions(e.g, absolute convergence
of Pourier series of its elements ) the P-factorization property
of ~ is reduced to the corresponding question about
] 0cb=
" iv% e 0
%({) e
(2)
where V > 0
C-~,~) . Assuming that card C~) < co
%i (i= ~'~' ) by the recurrence formula
and the spectrum ~ of %0~ ~ is contained in
define a.p. polynomials
(3)
It is supposed that the sequence ~ of leading exponents of ~i
strictly decreases and ~ ~ AP + ~. Analogously to the case of
the usual factorization of continuous triangular matrices E5] there
exists an algorithm (say, A ) for P-factorization of Bo which is
connected with the continuous fraction expansion of ~i/~0
or equivalently with the relations (3). Algorithm ~ unlike in
the continuous case does not necessarily lead to the aim. A suffi-
cient condition to make application possible i~K ~K4 ~ 0 for some
K EZ+ . In this case the factors B± are a.p. polynomials.
PROBLEM 5. Give conditions for the convergence of the algo-
rithm A to obtain a ~ - (or P- ) factorization of matrix
(2~. These conditions have,to be formulated in terms of entries of
Algorithm A can be applied to obtain a ~ -factorization
of matrix (2) if, for example, ~C(-$, 0] or the distances bet-
ween the points of ~ are multiples of a fixed quantity (in par-
ticular, if card CI~) ~ ~ ). Already in the case card ~= 3 , i.e.
there exist situations when algorithm ~ fails. One of such situa-
tions is : #=0 , ~= [ - & and ~ - & / [ ) is i r ra t iona l ,
282
In this case we found another algorithm based on the successive ±~ ~+
application of the transformation ~'-~ ~B~C~ (A~, CB g A )
preserving the structure of B~ and on the factorization of elements
close to the unit matrix. With the help ~f this algorithm it was es-
tablished that under the restriction IC~I # I~÷~I a ~-fac-
torization exists with 9~ = 9~ = 0 ., but B± are no more a.p.
polynomials. In the case IC~I =I $4+~I the P-factorization of (2)
does not exist. Thus, even in the case card ~= ~ the following
problem is non-trivial.
PROBLEM 6. Describe the cases when matrix (2) admits %P-(0 r
Pc-) factorization~ calculate its a.p. partial indices and construct ,,
if possible~ corresponding f actorizations.
The interest to matrices of the form (2) is motivated by the
fact that they naturally arise in connection with convolution equa-
tions on a finite interval with kernels ~ for which ~&(~)(~)
has an a.p. asymptotics at infinity for an &~ C [3].
REFERENCES
I. s a r a s o n D. Toeplitz operators with semi-almost periodic
symbols. - Duke Math.J., 1977, 44, N 2, 357-364. 2. C a r i~ H a m B ~ ~ ~ A.M. Cm~ryaapHae m~Terpaa~H~e ypaBHeH~ C
EOS~M!U~eHT82a~, mge~m~ paspMB~ noay-no~TH-nepHo~eoEoro T~na. -
Tp.T6mmc.Ma~eM. EH--Ta, 1980, 64, 84--95.
3. Eapao B ~ D.M., C n~ T E o ~ c E ~ H.M. 0H~Te--
pOBOOTE HeEoTop~x C~Hrym~H~X ~HTe~sa~H~X onepaTopoB c ~mTpmn~m EOB~HI~eHTaM~ ~acca~A ~ CB~S~x c ~ CECTeM ypaBHeHNl~
cBepTEH Ha EOHe~HOM npomeayTEe. - ~oEa.AH CCCP, 1983, 269, ~ 3.
4. KapaoB ~ D.H. , C n~ T ZO B C E ~ ~ H.M. 0 H~Te--
pOBOCT~, ~- E ~--HOpMa~BHOCTI~ CEHIVJL~pHRX EHTeI~BH~X onepaTo-
pOB C MaTp~m~M~ EOS~EIU~eHTam~, ~OnyCESm~m paspNBN noay-no~T~- nepEo~H~ecEoro THna. - ~Eoxa no Teop~H onepaTopoB B ~OHaXBRNX
npocTpaHCTBaX (TesHc~ ~OF~S~OB), MHHCE, 1982, 81--82. 5. q e 6 o T a p e B F.H. qacTHae HH~eEc~ EpaeBo~ salaam P~Ma~a c
Tpey~oJ~HO~ MaTpm/e~ BTOpO~O nop~Ea. - Ycnex~ MaTeM.HayE, I956,
II, h 3, 199-202. Yu. I. KARLOVICH
I.M. SPITKOVSKII
(M.M. CHMTKOBCK~)
CCCP, 270044, 0~ecca, Hpo~eTapcK~ 6y~BBap 29,
MopcKo~ rH~pOSHSMYecKM~ MHCTMTyT 0T~e~eH~e SKOHOMMKM M
~K0~OrHM ~p0BOr0 oKe~a.
283
5.10. TOEPLITZ OPERATORS IN SEVERAL VARIABLES
?or ~ the complex numbers, let ~ be a bounded domain in
C ~ with closure ~- and with 9~ the Shilov boundary of the uni- formly closed algebra A(~-) generated by all polynomials in the com- plex variables ~=(E~,Z~., ~Z~) on ~- . In general, ~ is a
closed subset of the topological boundary of ~ . When ~ is one of
the classical domains of Cartan or in other cases of interest, ~ is a compact manifold with a "natural" volume element ~ and the
space ~(~) of ~-square integrable complex valued functions is
the setting for our analysis.
The closure of ~(~-)in ~($~) is denoted by H~(~) and this (Hardy) space, together withthe (unique) orthogonal projection operator ~ from ~(9~) onto H~(~) , is a basic object in comp- lex analysis on ~ . For q essentially bounded on 9~(~) , the
~e~ I ib;TZ@ =~(j~)e.rThet ~*r-a2g~ebraiged~t ed fyral~lT! ~th
continuous I is denoted by ~ (~). l
" Even for ~--~x~x xD ( M~ times) where ~ is the open unit disc in C , many interesting questions about ~(~) remain
open after more than a decade of study. Note that for .
( MS times), S~=T ~, the M~-torus. The structure of ~(T~)is well-understood for P~= I ,2 ~1,2,3].
(T~n) particular, necessary and sufficient conditions for ~ in
to be ?redholm of index ~ are known [2~. It follows from
the analysis of [2] that every ?redholm operator of index % in~(V ~) can be joined by an arc of such operators to
c I .
Here, ~ is an integer and ~i - ~i
PROBLEM I. Classify the arc-components of Fredholm operators in
This question reduces to:
PROBLEM 2. Classif,y the arc-components of invertible elements
in
284
REFERENCE S
I. C o b u r n L.A. The C*-algebra generated by an isometry I,
II. - Bull.Amer.Nath.Soc.,1967, 73, 722-726; Trans.Amer.~ath.Soc.~
1969, 137, 211-217.
2. C o b u r n L.A., D o u g I a s R.G., S i n g e r !.~.
An index theorem for Wiener-Hopf operators on the discrete quar-
ter-plane. - J.Diff.Geom.91972 , 6, 587-593.
3. D e u g 1 a s R.G., H o w e R. On the C*-algebra of Toep-
litz operators on the quarter-plane. - Trans.Amer.~Aath.Soc.~1971,
i58, 203-217.
L.A.COBURN State University of New York
Department of Mathematics
Buffalo, N.Y. 14214
USA
285
5.11. SOME PROBLEMS CONNECTED WITH THE SZEG6 LIMIT THEOREMS
1. Any sequence of (%x%)-mat r i ces {Ci.~Z~+~ { determines a , sequence l,,of m a t r i x - v a l u e d T o e p l i t z m a t r i c e s ° .
If a~ ~I ~T~ ~ 0 for sufficiently large values of
then the question about the limiting behavior of A ~+4 /A 6 arises.
The analogous question arises about A~/~ ~+~- provided the non-
zero limit ~ ~ .... ~ &~÷, /A~ exists.
It was G. Szego who studied both questions for the first time.
He dealt with the case %=~ and supposed that { C i }ig z is
a sequence of Fourier coefficients of a positive summable function.
See [1] for the precise formulations, for the history of the problem
and for its natural generalization
A
C i = H ( j ) , (1)
M being a f i n i t e non-negative Borel measure on 7 . By the Riesz-Herglotz theorem the class of sequences sa t is fy ing (1) is the class of pos i t ive de f in i te sequences.
We consider here the case when %~ and { Ci} is an &-sectorial sequence for some & ~ [ 0,g/~) . The latter means
that every T~ is & -sectorial i.e. its numerical range (Haus-
dorff set) lies in the angle {E:ll~l~t~& ~8~} . It is
clear that { C i} is an &-sectorial sequence iff there exists a measure ~ satisfying (1) and taking values in the set of & - sec-
torial Q~%)-matrices on all arcs of 7 • The real part ~
of this measure ~ permits us to construct the Hilbert space
~ =~C~) consisting of ~-tu~les of functions and equip-
ped with the sesquilinear form A(~,~)=~(~)%M(~) ~*c~)
Employing the factorization theorems from ~,3] we have proved in [4]
the existence of the limit ~ in the case %~ , &~0 and have
obtained the following formula:
T
where
earlier results by A.Devinatz and B.Gyires). Formula (2) is valid in
=~M/~ (see [43 for details and the information about
286
the case ~ G ~ C too; this is the only case when ~=0 .
We propose the following as an UNSOLVED PROBLEM: find an exten-
sic n of the Sze~B second limit theorem to the case of & -sectori-
al sequences.
We CONJECTURE that the limit ~ ~°~ ~ ~/~+4
(finite or not) exists for every &-sectorial sequence satisfying
the regularity condition
We are somewhat encouraged in this conjecture by Theorem 2 of Devi-
matz [5] related to the case provided ~=~ , M is an absolutely
continuous measure and ~ satisfies some additional restrictions
(including the requirement G ~L ~ ).
We have proved in the case & =0 , ~>~ the existence of
and h&ve obtained a formula for its calculation using some geometri-
cal considerations from [4]. Before formulating the corresponding
result let us remind that under condition (3) there exist two cano-
nical factorizations of the matrix G : the left one G=G G and the right one G =G* G~ ( G~ and G~ are outer matrix func-
tions of the class ~ ~ ). Let us denote the Toeplitz operator with
the (unitary valued) symbol ~ =G* -4 G~ by T F .
THEOREM I. Let { Ci} be a positive definite sequence of (~×%)-
-matrices and let ~ be a measure connected with it b[ formula (I~,
G =~/~ . Then under con d! tic n %3,), there exists a limit
i (4 oo) of the sequence ~ / ~ + ~ ̂ . This limit is finit.e
if__ f M is absolutel continuous and K F <)II
If these conditions are fulfilled then ~=(~T~T~ ~ 4.
We do not know whether there is any kind of general result in
the case ~ ~0 , ~>4 . it is well understood now that the
existence of ~ (and formulae for its evaluation) may be proved un-
der some additional restrictions with a help of results obtained in
another direction (the rejection of the positive-definiteness with
simultaneous amplification of restrictions on the smoothness of G )
that we do not touch upon here, see [6] and references in it.
2. Considering an & -secto~ial matrix measure M concentrated
on the line, it is possible to introduce a continuous analogue of the
287
space ~ and to establish the following result.
THEORE~ 2. The two statements given below are equivalent:
, where ~ = ~ / ~
2) the subspace ~ of constant ~-tuples has zero interHec-
tien with the subspace ~$=V {e~:t~$1 for all (at
least for one) S > 0 .
If these conditions are fulfilled and & = 0 then the square of t h e
distance from ~ ~ t_~o ~S i.~n ~ metric equals to ~$~
~$ is a "dis tansematrix"which is calculated by the formula where
0
Here V ~ is the inverse Fourier transform of the matrix-
function from the left canonical factorization of G ( G=G+G a.e. on ~ , -- ~+ is outer and belongs to the Hardy class ~ in
the upper half-plane).
For the case %=~ Theorem 2 was already proved in [7]; the
discrete analogue of (4) was established in ~4] in the general case
&~[0~ ~/~) , ~ . We propose a natural
PROBLEM: ~eneralize the second part (concerning formula (4)) of
Theore m 2 to the case" of distances ~n th e skew A -metric C&>o).
In this case obscure points already appear after first attempts
to interprete the right-hand side of the formula of type (4). The
fact is that the inverse Fourier transform of the factors ~±
from canonical factorizations of & -sectorial matrix-functions
[2,3] in general are not elements of ~ .
The problem to find continuous generalizations for the Szeg~
second limit theorem admits different formulations and even in the
definite case corresponding investigations form an "unordered set"
(see [8] and the papers cited there). There are still more unsolved
questions in the case & ~ 0 but we shall not go into this matter
here.
288
REFERENCES
I. r o a E H c E ~ ~ B.JI., M O p a r ~ M 0 B ~I.A. 0 npe~ea~Ho~
TeopeMe r.cerS. - MsB.AH CCCP, cepml MaTeM., 1971, 35, BNII.2,
408-427.
Kp e ~ H M.F. , c n ~ T E o B C E ~ ~ H.M. 0 ~aaTOp~Ssam~
MaTpm/-~y~zLU~ Ha e~ZHEqHO~ oEp3~KHOCTI~. --~OIO~I.AH CCCP, 1977,
234, ~ 2, 287-290.
K p e ~ H M.r., c n E T E o B C E E ~ 14.M. 0 ~aKTopHsam~
% --CeETopI~a~IBHNX MaTp]alI-~yHEI/~I~ Ha e~141IIiqHO~ OEpjfaHOCTH. --
~aTeM.I~CCJIe~OBalIK~, 1978, 47, 41-62.
K p e ~ H M.r., c ii E T E O B C E I~ ~ I~.~. 0 HeEOTOpRX
0606~eHE~X HepBo~ npe~e~Ho~ Teope~ Cere. - An~l.N~th., 1983,
9, N 1.
5. D e v i n a t z A. The strong Szego limit theorem. - Illinois
J.Math., 1967, II, 160-175.
6. B a s o r E., H e 1 t o n J.W. A new proof of the Szego
limit theorem and new results for Toeplitz operators with discon-
tinuous symbol. -J.0per.Theory, 1980, 3, N 1, 23-39.
7. E p e ~ H M.F. 06 o~o~ sEcTpanox~xmo~o2 npo6xeMe A.H.Ko~Moro-
poBa. - ~oF~.AH CCCP, I945, 46, ~ 8, 306-309.
8. M E E a e x ~ H ~.B. MaTpH~H~e EOHT~ETyS~w_RPS~e aHa~oI~ TeopeM
r.Cer~ o T'@~eB~X ~eTepM~HaHTaX. -- M3B.AH ApMCCP, I982, I7,
4, 239-263.
.
.
.
M. G. KREIN
(M.F.EP~H)
I.M. SPITKOVSKII
(H. M. CnETKOBC~ )
CCCP, 270057, 0~ecca,
yJi.ApT~Ma 14, EB.6
CCCP, 270044, 0~ecca,
HpoxeTapcK~ ~yzbsap 29,
MopcKo~ rH~oo~s~qecxM~ HHCTXTyT
OT~e~eHMe ~OHO~XH X
sKo~orxM M~t-posoro oxeaMa
289
5.12. THE DIOPHANTINE MOMENT PROBLEM, ORTHOGONAL POLYN0~IAT.S AND
SOME MODELS OF STATISTICAL PHrfSICS
I. In [I ], [2] it was shown that in investigations of the Ising
model in the presence of a magnetic field the following one-paramet-
ric Diophantine trigonometrical moment problem (DTNP) appears,
PROBLEM. Describe all non-negative m~asures ~ (@, ~) on the ci-
rcle T = { ~ ' ~ - e ~ @ ~ ~[-~,~[]} even in ~ , de~en~in~ o n a
~arameter ~, 0~ ~ ~ ~ , and such that
0 and the moments
1 0
are pol.vnomials (in ~ ) of de~ree ~e~ with integer coefficients;
the parity of Mk(~) coincides with the parity of ~k . Here ~e
is an inte~e r >I ~ (~ is the number of ,the nearest nei~hbours in
the lattice).
It is known that the description of such measures can be reduced
to the description of the corresponding generating functions
2. Hk( )-Tk , where T k are Tchebysh~v polynomials, _ 4-~ ~
re polynomials,
290
V~a note that in examples I-3 the generating function is ratiomal, whereas in example 4 it is algebraic (this case corresponds to the
orm-dimensional Ising model).
QUESTION. Has the ~eneratin~function correspondin~ to a DTMP,
to be algebraic?
Fixing a rational value of the parameter ~ , ~ = ~, p, ~
integers, 0 ~ p ~ ~ , * ~ ~, we see that our DTMP implies the fol-
lowing "quasi-DT~P":
-- --~ ~ k e ~O(~)= ~ , CK being integers,
0
In particular for ~ = 0 or ~(~-|) we obtain the following moment
PROBLEM:
Describe non-negative even measures whose trigonometrical moments
are inte~er~.
This problem is solved by the known Helson theorem [31:
. - 4 ( 2~sj . - 4
d,o(e)= Z o.,,~', o-- '~w + ~ g ~ s sOlO $=0 $='0
under some additional conditions on @s,~s [2].
II. It was shown in [4] that the theory of Toeplitz forms and or-
thogonal polynomials is closely connected with some problems of s~a-
tistical physics and in particular with the Gauss model on the semi-
axis. In this connection some mathematical problems appear whose so-
lution would be useful for the further investigation of such models.
I, Let ~C~) be an even non-negative summable function on T satisfying the Szeg~ condition
1 C el -
0
We define the function
0
291
- z k I z l < - N ~K , k-O
PROBLEM. Find necessary and sufficient condition s for ~K-~0 a_gs
k-~ (.physically the last condition means the absence of a long-
@rderDara~eter).
L ° It was shown in [4] that the condition ~ q implies ~--0 as k--co and
@O
k:o i + k
2, Let ~ (3 (~) be an even non-negative measure on T
~Ik-il = ]-~ i'~S (K-i)~O(@) ', k, i= 0,4,... 0
i s the co r respond i r~ T o e p l i t z m a t r i x . We denote by ~ k i / T (N) )N verse m a t r i x f o r JIK-jl : ( I I k -~ l k, i -O
PROBLEM. ~,ind an asymptotics for
and
the in-
N T(N) )_~ i,k-0
This problem appears in the study of the free ener~T in the Gaus-
sian model on the semi-axis with an external field (see 141 )° For exa-
mple, when ~O(~)-~ t ~@~(~), @ >0, this expression tends to ~/~
The numbers (T(N)) -~ ~ok , k =0,~,. ,N, are proportional to the coef-
ficients of the orthogonal polynomials ~N (Z) (see [41, [5]). This
leads to a PROBL~ of a more detailed investigation of the asymptotics
of ~N (e~@) as N-~ ~ in the presence of non-zero singular part
of the measure ~ . As we know only the case of an absolutely con-
tinuous measure was considered in detail (see, e.g. [6], [7I)°
3. The multidimensional Gaussian model.
Calculate the free energy and correlation functions under less
restrictive conditions than in [41, [81~.
292
REPERENCES
I. B a r n s 1 e y M., B e s s i s D., M o u s s a P~ The
Diophantine moment problem and the analytic structure in the ac-
tivity of the ferromagnetic Ising model. - J,Math.Phys., 1979, N 4,
20, 535-552-
2. B a a ~ a M a p O B B.C., B o a o B ~ ~ H.B. Mo~eJ~ Hsasra c
Ma~TImM noaeM ~ ~O#aHToBa nloo6aeMa MOMelITOB. - TeOlO.Ma~eM. ~Hs.,
1982, 53, ~ I, 3-15. 3, H e 1 s o n H. Note on harmonic ftulctions~ - Proo.Amer.Math.Soc.,
1953, 4, N 5, 686-691
4. Bm a ~ ~ M ~ p o B B.C., B o a o B ~ ~ H.B. 06 O~HO~ uo~e~
O~aT~C~ec~o~ ~s~FJ. -Teop. MsTeM. ~Hs., 1983, 54, ~ I, 8-
22.
5~ Ba a ~ ~ M ~ p o B B.C., B o a o B ~ ~ H.B. YlmB~eH~e BzNe-
- Xo~a, sa~a~a l~a~a - ~a~6epwa ~ op~o~oHaa~e ~o~o~e~.
-~oEa.AH CCCP, I982, 266, ~ 4, 788-792. 6. S z e ~ ~ G. Orthogonal polynomials. AMS Coll,Publ., 23, 2
ed., 1959,
7. 1 ~ o a ~ H C ~ ~ ~ B.J I . A c m ~ n ~ o ~ e c ~ o e i i pe~c~aBaeH~e ol~oz'o]~aa.l~- HSX MHOI'O~e~OB. -- Ycnex~ Ma~eM.~sy~, 1980, 35, ~ 2, 148-198.
8. Jl ~ H - ~ ~ H.D. l~oPo~el:~m~ a~maoP ~eol~m~ Ce#~. - HsB.AH CCCP,
cep.~m~eu., 1975, 39, ~ 6, 1393-1403.
V,S.VLADIMIROV
(B. C. B~A~I~4POB)
I.V.VOLOVICH
(~.B.BoaOB~)
CCCP, 117966, Moomm
ya. BaB~aOBa, 42
Ma~esaT~eoF~ ~ H C ~ T AH CCCP
293
5.13. THE BANACH ALGEBRA APPROACH TO THE REDUCTION METHOD FOR
TOEPLITZ OPERATORS
Let H ~ denote the Hardy subspace of L Z = L £ (T) , consisting
of the functions { with ~(~)=0 , ~<0, and letP be the or-
thogonal projection f rom [~ onto H ~ . For ~ @_ L ® = L " (T) the
Toeplitz operator with symbol & is defined on H i by T(~)~ = P(@@) •
Let ~ { H ~) be the Banach algebra of linear and bounded opera-
tors on H ~ . Given a closed subalgebra. B of ~ denote by ~T(B)
the smallest closed subalgebra of ~(H '~) containing all operators
T(¢) with Cb6 ~ . Furthermore, let Q(~) denote the so-called
q u a s i c o m m u % a t o r i d e a 1 of ~T(B) , i.e. the
smallest closed twosided ideal in ~T(B) containing all operators
of the fo= T(~)-T(~)T~) (~,~ 6~). It is a rather sur- prising fact that this ideal plays an important role not only in
the ~red/aolm theory of Toeplitz operators, but also in the theory of
the reduction method for operators A~T(~) .% (with respect
to the projections PIT defined by p~(~)~k I(~)~k) k=O k=O
For /~e~ IH ~) write A~rI{P~I i f the reduction method is ~,ppli- cable to A (see [3] for a precise definition). Finally, put
Q~=I-P~ and denote by ~ the group of invertible elements
of a Banach algebr~ ~ with identity.
For A 6~(H ~) , the following statements are easily seen to be
equivalent:
( i i i ) P AP,÷Q G (H ( ~ 7/I~ 0 ), and
(iv) Q,,A Q~, +P,, ~G'~(H ~)
t ' l ,~tT o
294
(.) A~GB{H~), V_~A'~V~EGB{H ~) (~,) and
~I(V_~A V~) i<oo, where V~=T(~),
V_~=T(~ -~) (~eB). There is an important estimate closely related to (v) (see [I]*)):
k (~)
which holds for every finite collection of functions
Now, given a closed subalgebra of ~ it follows
k ~, k~,
Gtq~ E L ~ from (I) tha;
defines a bounded projection S on ~T(B) . One can show that
5 (A)- ~-~ V_~ AV~.
zf A e ~ T ( ~ ) nG~cH ) , then A G ~ T ( ~ ) , since ~ T( ~ ). is a C -algebra. Thus S ( A ) makes sense and belo~Vs to #~T(~) . ~oreo~er, *~ A~R{P~ t h e n (ii) and
(v) imply the invertibility of S (~ ] .
i~ /~ ~nd S(A 4) ~r~in G'~(H~). The following special cases are of particular interest;
(a) A=T(~)~
, r i i u Ji , ll,iiii
~) See also H.K.H~ox~c~, 0nepaTop~ ra~e~ ~ T~_~a. CneET-
pax~a~ Teop~. - Hpenp~ ~0~ P-I-82, 2eH~rps~, 1982. - Ed.
295
(b) A~-~ .~Tq~(~ i) , where ~'i e {-4, ~} and,
of course, for ~ =-~ the invertibility of T ( ~ ) is part of the hypotheses.
For ~ ~ PC (~is the algebra of piecewise continuous functions on with only finitely many jumps) the case (b) is of importance in
connection with the asymptotic behavior of Toeplitz determinants ge- nerated by singular functions (cf.~S]). In the case (a) the conjectu-
re 1 is confirmed for A~C +~
in the case (b) for ~=(T~(@)) ~
(see [4], [7] ).
or (b6~X~PC (see [3],[6]), and
One possible way to attack these problems concerned with the re- duction method is to formulate them in the language of Banach algeb- ras and then to use localization techniques (cf.~6]). Define
W~ : H ~ "H%y
[A ® and denote by ~ the collection of all sequences ~}~=0
~ $ ~ ~$~ having the following property: there exist two
operators ~, ~ ~(H~) such that
(strong convergence . e i=t on {A I +[ = {
nach algebra with identity. If A ~ ~ (~ is the ideal o f compact
H ~ operators on ) or even if A ~ ~T(~ ~) , thenIP~AP~
(see [5] ). Notice that this is obvious for ~=T(@) , ~ ~ ,
since W~T(~)W~ = P~T(~)P~ , where ~(~)=~(~/~) . It can be proved that the set
296
J={{ A,}. AcP, TP~ ,WGW~+C~,T,?aU. I c,, p,l-,,o}
actually forms a closed two-sided ideal in ~ , and that the prob-
lem of the applicability of the reduction method to ~ ~ ~ T (~)
admits the following reformulation (see [6]):
A ~n{P~I~A,~ ~(#) and the c o ~ e t o f ~/j o o ~ t a i ~ i n g
{P~AP~} i , invertible i n ~ / J .
Note that now localization techniques can very advantageously be ap-
plied to study invertibility in the algebra A/~ .
There is a construction which is perhaps of interest in connec-
tion with the case (a). Denote by ~{~ ~T(~) the smallest
closed subalgebra of ~ containing all ~elements of the form
{P~T(~)P~t ,where @EB . I f C(T) c B then ~oeC~T(B ) and J c ~ t M T ( I ~ ) (of.l:2]). A,si~ t o each sequsnoo
{~I~}~G'~{PI,,,}T4~) n. ,~ t h e c o s e t [A]F~ ~ T ( B ) / ~ o o containing . ~ - ~ - ~ ~ . In this way a continuous homomor-
phism ,R: _~.,{P,,} T{B) ~ 4~ T (B) /T® is produced, and one has ~;DJ
CONJECTURE 2. If B= L ~ then ke~, ~ = ~ .
A confirmation of this conjecture would imply that
which, On its hand, would verify conjecture I for ~--|(~), ~L.
It is already of interest to find sufficient conditions for the va-
lidity of (2) in the case ~ ~ m ~° . Note that (2) was proved for
B =C+H ® or B = d ~ P C in [2].
REI~ERENCE S
I~ B o t t c h e r A., S i I b e r m a n n B. Invertibility
and Asymptotics of Toeplitz Matrices. Berlin~ Akademie-Verlag,
(to appear).
2. B o t t c h e r A., S i I b e r m a n n B. The finite sec-
tion method for Teeplitz operators on the quarter-plane with
297
piecewise continuous symbols. - Math.Nachr. (to appear).
3. r o x 6 e p r H.~., ~ e ~ ~ ~ M a H H.A. YpaBHeH~ B cBepTEaX
npoeFa~oHH~e MeTO~N EX pemeH~. MOCEBa, HayEa, 1971. (Transl.
Math.Monogr., Vol.41, AMS, Providence, R.i., 1974).
4. R e c h S., S i I b e r m a n n B. Das Reduktionsverfahren
f~r Potenzen yon Toeplitzoperatoren mit unstetigem Symbol. -
Wiss. Z. d. TH Karl-Marx-Stadt 1982, 24, Heft 3, 289-294.
5. R o c h S., S i I b e r m a n n B. Toeplitz-like Operators,
Quasicommutator Ideals, Numerical Analysis. - Math.Nachr. (to
appear).
6. S i 1 b e r m a n n B. Lokale Theorie des Reduktionsverfahrens
fur Toeplitzoperatoren. - Math.Nachr. 1981, 104, 137-146.
7. B e p 6 ~ ~ E ~ ~ H.8. 0 MeT0~e pe~Jz~H~ c~eneHe~ T~nn~e-
B~X MaTpHS. - MaTeMaTH~ecEHe Ecc~e~oBaHE~, 1978, BHH.47, 3--11.
B. SILBERNANN Technische Hochschule
Karl-Marx-Stadt
SektionMathematik
DDR-9010 Karl-Marx-Stadt
PSE 964
298
5.14. STARKE ELLIPTIZIT~T SINGUL~ INTEGRALOPEP~ETOREN UND SPLINE-APPROXI~T ION
Sei ~ eim Kurvensystem in ~ , das aus endlich vielen ein- fachen geschlossenen oder offenen Ljapunowkurven besteht, die keine gemeinsamen Punkte haben. Des weiteren seien ~, , ~ ~ paarT~ weise versehiedene Punkte, -4<&k<~ ( k=4 , . . . , M"I,) u.nd ~(~)
I'11,
I~--~ k I Ak , ~it ~.~(~) bezeichnen wir den Hilbertraum k'-~ auf ~ me~b~ren ~unktionen ¢ sit ~ 415~ ~ h ~ (F) . Wir aller
betrachten die singul~ren Integraloperatoren der Gestalt
F
mit stuckweise stetigen Koeffizienten ~,~ 6. PO (F) . Bekan- ntlich gilt ~r ~ Z ( ~.~ ( F, fl )). Mit ~ (P~) bezeichnen wir /ie kleinste abgeschlossene Teilalgebra yon ~ (~.~ ( ~, 2)) , die alle Operatoren A. sowie das Ideal ~ der kompakten Operatoren in [ ( F, ]~) enthalt, und mit ~ A das Symbol eines Opera- tors A £ ~(~C) (vgl. [6 ] oder [IO~).
I~ stetige Koeffizienten P~, ~ £ C ( ~ ) gilt
umd %~ SF ist eine stetige 9hniktion auf einer gewissem Raum- kurve "A = ~ (~) [7]. Im Falle des Intervalls ~ = [ ~, ~ ] ist ~ der Rand des Reohtecks [@,~]x[-~,4] . Wenn ~ nur aus geschlos~enen Kurven besteht und ~ ~---4 ist, dazm gilt
Die Abbildung Sym ist ein isometrischer Isomorphismus der symmetri- schen Algebra ~(C)/~ auf G(A) mit ~ A* - ~A
Der Operator ~ ~ ~ ( ~ . ~ ( F , ~ ) he is t s t a r k e 1 -
i i p t i s c h, gilt
wobsi D positiv definit und T ~ T~ ist. Wir nenmen A @ -s t a r k e 1 1 i p t i s c h, wenn eine 9kmktion @ @ C(~) @(%) @ 0 (~ % £ ~ ) , existiert derart, d~ 8Astark elliptisch ist.
299
~r 0peratoren der Algebra ~(6) gelten folgende Kriterien: 1°. A £ ~(C) ist genau dann stark elliptisch, wenu
"Re A ;'o. Ar = ~I + ~ ~r ( ~v,6 ~ C (I ~ )) ist ge:z~a1~
dann ~ -stark elliptisch, wenn
wobei K~({) die konvexe Hulls der }Ienge {~ ~F({,~)}(~,~)£A
bei fsstem ~ 6 ~ bezeiohnet. Im Falls f =- ~ ist K4(%) = [-~, 4] Die Hinlanglichket der Bedingung ~£ ~ A F 0 folgt leioht
aus den obengenannten Eigenschaften der Abbildung ~ A (vgl. [9] ); ihre Notwendigkeit ergibt sich aus der Hinlanglichkeit und der Eigenschaft, da2 der Operator ~ £ ~(C) gen&u dann sin Fred- holmoperator ist, wenn ~ ~ ~ 0 [6], [10]. Die Notwendigkeit von( 1 ) ist sine direkte Folgerung der Eigensch~ft 1°; ihre Hinlan- glichkeit kann man mit Hilts einer Einheitszerlegung der Kurve beweisen [3]. Wegen 6~£~s (.~p) = { ~ Sp(~,~)} (~,Z) ~ (vgl. [7] ) zieht die Bedingung
(2)
die Bedingung (I) nach sich; f~r konstante Koeffizienten ~ , ~ sind beide Bedingungen (1) und (2) aquivalent.
Die starke Elliptizitat ist sine notwendige und hinreichende Bedingung dafur, d~ f~r den invertierbaren Operator A die Reduk- tionsmethode bezuglich einer beliebigen 0rthonormalbasis konvergiert [4]° Wenn U =~ der Einheitskreis ist, so konvergieren fur den singularen Operator A T in h% (T) gewlsse Projektionsmethoden mit Spline-Basisfum/~tionen genau darm, we~ A~ @ -stark elllp- tisch ist [12], [13].
Wit betrachten auf
f -- -{)({k+ "4
0
wobei ~k (k= 0 , . . . , * -4) ' bezelchnen wir den Orthoprojektor in
die stuckweise linearen Splines
{k far { £ {k{k,~ SONS%,
mt P., L~(T) auf die lineare H'ulle
300
... . (~) ~ ~ $@~ ~ ) + . . , ~ ~.~ } und mit O~ den Interpolations- projektor, der jeder beschr~ten F~nktion ~ den Polygonzug
~'~ c~) (+,1
zuordnet. Wenn die Operatoren A~ = H ~ c h £ ( T ) (fur alle ~ 7 ~o ) i~L~er%ierb~r sind trod
~ t t ~ - ~ l l <~o ist, dann schreiben wir AT~]]~P,,,Q~}'~ in diesem Falle g i l t ~-~ ~ ¢ ~ ~'; ~ , .--~oo , ~ l l e ~ ~ i t Q+~ ~ ~ , i ~ e o o ~ d e r ~ e ~ ~ l l ~ ~ i e m ~ t e ~ i e : ~ - r e n 1~m~tionen [12]. Das soeben besohriebene 2rojektionsverfahren heist Kollokationsmethode (auch Polygonmethode). Analoge Bede~tung hat ~{P~, P,,} (Reduktionsmethode). Es gilt folgender
SATZ ([12],[13]). Se~ AT= @I +~ (~=~I¢~)
mit ~, ~ E PC ( r ) . Dann gilt A T ~ rl [ P~, O~} gena~
dann~ wenn folgende Bedingung erfS_llt ist:
C3)
+&c~.o)~,(-r.-o)(,t-.;)~o, V.'r.~r, '~y. ,~[o,q
Wir bemerken, da~ aus (3) die Invertierbarkeit von A T in L~(~) folgt [7], [10]. Im Falle ~ ~C(~) bedeutet (3)
gerade die ~-starke Elliptizitat yon A T-
HYPOTHESE I. Bedingung (3) ist ~quivalent tier ~ -starken
Elli~tizit~t ' des Operators AT im Raum L ~(~) (0 ~ ~C (T))°
Aus der G~ltigkeit dieser Hypothese wurde insbesondere A T ~ I PI, I,, PK. t folgen. Die Schwierigkeiten beim ~-berpr~en der
Hypothese I bestehen darim, da~ ~ AT eine Matrixfunktien und ~ (PC) eine nichtsymmetrische Algebra ist.
Von gro~em theoretischen und praktischenInteresse sind Bedingtm- ge~, die ~ Konvergenz entsprechender Kollokationsmethoden mit gew- ichteten Splines auf offenen Kurven garantieren (s.z.B. [2~, [8] ). Sei der Einfachheit halber ~= [0,~] , ~)=j/~t (i =0~,..,~)
und ~ ' ) die entsprechenden st~okweise lineare~ Splines ~ !~) - |~ -J
301
-41~ (~) =~ ~, . Nit ~ bezeichnen wir den 0rthoprojektor in
" C~) (~ ~ und mit
~ den entsprechenden Interpolationsprojektor.
2. Sei A~=~I+65 r- (~,%)£0C~)) ,~in E'fgOTKESE
-stark elliptische r Operator. Eann silt ~ 6
Im ~alle ~==~ ergibt sich die Richtigkeit der Hypothese 2 aus dem obengenannten Satz dutch Abbildung yon ~ auf eine Halfte
yon ~ und anschlie~ende Fortsetzung der Koeffizienten auf ganz
(vgl. ~0], Seite 86).
HYPOTHESE 3. Hypothese 2 ~ilt fur be!iebi~e Dol.ynomiale Splines
i ~erade,n Grades.
~r gesohlossene Kurven ~ wurde die Konvergenz der Kolloka-
tions - tmd Reduktionsmethoden mit Splines beliebi~ea Grades in nicht-
gewichteten Sobolewr~umen in [I], [11] ,[I~ untersucht.
LITERATUR
I. A r n o 1 d D.N., W e n d 1 a n d W.L. On the asymptotic con-
vergence of collocation methods. - Math.of Comput., 1983.
2. D a n g D.Q., N o r r i e D.H. A finite element method for
the solution of singular integral equations. - Comp~Math.with
Appl., 1978, 4, 219-224.
3. E 1 s c h n e r I., P r o s s d o r f S. ~ber die starke
Elliptizitat singularer Integraloperatoren. - Nath.Nachr. (im
Druck).
4. ro x 6 e p r M.~., * e a ~ ~ M a H M.A. YpaBHeH~ B cBepT~ax
npoemmomme MeTO~H ~X pemeH~a. M., HayEa, I97I.
5. r o x 6 e p r M.~., Kp y n H ~ ~ H.H. 06 aareOpe, nopo~t~eHHO~
O~HoMepH~ C~I~JIHRH~]M~ ~HTerpazBmes~ onepaTopa~m c EyCO~HO-He- npep~me~ ~os~mmeHTa~m. - ~HKA.aaaa. ~ ero npza., I970, 4,
.~ 3, 26-36 . 6. r O X 6 e p r M.H. , K p y n H ~ E H.H. C~I'y.Tr...Ep~e ~ T e z p a x ~ -
m~e onepaTop~ c Eycovao-~enpepmmm~ ~o~mmeHT8~,~ ~ ~x cm~oza.
- E s B . A H CCCP, cep.MaTeM., I97I, 35, ~ 4, 940-96Io
7. r 0 X 6 e p r Mo~., Kp y n ~ ~ E H.H. BBe~eH~e B Teopm0
O~HoMepH~X cm~yaapH~x EHTerpa~H~X onepaTopoB. Kmm~eB, ~T~mam,
1973o
302
8. I e n E., S r i v a s t a v R.P. Cubic splines and approxi-
mate solution of singular integral equations. - Nath. of Comput.,
1981, 37, N 156, 417-423.
9. K o h n I.I., N i r e n b e r g L.I. An algebra of pseudo-
differential operators. - Comm.Pure and Appl.Math., 1965, 18,
N 112, 269-205.
10. M i c h i i n S.G., P r o s s d o r f S., Singulare Integ-
raloperatoren. - Akademie-Verlag, Berlin, 1980.
11. P r o s s d o r f S. Zur Splinekollokation fur lineare Opera-
toren in Sobolewraumen. - Teubner - Texte zur ~ath. "Recent
Trends in Math.", 1983, Bd.50, 251-262.
12. P r o s s d o r f S., S c h m i d t G., A finite element
collocation method for singular integral equations. - Math.Nachr.~
1981, 100, 33-60.
13. P r o s s d o r f S., R a t h s f e 1 d A. Pinite-Elemente
Methoden fur singulare Integralgleichungen mit stuckweise
stetigen Koeffizienten. - Math.Nachr. (im Druck).
14. S c h m i d t G. On spline collocation f~r singular integral
equations. -Preprint P-Math.-13/82, Akademie der Wissenschaften
der DDR, Inst.f.Math., 1982.
S.PR~SSDORF Institut fur ~thematik AdW
Mohrenstra~e 39,
1086 Berlin,
DDR
303
5.15. HOW TO CALCULATE THE DEFECT IW3MBERS OF THE GENERALIZED
R~EMANN BOUNDARY VALUE PROBLEM?
The question concerns the problem of finding functions ~ ~P
satisfying the boundary condition
p Here ~,~ £L , k£ U ,4<~<~, ~ is a non-singular orientation preser-
ving diffeomorphism of T ("the shift") with the derivative in
~p~,0<~<~ . The case of orientation-changing shift & comes to
this'J~by an evident replacement of ~(~) by A~[~ ; & by ~ ; ~ by ~ •
The investigation of (I) and of its generalizations is connected
with a number of questions of elasticity theory ([1],Ch 7), the rigi-
dity problem for piecewise-regular surfaces [2], etc., and has alre-
ady a rather long history, starting with A.I.Markushevitch's work of
1946 (see [3] and a detailed bibliography contained therein).
Fredholmness conditions and the index of the operator correspon-
ding to the problem (I) are known and don't depend upon "the shift"~
If ~4~ is sufficiently small, then under certain additional conditi-
ons on & (e.g. II~ ~<~Cp,-~ ~E~ ~ being the class
(introduced in [5]) of multipliers not affecting the factorizability)
one of the defect numbers of (I) is equal to zero, and therefore de-
fect numbers don't depend upon "the shift". I.H, Sabitov's example (see
[3], p.272) shows that this is not the case in general.
PROBLEM. Calculate the defect numbers of the problem (1). Find
the c.onditions on the ' coefficients ~,~ , under which the defect num-
bers do not depen d upon "the shift".
The defect numbers ~ and ~' of the problem (1) without "shift"
(~(~)={, { £T ) are connected [3,4] by formulas ~= ~(~i~)+ e~uX(~,0)-I~ 61= ~4 +~'~'~ with partial indices ~4,~ of
the matrix
(here the defect numbers are calculated over ~ ).
The problem to calculate partial indices of matrix-valued functi-
ons even of this special kind, however, is far from final solution.
S04
Under assumptions
(2)
(where ~ = @-@I 6, ~ . i s an o u t e r f u n c t i o n wi th I~+1 = lml a . e . ) , as it is shown in [6], ~I and ~ are expressed in terms of the multi-
plicity of S-number I of the Hankel operator 0~P ( P is the Ri-
esz projection of ~P onto H P, Q-- ~- P ). using this fact and
the results of V.M.Adamjan, D.Z.Arov and M.G.Krein ~7,8,9] the defect
numbers of problem (I) are expressed in [6] in terms of approximation
characteristics of its coefficients. The elimination of restrictions
(2), and the generalization of the above-mentioned results to the
weighted spaces seems to be of interest.
In the case ~({)~ the calculation of defect numbers of (I) may
reduce to the problem of calculation of the dimension of the kernel
of the block operator
Pw2 '
composed by "slLifted" H~nl~el and T o e p l i t z ope ra to r s (here W,~= ~0~
is the so-called "shift" (translation) operator). It is interesting
to remark, that these operators also appear while investigating the
so-called one-sided boundary-value problems, studied in [I0] .Thus,
the investigation of the problem ~(%)=0v~ +~, ~6 ~?, ~
with an involutory orientation-changing "shift" ~ under ordinary
conditions ~3] ~" ~=~ ~@~)a 0 can reduce to the study of the
operator ~W~ 0 : both have the same defect numbers, their images
are closed simultaneously and so on. Thus, the new information about
"shifted" Toeplitz and Hankel operators may be employed in the study
of the boundary value problems with the shift.
In conclusion it should be remarked, that by an analytical conti-
nuation of ~ into the domain {~£C :I~I>4] and by a conformal map-
ping, the problem (I) can be reduced to the problem of finding the pair of functions in Smirnov classes Ev+, satisfying a "nonshifted"bou- ndary condition on a certain contour. We note, by the way, that the
related question of the change of partial indices of matrix-valued
functions under a conformal mapping (i.e., practically, the question
of calculating defect numbers of vectorial "shifted" Riamann boundary
value problem) put by B.V.BoJarski~ [11], has received no satisfacto-
ry solution so far.
The authors are grateful to I.M.Spitkovskil for useful discussi- ons.
305
RE~ERENCES
I. B e K y a H.II. C~0TeM~ C~HI~JIS!0H~X BHTST!0aalSHSX y!08BHeHm~ ~ He-
EOTOI~e I"!08H~Hse sa~a~, M., HayEs, 1970.
2. B e z y a N.H. 06o6~essue aHa.Tl~T~l~ecF,~e ~ . r ~ a a , M., (I:~, 1959. 3. ~I a T B ~ H ~ y Z r.c. KpaeBue sa,~aqa a caHr~Slo~e aHTer!oaa~-
Hue ylmBSe~s CO c~Baroa, M., HayEs, 1977. 4. C II a T K O B 0 K ~ ~ H.M. E Teop~H O606~esHo~ KpSeBO~ Sa~SR~
PaUSHa B F~accax L P . - Yl~p.~aTem.mypH., I979, SI, ~ I, 63-78.
5. C n z T E 0 B C E Z H H.M. 0 ~HOZZTe2SX, He B~lZ~0~X Ha SSETOpS-
SyemOCTB. - ~oF~.AH CCCP, I976, 231, ~ 6, I300-I308. 6. ~IRTB~HNyE r.c., Cllz TEOBCE~R 14.M. TOWHee
o~eHE~I ,~e~e~T~Ux ~ac8,,'l 060611~eEHo#r KIBeBOf~ sa,~a~a Pa~aHS, ~8KTOp~-
sa~zs @I~ZTO~UX ~aTp~-Q~y~t ~ ~eEOTOl~e npo6~e~u np~6~eHas seposOl~Uz ~ySF~ZS~Z.- MaTe~.c6OpH., 1982, 117, 2 2, 196-214.
7. A ~ a ~ s H B.M., A p o B ~.3., K p e ~ H ~.r. 0 ~ecKoHe~-
HUX ~as~e2eBux aa~pa~ax ~ 0606~eHH~X 3a~sNax EapaTeo~opa-@e~epa
*.Pacca. - ¢~HF~.a~a~. zero n!oz2., I968, 2, ~ I, I-Ig. 8. A~ a u s H B.~., A p o B ~.3., Kp e ~ H ~.r. Bec~ose~sue
~am~e~eBs MaTp~II~ a o6o~meHm~e sa~m~ KapaTeo~op~-~e~elm z H.~Ie.
-~H~s~.aHa~. a ero nlO~., 1968, 2, ~ 4, 1-17.
9 . A ~ a M s H B.M. , A p 0 B ~ I .3 . , K p e ~ H M . r . A ~ m ~ T a ~ e c -
E~e CBO~CTB~ nSp ~aATa raH~e~eBs onelmwop8 ~ o6o6=eHHms sa~a~a
~ypa - TaEsr~. - MSTe~.c6opH., IgVI, 86, ~ I, 33-73.
I0. 3 B e p o B ~ ~ 3.~., ~I ~ T B ~ H ~ y ~ r.C. 0~HOCTOpOHHae
KI~SBHe 38~8~Z Teop~ 8Ha~T~eCE~X ~ H . - HsB.AH CCCP, cep.
~STe~., 1964, 28, ~ 5, I003-I036. II. B o , p C E a ~ B.B. AHS~I~S I)SSI~8~ISMOOT~ I~IOSHZEHRX sa~aq TSOID~
~HE~R~. - S EH. : HCCAe~0BSHRS no COB!O.IlpOO~leua~ TeoI~ ~SyHE~R
~o~n~ezcHoro nepe~esHoro, M., ~ , 1961, 57-79.
Yu° D. LATUSHKIN
(~.~.~~)
G. S. LI TVINCHUK (r.c.m4TBHRNYK)
CCCP, 220000, 0~ecca,
~.HeTpa Be~oro 2,
0~eocE~ roc?~spc TBeH~u2
yHMBepC~ Te T
306
5.16. POINCAPd~-BERTRAND OPERATORS IN BANAOH ALGEBRAS
Let A be an associative algebra over ~ . A linear operator .
~E#~ A is said to satisfy the Pomnoare-Bertrand identity if
for all X,Y~A
~CX" EY+ EX.Y) =~X. ~Y+XY . (~)
THEOREM. Suppose ~ satisfies ' (1)~Then
(i) the formula
X x y = gX'Y+X'gY
defines an associative product in A (We denote the corresponding al~ebra by A~ );
(ii) The mappi~s ~ ± { are homomorphisms from A& into
A - ~et A± =J~C~tD , N±=Ke~C~4) • ~hen A± ~ A is a sub-
algebra apd N!~ A± is a ~wo-sided i dgal, Also, A = A+ + A_ ,
N+~N-=O ( i i i ) The mappi~ of the q,uotient ' a,,lgebras
e: A+/N+--- A_/N_
~iven by e :(~+0X - - ~ - ~ ) X is an al~eb~ isomorphism;
(iv) each X~A can be tmiquely decomposed as
X=X+-X_ ,X± cA±, e~+)=X_ (we denote by X~ the residue class of X±
by
modulo N
E~JL~I~o Let A =W be the Wiener algebra. Define
e
~×= I X, if X is analytic in ~ ,
[ -X, if X is antisa~alytic in
~hen ~ satisfies (I).
PROBLEm. Po r A = W ® M@t C ~ c) o~ A = L ® M~t c~ c) describ~ alllinear operators satisfyiz~ (I~.
307
NOTES. I. This problem arises as a byproduct from the studies
of completely integrable systems. This connection is fully explained
in [1]. For partial results in the classification problem cf. [I],[2].
Inmost papers the problem is considered in the Lie algebraic setting
In that case equation (I) is replaced by
+ y]+Cx,y]
which is commonly known as the (classical) Yang-Baxter equation.
2. Given a solution of (I) an operator X ~-~-X ×g Y can be
regarded as an analogue of the Toeplitz operator with the symbol Y .
It seems interesting to study the corresponding operator calculus in
detail.
REFERENCES
I. C e M ~ H O B -- T ~ H -- ~ & H C E E ~ M.A. I~TO ~aEoe ExaccH-
~ecEa~ t-MaTp~a. -~.a~a~. ~ ero np~., 1983. 17, ~ 4.
2. B e ~ a B ~ R A.A., ~ p ~ H ~ e ~ ~ ~ B.r. 0 pemeH~i~x
~acc~ecEoro ypaBHe~ H~r~a-Ba~cTepa ~ npOCT~X a~re6p ~. -
@yH~.a~a~. ~ ero np~., I982, I6, ~ 3, 1-29.
~.A. SE~'NOV-TIAN-SHANSKY (M.A. CE~0B-THH~CK~)
CCCP, I9IOII, ~eH~Hrpa~,
~oHTa~Ea 27, ~0~
CHAPTER 6
SINGULAR INTEGRALS, BMO, H p
This c h a p t e r i s a n a t u r a l c o n t i n u a t i o n o f t he p r e c e d i n g one:
e i g h t problems opening the c h a p t e r dea l w i th s i n g u l a r i n t e g r a l s . The
two f i r s t a r e "o ld" (and a r e e s s e n t i a l l y i n f l u e n c e d by C a l d e r 6 n ' s
1977 b r e a k t h r o u g h i n f ,~ -es t imates of Cauchy- type i n t e g r a l s on
Lipsch~tz curves (see S.5 and Commentrary therein). Others cover va-
rious aspects of the theory of singular integrals (continuity, two-
sided estimates as in 6.8 and even the exact values of their norms
as in 6.6).
Unlike the preface of the "old" Chapter9 we dispense here with
emotions caused at that time by the very appearance of BHO and the
real HP-spaces. The HP-BMO ideology has shared the destiny of
all sis~lificant theories (see Introduction to Chapter I) being now -
together with ~,P's or C - a necessary prerequisite for analytical
activity, as though they (i.e. BMO and H P )"existed always" but pas-
sed unnoticed for a period of time.
BMO is ubiquitous as is seen, e.g., from the items of this (and
not only of this) Ghapter. In Problem 6.10 BmO is intertwined with
famous coefficient problems for univalent functions. In some problems
its presence or influence is not so explicit (as e.G. in 6.11, 6.12~
6 14 or in Problem 6.13, dealing with a quantitative variant of
the John - Nirenberg inequality) but nevertheless undeniable The sa-
309
me can be said about VMO-setting of the "old" Problem by Sarason
(S°6). Various aspects of the ~P -theory (real or complex) are dis-
cussed in 6°4, 6,9, 6~13, 6,15-6,19. Other interesting connections
are represented by items 6.9 aud 6.10. These problems are of importan-
ce for Toeplitz operators (see Chapter 5)~ The solution of S,6 found
by T.Wolff (see Commentary in S,6) yields a useful factorization of
unimodular functions in ~+ C leading to a factorization of Toeplitz
operators with (~$ G) -symbols, The prediction contained in the last
phrase of Section 2 in 6.9 was more than justified: the n e g a t i v e
solution obtained by T.Wolff (see Commentary to 6~9) also has an impor-
tant application, namely, the existence of a non-invertible Toeplitz
operator whose symbol has a Poisson extension bounded away from zero
This disproves the famous conjecture of Douglas.
We conclude by the indication of Problem S.11 inspired by the abs-
tract HP-theory of Coifman - Weiss. (We first included the problem
into this Chapter, but people became aware of it before the volume was
ready and got so interested that we had - at the last moment - to re-
move it to "Solutions".)
6.1.
old
a function defined on
defined as:
310
ON THE CAUCHY INTEGRAL ANDRELATED INTEGRAL OPERATORS
Let P be a rectifiable curve in 6 . The Cauchy integral of
and integrable relative to arc length is
Recently A.P.galderon [I] has proved the existence almost every-
where of nontangential boundary values for the function C(~) (de-
noted as C(~) (~) ). This poin%wise existence theorem follows from
the following estimate proved in [I] by an ingenious complex variab-
le method.
THEOREM I (A.P.Calder6n). There is a constant ~o~ ~o > O,
mlllllfO= II 'll < there exists a
constant ~ for which:
It is not hard(by using singular integral techniques) to re-
duce the existence a.e. result mentioned above to theorem I. SEVERAL
IMPORTANT QUESTIONS remain open.
L Is the restriction II~'I~ < T0 necessary to obtain the ,es-
, ,, • ! timate of theorem 1? Calderon s method as well as other techniques
are unable to eliminate this restriction.
IX. Since the operator 0(#)(~) exists almost everywhere for
all functians in L~(P, I~I) , it is natural to conjecture the
existence of a weight cop (~) ( > 0 a.e.) for which
311
(The existence of such a weight for a weak ~ estimate is guaranteed
by general considerations related to the Nikishin-Stein theorem.)
IIL The integral operator appearing in Theorem I is related to
a general class of operators like Hilber~-transforms, of which the
following are typical examples.
a) The so called commutators of order
- f , -~
b) ! Ac o+'b- Ac.-lb ( -lb
~>0 (Here ~ l~ ),,
It is easily seen that ~heorem 4 is equivalent to the following
estimates on the operators A~ : A
for some constant C . The boundedness in L ~ of the operators in
a) b) c) has been proved in E2], E3~ by using Fourier analysis and
real variable techniques (which extend to ~ ). Unfortunately the
estimate obtained (by these methods) on the growth of the constant
in (,) is of the order of ~V (and not 0 h ). It will be h$~hl~
desirable to obtain a ~roof of Calder~n's result which does not de-
~end on special tricks or complex variables. Any such technique will
extend to higher dimensions and is bound to imply various sharp esti-
mates for operators arising in partial differential equations.
312
REFERENCES
I. C a I d e r 6 n A.P. On the Cauchy integral on Lipschitz cur-
ves and related operators. - Proc.N.Ac.Sc.1977, 4, 1324-27.
2. C o i f m a n R.R., ~ e y e r Y. Com~utateurs d'integrales
szmgulieres et op~rateurs multilinealres.--Ann, Inst.
Fourier (Grenoble), 1978, 28, N 3, xi, 177-202.
3. C o i f m a n R.R., M • y • r Y. Multilinear pseudo-
differential operators and commutators, to appear.
R.R.COII~N Department of ~athematics
Washington University
Box 1146, St.Louis, M0.63130
USA
YVES ~EYER Facult~ des Sciences d'Orsay = ° r
Universlte de Paris-Sud
Prance
CO~ENTARY
The solution of Problem I is discussed in Commentary to S. 5.
313
6.2. SOME old
PROBLE~gS CONCER/~ING CLASSES OF DO~IA~TS DETER/gINED
BY PROPERTIES OF CAUCHY TYPE INTEG~%LS
Investigation of boundary properties of analytic functions rep-
resentable by Cauchy-Stieltjes type integrals~ni a given planar do-
main G (i.e. functions of the form ~b-~ (~-%f~); if ~G
~ = ~ we denote this function by ~ ), as well as some
other problems of function theory (approximation by polynomials and
rational fractions, boundary value problems, etc.) have led to intro-
duction of some classes of domains. These classes are defined by con-
ditions that the boundary singular integral ~V (D (~ = ~) should
exist and belong to a given class of functions on ~ or (which is
in many cases equivalent) that analytic functions representable by
Cauchy type integrals should belong to a given class of analytic fun-
ctions in G . see [I] for a good survey on solutions of boundary
value problems. An important role is played by the class of curves
(denoted in [I] by ~p ) for which the singular integral operator is
continuous on mP(r)~ ~ ~ ~p if and only if
Vco LPCr) llSrC °)ll Cpll°°ll P • (I)
Ep (,G) functions
some system of closed curves
are bounded. That is~ ~p can be
This means that the M.Riesz theorem (well-known for the circle) holds
for r . Some sufficient conditions for (1) were given by B.V.Hvede-
lidze, A.G.D~varsheishvily, G.A.Huskivadze and others. I.I.Danilyuk
and V.Yu°Shelepov (a detailed exposition can be found in the mono-
graph [~ ) have shown that (I) is true for all p>~ for simple
rectifiable Jordan curves r with bounded rotation and without
Some general properties of the class ~p were described by c u s p s .
V.PjHavin, V.A.Paatashvily, V.M.Kokilashvily and others. It was
shown, e.g., that (I) is equivalent to the following condition:
being the well-known V.I.Smirnov class (cf. e.g, [~) of
# analytic in 6 and such that integrals of I~I P over
( w i t h , )
characterized by the property
314
where ~ is a conformal mapping of ~ onto ~ .
Another class of domains (denoted by ~ ) has been introduced and
investigated earlier by the author (cf.[4] and references to other
author's papers therein). We quote a definition of K that is close-
ly connected with definition (2) of ~p : ~ K if for any func-
tion ~ in ~ , analytic and representable as a Cauchy type integ-
ral, the function C~o~) ~' ( g being a ccnformal mapping of D
onto 6 ) is also representable as a Cauchy type integral: K)
Note that by Riesz theerem it is sufficient for F~p that~he
function ( ~o~)~ in (2) be representable in the form
with ~LPCT) . This allows to consider K as a counterpart of
~p for p= I (it is well-known that to use (2) directly is impos-
sible for p=1 even for F= T ). It is established in [5]
(see also ~]),using Cotlar's approach, that the classes ~p coinci-
de for p > 4 . Thus the following problem arises naturally.
PROBLEM I. Do the classes ~p <p > ~) an___~d ~ coincide?
If not, what..~eometric conditions guarantee ri~_.~p ~ ~4 ?
Note that for ~ ~ ~ it becomes easier to transfer many theo-
rems, known for the disc, on approximation by polynomials or by ra-
tional frsctions in various metrics (of. references in [4]). It
is possible to obtain for such domains conditions that guarantee con-
vergence of boundary values of Cauchy type integrals [4]. As I have
proved, ~ is a rather wide class containing in particular all do-
mains ~ bounded by curves with finite rotation (cusps are allowed)
[4]. At the same time, it follows from characterizations of
proved by me earlier that ~ coincides with the class of F~ber do-
mains, introduced and used later by Dyn'kin (cf. e.g.[6],[7]) to in-
vestigate uniform approximations by polynomials and by Anderson and
~) In virtue of a well-known V.I.Smirnov theorem, the analog of this property for Cauchy integrals is always true.
315
Ganelius [8] to investigate uniform approximation by rational frac-
tionswith fixed poles. This fact seems to have stayed unnoticed by
the authors of these papers, because they reprove for the class of
Faber domains some facts established earlier by me (the fact that
domains with bm~uded rotation and without cusps belong to this class,
conditions on the distribution of poles guaranteeing completeness,
etc.). The following question is of interest.
PROBLEM 2. Suppose that the interior domain ~+ of a curve r
belongs to K (=~) . Is it true that the ̀ extgrior domain 6 als__._~o
belongs to ~ ? (Of course, we use here a conformal mapping of
6- o to {lwl> } ). For ~p with p >J the positive answer to the analogous que-
stion is evident.At the same time the similar problem formulated in
[9] for the class S of Smirnov domains remains still open.At last
it is of interest to study the relationship between the classes S
of Smirnov domains andAoof Ahlfors domains (bounded by quasicircles
[SO]), on the one hand, and K and ~p (considered here) on the
other. See [9] for more details on ~andAo . It is known that
~pcS , K~:=~ ([4],[11]). At the same time there exist domains
with a rectifiable boundary in A o which do not belong to S (of.
[3],[~). Simple examples of domains bounded by piecewise differen-
tiable curves with cusp points show that K\ Ao ~% ~ .
PROBLEM 3. Pind ~eometric conditions ~uaranteeing
G KD2o A°. Once these conditions are satisfied, it follows from the papers
cited above and [12], ~3] that many results known for the unit disc
can be generalized.
One of such conditions is that ~ should be of bounded rotati-
on and without cusps.
RE FEREN CE S
I, X B e ~ e a E ~ s e B.B. MeTo~ HHTerpa~OB w~na Ko~ B paspHB--
HRX l~aH~l~Ix s~a~ax Teop~Gi rOJIOMOp~H2X ~yHEL~ O~J~io~ EOM!DIeEc--
HO~ HepeMeHHO~i. "COBpeMeHS~e npo6xeM~ MaTeMaTHEH", T.7, MOOEBa,
1975, 5-162.
2. ~ a H ~ a ~ E ~.H. Hepei~yx~H~e rpa~m~H~e ss~a~ Ha IL~OCEOCT~,
MOCEBa, HayEa , 1975.
316
3. D u r e n P.L., S h a p i r o H.S., S h i e 1 d s A.L.
Singular measures and domains not of Smirnov type. - Duke Math.
J., 1966, v. 33, N 2, 2%7-254.
4. T y M a p z ~ H r.~o PpaHH~e CBO~CTBa a~axHT~ec~Hx ~ ,
npe~cTaBm~X ~HTerpa~a~m T~na Eo~. -MaTeMoC6., I97I, 84 (126),
3, 425--439.
5 .H a a T a z B ~ x ~ B.A. 0 cmI t~yz~ap~ HHTerpaxax Ko~m. - Coo6~. AH r p y s . c c P , I 9 6 9 , 53 , ~ 3, 529 -532 .
6.~ H H ~ K ~ H E.M. 0 paBHoMepHoM np~6~eH~ MHOrO~eHS~m B
EOMr~eEcHo~ ~OCEOCT~.- 8a~.Hay~H.CeMEH.~0M~, 1975, 56, 164--165,
7.~ H H ~ Z ~ H E.M. 0 paBHOMepHoM np~6~eH~ ~ B mop~a-
HOB~X 06~aCT~X. -- C~6.MaT.~. 1977, 18, ~ 4, 775--786.
8. A n d e r s s o n J a n- E r i k, G a n e I i u s T o r d.
The degree of approximation by rational function with fixed
poles.- Math.Z., 1977, 153, N 2, 161-166.
9. T y M a p E ~ a r.h. I~aH~e CBO~CTBa EoH~pMm~ OTO6ps~e--
H~ HeEoTopRx E~aCCOB o6~aoTe~.-c6."HeEoTopHe BonpocN CoBpeMeH--
HO~ Teopm~ ~yHEn~", HOBOCH6HpcE, I976, I49-I60.
I0. A x ~ ~ o p c ~. ~eEn~ no EBaS~EOH~OpMmm~ oTo6pa~em~M.
MOCEBa, M~p , 1969.
II. X a B ~ H B.H. l ~ ~ e CBO~OTBa ~HTe#ps~OB TEa KO~ ~ IBp--
Mo~eoE~ conp~l~eHs~X ~yHE~ B o6XaCT~X CO cnp~eMo~ rpaH~-
~e~. -MaTeM.C6., 1965, 68 (II0), 499--517.
12. B e x N ~ B.H., M ~ E x ~ E o B B.M. HeEo~opwe CBO~CTBa
EOH~Op~x ~ EBaS~EOH~Op~X OTo6pa~eHm~ ~ np~e Teope~ EOHCT--
pyET~BH02 Teop~E ~y~. -- HsB.AH CCCP, cep~ MaTeM.~I974, ~ 6,
1343--1361.
13. B e x ~ ~ B.H. EOH~OpM~e OTO6pa~e~ ~ npH62eH~e aHaJmT~e-
CKEX ~ B o6~a0T~X 0 EBaS~EOH~Op~Ho~ rpaH~e~. - MaT.c6.,
1977, 102, ~ 3, 331-361.
G. C. TUMARKIN
TYMAPEEH) CCCP, 103912, MOCEBa,
npocn.MapEca 18,
MoCEOBO~ reo~oro-
pasBe~o~ ~CT~TyT
CO~NTARY
A complete geometric description of the class ~p, ~<~<oo, has be-
en obtained by Guy David. See Commentary to S.5 for more information.
317
6.~. BILINEAR SINGULAR INTEGRALS AND N&XI3~bL FUNCTIONS
While the boundedness of Cauchy integrals on curves is now fair-
ly well understood D], there remain some difficult one dimensional
problems in this area~ One such example i8 the operator
p.v. T
I~s t~ a bounded operator fro m L~X L~ t_~o A. P. Calder6n ?
first considered these operators during the 1960's, when he noticed
(unpublished) that the boundedness of T I implies the boundedness
of the first commutator (with kernel 8(x;-A~) , A ~ ~ )
as an operator from ~ to Lw • In order to make sense out of T~, it
seems that one must first study the related maximal operator
f
and see whether T~ is a bounded operator~from h ¢ x h ¢ to h ~
It is easy to see that ~. maps to weak
REPERENCE
I. C o i f m a n R.R., M c I n t o s h A., M e y e r Y.
L'int@grale de Cauchy d@finit un op@rateur born@ sur L ~ pour les
courbes Lipschitziennes. - Ann.Math.,1982, 116, 361-387.
PETER W.JONES institut Mittag-Leffler
Aurav~gen 17
S-182 62 Djursholm
Sweden
Usual Address:
Dept.of Mathematics
University of Chicago
Chicago, Illinois 60637
USA
318
6.4. WEIGHTED NORM INEQUALITIES
The problems to be discussed here are of the following type.
Gzw~ p ~TISH~G ~ < p < ~ A~D TWO OPen,ORS T A~D $ , D~TE~n~
ALL PAIRS OF NONNEGATIVE ~NCTIONS U, V SUCH THAT
throughout this paper C denotes a constant independent of ~ but
not necessarily the same at each occurrence. There is a question of
what constitutes a solution to this sort of problem; it is to be
hoped that the conditions are simple and that it is possible to de-
cide easily whether a given pair U • V satisfies the conditions.
In some cases, particularly with the restriction ~ : V , this prob-
lem has been solved; for a survey of such results and references to
some of the literature see [3~. Some of the most interesting unsolved
and partially solved problems of this type are as follows.
p <cO ~ind all n0nne~at~ve pairs U and V suc___~h I. Por <
that
O0 O0 oo
(I)
This tw~ dimensional version of Hardy's inequality appears easy be-
cause T can be assumed nonnegative and no cancellation occurs on
the left. The solution of the one dimensional case is known; the ob-
vious two dimensional version of the one dimensional characterization
is
$~ O0
for 0 ~ $~ <~ , This condition is necessa~j for (I) but not
sufficient except for e=~ . see [7] for a proof that (2) is not
su_fficient for (I) and for additional conditions under which (2) does
imply (I).
2. For ~ < ~ <@@ find a simple characterization of all non-
319
negative pairs U , ~ such that
where
wood maximal function.
[ MI(m)] P U(~)%~ .<C II]~(~)l P Vm %~ , (~)
M~(,):~(~ ~)~!I ~(~)l%t is the Hard,y-Little-
This problem was solved by Sa~nJer in [5];
his condition is that for every interval I
I[ M (~I(,)V(~)~)l p U(,)~ C IV(~) P~ I I
with C independent of I. It seems that there should be a characte-
rization that does not use the operator M . one CONJECTURE is
that (3) holds if and onl~ if for ever~ interval I and ever~ subset
E of I with
I
iEl-llll~ we have
SV(,i I E
(5)
E T with ~ independent of and I. Condition (5) does give the
right pairs for some of the usual troublesome functions and is not
satisfied by the counter example in [5] to an earlier conjecture.
3. For ~ ( p <0o find a simple characterization of all non-
negative pairs U , V such that
1 IH~(~)l P U(~)~,, cl I~(~)IP V(,)~ (~
where HI(~)=c~. I ~(X-~)/~ ~ is the Hilbert trans-
form. There i icated solution to the periodie version ef
this by Cotlar and Sadosky in [I]. One CONJECTURE here is that a
320
]:),air U , V ss, t i s f ies (6~) i f and onl~ i f U , V satisfy (3) and
(7)
I where p-P/(P-h
4. ~or J ~< p<+~ find a simple characterization of all non-
negative pairs U , V for which the weak t~ype inequality
IH,~(=)b,~ -"
is valid for ~>0 • A CONJECTURED SOLUTION is that (7) is a ne-
cessar~ and sufficient condition for (8).
5. For ~( p <~ find a characterization of all nonnegative
functions U such that
I, I P A necessary condition for (8) is the existence of positive constants
C and ~ such that for all intervals I and subsets ~ of I
I--tq-J i IIl~+l~;-~h:l"' ' E
(lo)
where ~v denotes the center of I and that if (10)
CONJECTURED
6. FO r
which
~-p . In [6] it is shown
holds for some ~ >p , then (9) holds. It is
that (I0) with ~--p is also sufficient for (9),
< p~<~<~ find all nonne~ative pairs U ' V for
.< C [ t~ (c'~ I P Vm ~']4/P (11)
321
It was shown by Jurkat and Sampson in ['2] that if for
0 0 (t2)
where indicates the nonincreasing rearrangement and p=p/(p-~), then (11) holds. Furthermore, if (11) holds for all rearrangements
of U and V , then (12) is true. However, (12) is not a necessary
condition for (11) as shown~in [4]. This problem is probably diffi-
cult since if p=~=~ and V(~)=]zl @, O<a/<~ , then the ne- cessary and sufficient condition on U is a capacity condition. Its
difficulty is also suggested by the fact that a solution would pro-
bably solve the restriction problem for the Fourier transform.
REFERENCES
1. C o t I a r M., S a d o s k y C. On some ~ versions of the
Helson-Szego theorem. - In: Conference on Harmonic Analysis in
Honor of Anteni Zygmund, Wadsworth, Belmont, California, 1983,
306-317.
2. J u r k a t W.B., S a m p s o n G. On rearrangement and
weight inequalities for the Fourier transform, to appear.
3. M u c k e n h o u p t B. Weighted norm inequalities for classi-
cal operators. -Proc.Symp. in Pure Math 35 (1), 1979, 69-83.
4. M u c k e n h o u p t B. Weighted norm inequalities for the
Fourier transform. - Trans.Amer.Math.Soc., to appear.
5. S a w y e r E. Two weight norm inequalities for certain maxi-
mal and integral operators. In: Harmonic Analysis, Lecture Notes
~th. 908, Springer, Berlin 1982, 102-127.
6. S a w y e r E. Norm inequalities relating singular integrals
and the maximal function, to appear.
7. S a w y e r E. Weighted norm inequalities for the n-dimensio-
nal Hardy operator, to appear.
BENJAMIN MUCKENHOUPT Math. Dept.
Rutgers University
New Brunswick
N.J. 08903, USA
322
6.5. A SUBSTITUTE FOR THE WEAK TYPE (I,~) INEQUALITY ~OR MULTIPLE RIESZ PROJECTIONS
Let C A (T~)denote the poiydisc algebra, i.e. the subspace of C(T ~) consisting of the restrictions to the ~-dimensional toras ~ of f~nctio~ ~alytic in the open poly~so ~ d continuous in ~D. ~ By H (T)we denote the closure of ~A ITs) in ~(T~)and by i:C A C~ ~) -~ ~ (T ~) the identity operator. The space H~(T~) ~ will be iden- tified with H~(T~), the bar standing for the complex conjugation (we
use throughout the duality established by the pairing < {,~ >-
Irl(0) 9(s)%0) PROBLEM I. Does there exist a positive function ~ on (0,~] with
~(~)~0 a__~ ~0 such that for e~ch ~ H~(T ~) with II~II~=4 the
fol!owi ,n~, ,%nequality holds:
(throughout ~' ~p denotes the ).P -norm).
If ~4 the answer is evidently "yes". Indeed, in this case the Riesz projection P. (i.e. the orthogo-
hal projection of ~(T 4 ) onto ~4) ) is of weak type (1,1) and
so the above function ~ satisfies the estimate
Using this and ~=~ it can be shown by means of a simple calcula- tion that (~) holds with ~(~)- ~£ ( ~, 6~ ~-~ ) (and moreover, for all .~' 4<p6~ we have lflIl̂ l a ~<0(O-|)'4{{~*~II~ with
given by ~-4__ @+ (4-0)/~ ). ~ !
Pot ~ the orthogonal projection of ~@(~)onto ~(~ (which is nothing else as the ~-fold tensor product ~. ®... @ P- ) is no
longer of weak type (1 ,I) and the above argument fails. Nevertheless for ~=~ PROBLEM I also has a positive solution. This was proved by the author [1] with q a power function. Using the same idea as in [I] but more careful calculations it can be shown that for ~=~ and
~< ~< ~ we have
{1 11 .< 11 '9 ( ~ is the same as for ~=~) provided ~ £ ~ and 11~11-- =~- Consequently, (I) holds for ~=~ with. q(£) = ~ ~ ((+ ~ ~-4~ (to see this set ~= *~+(~ 1~'~'/'~ ~-4)'I in the preceding i" °nequality).
Nothing is known for ~ . It seems plausible that for such ~v
323
PROBLEM I should also have a positive solution. Moreover, I think
that for I<~<~ the inequality ll~Ip~<C~ (p-()~ II~*~ ]J' should
be true (and so (I) should hold with ~(~)=C~6 (~+(~'|)~) a .,,)"
The estimate (I) for ~=~ was used in [I] to carry over from 0~I)
T ~ some r C, "~ to ~A ( ) = esults whose standard proofs for A( ) use the weak type (1,1) inequality for ~.. (For example, it was established in [I]
5 that, given a A.-subset E of (Z+) , the operator ~, ~ =
= [~(K)~ ~EE maps ~A(T ) onto ~ (E) . It is still unknown if the
same is true with ~ and(~)%eplaoed by ~s/id (A) ~! ~ ~/£" )
So (I) is really a substitute for the weak type inequality~ Profound
generalizations of inequality (I) for ~=~ with very interesting ap-
plications can be found in [2] (some of these applications are quoted in COMMENTARY to S. I ).
The proof of (I) for ~=~ in [I] is based on the weak type (1,1)
inequality for ?- and a complex variable trick, and essentially the
same trick appears in [2]. Investigating the case ~ one may seek
a more complicated trick that also involves analyticity. But to seek
a purely real variable proof is probably more promising from diffe-
rent viewpoints. The solution of the following problem might be the
first step in this direction.
PROBLEM 2. Fi~d a real-variable proof of (1) for ~=~ .
In connection with PROBLEM 2 we formulate another problem which
is also rather vague but probably clarifies what is meant in the for-
mer, The inequality (I) is clearly equivalent to the following one:
provided ~ e ~ (T ~) and ~( P- ® . . . P- )k llm = 4. PROBLEM 3. At least for ~=~, , find and prove a "right" analo~ of
the above inequality involvin~ ~-fold tensor products of operators
of the form ~ ~(~ ~ ~)~ beir~ a measure on a multidimensional
torts and ~ bein~ a Oalderon - Zy~mund kernel on the same to~as,
rather than tensor products of Riesz pro,jections,
RFYERENCES
I . K m c z s K o B C.B. E o ~ s ~ e H ~ ~Yl~e r l ~ H S m ~ x s ~ e H u ~ ~ - - ~ d , a H a ~ T N ~ e o ~ x B ~oyre H B d z ~ o ~ ¢ e . - T loy~ MaTeM.~H--Ta BM.
B.A.CTeF~OBa, I98I, I55, 77-94.
324
2, B o u r g a i n J. Extensions of ~ -valued functions and
bounded bianalytic functions. Preprint, 1982.
S. V. KISLIAKOV
(C.B.K~C~0B) CCCP, 191011, ~e~Im~
@oH~aHEa 27, ~0MH AH CCCP
325
6.6. THE NORM OF THE RIESZ PROJECTION
The operator of the harmonic conjugation ~ and the Riesz pro-
jection P ( i .e . the orthogonal project ion onto H ~ in L~(Y) ) are connected by the simple formula S:~P-I . It has been proved
in [I] that
,[~"~zp (P=P) ' ISlL~4~I~ ' IPI~,,b-L~ (,~-~(p,p')). (1)
In [ I ] it has been also conjectured that the inequalities in (I) can
be replaced by equalities. In the case of operator ~ this conjec-
ture has been proved in [2,3], but for ~ the question remains open.
The following refinement of the main inequality of [2] has been ob-
tained in [4]:
t ii~t~llLP ~ iil~llHP~ t (2)
where ~ ~P , im / ~ ( 0 ) : 0. The right-hand s~de of (2) gives the norm of the restriction of ~ onto the space of all real-valued
functions in LP satisfying ~ (0) = 0
The same situation occurs for the weighted ~P spaces
T
where to ~ I , - I < ) < P-~ . The formula for the norm of LP(~) has been obtained in [~]. ~or P it is known o~ly that
(see [6])
in
(3)
The conjecture holds for p=~ because in this case the prob-
326
lem can be reduced to the calculation of the norm of the Hilbert
matrix [i+k+X }j,k)o ~ch =F ~/~,
a~d T~=P~ IH ~ is invertible in H ~' and Tj ~
IT~'I-- I~, P
. Here is a SKETCH OP THE PROOF. Let
. It is known [6] that the Toeplitz operator ~A
= ~+ P ~;~ . Consequently
IL~ =IPlEc#).
The ope~t~r T t is invert ible and I@I--~ , therefore (see [7]> lq I~=_ I P~QI £. Here ~=~-P and P~e= =(@(i*k+O)~,k>,0 is a Ha~e~ ope=tor. Let us note
that
k+~ +~I~ '
I t is known [8] that the norm of matrix [ ~ ) 4 , k>/0 equals 4 0
IPI,~,0~ =~-4~ ~I~ =~-~/~. • be a simple closed oriented L~punov curve; ~4~... ,~
be points on P IPloss( IP '~K) , ,~$5 ) be the essential norm of P
in the space L ~ [ ~ (L ~ (r, ~k~ on P with the weight ~ ~T)~
i t .as p~-o.,ed tha~" I P Io.~,.>.. m.o..~ ~ (p,~,.~ (~ (p,.~)~-~.~i.,¢~, ~.~.~i~ deZi,,,~ ~,y (3)) • Then "i-Z[ 5] i t ~ p r o ' V e d t'~'0 "P'I Io~. t.~= ~,~,,, I Pl ~'j'~ If our Conjecture is true then l P l ~ = ~ (p, ~ , ) .
In conclusion we note that in the space ~ on the circle T (~ thout , e i ~ t ) IPlo~ - I P l ( [3 ] ) . But i~ ~ene~l the ~o~ I P l depends on the weight and on the contour r (E3], [5] ).
327
REFERENCES
I. r o x d e p r H.ll., E p y ~ H ~ E H.H. 0 BopMe npeo6pasoBa-
~ r~depTa B npocTpaHcTBe L P . - SyH~.aaax. ~ ero np~.,
I968, 2, ~ 2, 91-92. 2. P i c h o r i d e s S.K. On the best values of the constants in
in the theorems of M.Riesz, Zygmund and Kolmogorov. - Studia Kath.,
1972, 44, N 2, 165-179.
3. E p y n R x ~ H.~., H o x o ~ c K ~ ~ E.H. 0 sopMe onepaTopa
cm~ryxspnoro ~Terp~poBan~s. - ~ya~. aRax. • ero np~x., 1975, 9,
4, 73-74.
4. B e p d ~ ~ E ~ i~ H.B. 0nem~a Hop~g~ ~EI~ HS IIpOOTI~OTBa Xap2 ~epes HopMy ee BemecTBemmo~ ~ M~o~ ~ac~z. - B cd."Ma~eM.
~cc~e~o~a~zs", l~m~eB, ~]T~a, 1980, J~ 54, 16-20.
5. B e p d • ~ • ~ ~ H.B., E p y n ~ • ~ H.~. To~e EOHCTSHTI~
TeopeMax E.H.Bade~o n B.B.X~e~se od OI~paH~IeHHOCTI~ C~Hry~p-
moro onepaTopa. - Cood~.AH rpys.CCP, 1977, 85, ~ I, 21-24.
6. r o x d e p ~ H.II., Ep y ~ ~ n ~ H.H. B~e~e~e ~ Teop~
c~jxsp~x ~e~pax~x onepa~opoB. - E~m~eB, ~ua, 1973.
7. H ~ ~ o x ~ c ~ z ~ H.E. ~e~ o6 onepaTope c~m~a. M. : Hay-
~, 1980.
8. H a r d y G.H., L i t t I e w o o d J.E., P ~ I y a G.
Inequalities. 2nd ed. Cambridge Univ.Press, London and New York,
1952.
I. E. VERBITSKY
(H. B. BEPBI~II~)
N. Ya. ERUPNIK
(H.~. KPYnHHK)
CCCP, 277028, Emmme~,
H~C~TyT reoalmsmrz ~ reo~o~
AH MCCP
CCCP, 277003, I~eB,
ElmmKeBc~ rocy~apcTBe~
Ym~BepcxTeT
328
6.7. IS THIS OPERATOR I~RTIBLE?
Let ~ denote the group of increasing locally absolutely con-
tinuous homecmorphisms k of ~ onto itself such that ~t lies in
the Muckenhoupt class A ~ of weights. Let Vk denote the operator
defined by V ~ ( { ) - - t o k , so that V~ is bounded on BM0(~) if
and only if ~ CG (~ones [3]). Suppose that P is the usu~l
projection of BMO onto BMOA. Por which k~G is it true that
there exists a C~0 such that II~V~(1)~M0 ~(~B~O
for all ~ BMOA? Is this true for all W~G ?
This questions asks about a quantitative version of the notion
that a direction-preser~¢ing homeomorphism cannot take a function of
analytic type to one of antianalytic type. For nice functions and
homeomorphisms this can be proved using the argument principle, but
there are examples where it fails; see Garnett-O'Farrell [2].
We should point out that, the natural predual ~ormulation of this H' U£#~#114 . I I14 l
J "
This also has the advantage of working with analytic functions whose
boundary values trace a rectifiable curve.
An equivalent reformulation of the problem is to ask when
V- HV ~4 ,,+ k ~ is inve~ible o~ ~o, if ,. denotes the Hilbert transform. This question is related to certain conformal mapping es-
timates; see the proof of Theorem 2 in [I]. In particular, it is
H this operator is invertible if ~ WU~M0 shoIAr~X there that
is small enough.
REFERENCES
I. D a v i d G. Courbes corde-arc et espaces de Hardy generallses.
-Ann.Inst.Fouzier (Grenoble), 1982, 32, 227-239.
2. G a r n e t t J., 0 ' P a r r e 1 1 A. Sobolev approxima-
tion by a sum of subalgebras on the circle. - Pacific J.Math.
1976, 65, 55-63.
3. J o n e s P. Homeomorphisms of the line which preserve BMO,
to appear in Arkiv for Natematik.
STEPHEN SEMMES Dept. of Mathematics Yale University
New Haven, Connecticut
06520 USA
329
6.8. AN ESTI~LiTE OF B~O NOR/{ IN TER~S OF AN OPERATOR NORM
Let ~ be a function in B~IO (~f~) with norm II~ I, and let K
be a Calderon-Zygmund singular integrel operator acting on -----L~(~ ~) .
Define ~ by Kg(~)--g~ K(Cg~) . The theory of weighted norm
inequalities insures that ~6 is bounded on ~ if II 6 II, is small.
In fact the map of 6 to ~6 is an analytic map of a neighborhood of
the origin in BMO into the space of bounded operators (for instance,
by the argument on p.611 of [3]). ~Iuch less is known in the opposite
direction.
QUESTION: Given 6~ ~ ; if I~- K61 is small I must
The hypothesis i~ enough to insure that I16 ~ is finite but the
naive estimates are in terms of ~] + ~ .
If ~ = 4 and ~ is the Hilbert transform then the answer is
yes. This follows from the careful analysis of the Helson-Szego theo-
rem given by Cotlar, Sadosky, and Arocena (see, e.g. Corollary
( I I I . d ) of [1] ). A similar question can be asked in more general contexts, for
instance with the weighted projections of [2]. In that context one
would hope to estimate the operator norm of the commutator [~,p]
(defined by [M~,P](~)=6P~-P(~-).,_,,~ ) in terms of the operator norm
of P- P 8 .
RE~ERENCES
I. A r o c e n a R. A refinement of the Helson-Szego theorem and
the determination of the extremal measures. - Studia ~¢eth, 1981,
LXXI, 203-221. 2. C o i f m a n R., R o c h b e r g R. Projections in weight-
ed spaces, skew projections, and inversion of Toeplitz operators.
- Integral Equations and Operator Theory, 1982, 5, 145-159.
3. ¢ o i f m a n R., R o c h b e r g R., W e i s s G. Fac-
torization Theorems for Hardy Spaces in Several Variables. - Ann.
Math. 1976, 103, 611-635.
RICHARD ROCHBERG Washington University
Box 1146
St.Louis, MO 63130
USA
330
6*9. old
some OPE~ PROBnEMS CO~CER~n~G H ~ Am~ BM0
I. A n int erp ola t ing BIs e chk e pro-
d u c t is a Blsschke product having distinct zeros which lie on
an~ ~ interpolating sequence. Is~ ~ the un~fgrmly closed linear
span of the interDolatin~ Blaschke pr0duc%s? See D],~]- It is
known that the interpolating Blasohke products separate the points
of the maximal ideal space (Peter Jones, thesis, University of Cali-
fornia, Los Angeles 1978),
2. Let ~ be a real locally integrable function on
that for every interval
[5l snd [8~.
. Assume
REFERENCES
1. M a r s h a i I D. Blsschke products generate ~
Math.Soc.,1976, 82, 494-496.
2. M a r s h a I i D. Subalgebras of L °° containing H ~ . - Acts
Math.~ 1976, 137, 91-98.
3, H u n t R.A., M u o k e n h o u p t B,, W h e e d e n R.L.
Weighted norm inequalities for the conjugate function and Hilbert
transform, - Trans,Amer.Math, Soc.,1973,176, 227-251.
. - Bull.Amer.
where ~I is the mean value of ~ over l , and where C is a con-
~tant. Does it follow that ~= ~ ~ H~ . whoso ~ L ~ snd~l -< ~ ?
(H denotes the Hilbert transform). This is the limiting case of the
equivalence of the Muckenhoupt (A~) condition with the condition of
Helson and SzegS. See [3] and [4] • This question is due to Peter
Jones. A positive solution should have several applications.
3, Let ~ be a function of bounded mean oscillation on ~ .
~CII~IIBM 0 with C a constant not dependln~ on ~ .See [5] and
[6]. R ~ 4. Let ~%,,..~ be singular integral operators on . See
[7], Pind necessary and sufficient conditions on {~T~.., ~Tw } such
+=~ I~ ~IEL~(~) see if and only ifl~l I
331
4. H e I s o n H., S m e g 5 G. A problem in prediction theory.
- Ann.Math.Pure AppI.~1960, 51, 107-138. 5. F e f f e r m a n C., S t • i n E.M. ~ P spaces of several
variables. - ActaMath.,1972, 129, 137-193o
6. C a r 1 e s o n L. Two remarks on H ~ and BMO. - Advances in
Math., 1976, 22, 269-277.
7. S t e i n E.M. Singular integrals and differentiability pro-
perties of fuuctions. Princeton N.J.~1970.
8. J a n s o n S. Characterization of ~I by singular integral
transforms on martingales and ~w . - Math.Scand.,1977, 41,140-
-152.
JOHN GARNETT University of California
Los Angeles, California
90024 USA
COMMENTARY
QUESTION 2 has been answered in the negative by T.Wolff [9S.
QUESTION 3 has been solved by P.Jones DO]. Other constructive
(and more explicit) decompositions were given later in [11~, [12S and
[13]. One more constructive decomposition of BM0 functions can be ob-
tained from a remarkable paper ~4] See also ~5~, ~7],
QUESTION 4 has the following answer found by A.UchJyama in [12]
(he obtained a more general result). Let T%~-~-~ *~, ~ ~ ~
M~ be the ~ourier transform of ~ . Su~ppose "M~ are homogeneous oo ~-4
of degree zero and 0 on the unit sphere ~ of ~ . Then
if and only if the matrix
I M,(i), ..., 1 •,
is of rank 2 everywhere on , The "only if" part is essentially
due to S.Janson [8]. In particular
332
H'{ I 'rj ,rl (l 'b
In connection with this result
iff for any ~G ~-~ there exists
see also PROBLEm6.16.
RF~ERENCES
9. W o i f f T. Cottuterexamples to two variauts of the Helson -
Szego theorem. Preprint, 1983, Institut Mittag-Leffler, 11.
lO°J o n e s P. Carleson measures and the Fefferman - Stein decompo-
sition of BMO(~). - Ann. of Math., 1980, 111, 197-208~
11.J o n e s P. L~-estimates for the ~ -problem. To appear
in Acta Math.
12.U c h i y a m a A. A constructive proof of the Fefferman - Stein
decomposition of BMO(~). - Acta Math., 1982, 148, 215-241.
13,S t r a y A. Two applications of the Schur - Nevanlinna algorithm.
- Pacif, J° of Math., 1980, 91, N I, 223-232.
14.R u b i o d e F r a n c i a J~L. Factorization and extrapo-
lation by weights. - Bull.Amer.Math.Soc., 1982, 7, N 2, 393-395.
15oA m a r E. Repr@sentation des fonctions de BMO et solutions de
l'~quation ~ . Preprint, 1978, Univ. Paris XI Orsay.
I£.C o i f m a n R., J o n e s P.Wo, R u b i o d e F r a n-
c i a J.L. Constructive decomposition of BMO functions and fac-
torization of A p weights. - Proc.Amer.Math. Soc., 1983, 87,
N 4, 675-680.
333
6.10. TWO CONJECTURES BY ALBERT BAERNSTEIN old
In [I~I proved a factorization theorem for zero-free univalent
functions in the unit disk ~ . Let ~0 denote the set of all func-
tions ~ anal~ic and ~--~ in ~ with 0 ~ ( ~ ) , ~ ( 0 ) = ~ .
THEOREM 1. I f ~e ~o , then, f o r eac h i , ~ e ( O , ¢ ) , there '
exist functions B and Q ang, 1,7%i0, in D such that
where B'N" , # I B - ~ ~ , ~ d I ~ T Q I ~ ~ .
The "Koebe function" for the class ~a is I~(~,1~- _
which maps ~ onto the s l i t p l a n e l W e ~ : ' . ~ % ~ W l < ~ } . This suggests that it might be possible to let ~ in Theorem I.
CONJECTURE I. l.~f ~ ~o , then there, exist func%ioms B and
anal~ic in D such that
. h e r e B ~ " , ¢ / B ~ " , ~ I ~ Q I ~ ~ "
We do not insist that ~ or Q be univalent, nor that
Q~0~----I . However, when the f~ctions are adjusted so that IQ(0>l--1,
then I] ~ [loo and llB-'llooshould be bounded independently of Y .
Using the fact that QI/~ has positive real part, i% is easy
to show that the power series coefficients I ~} of ~ satisfy
I~I<~ $~, ~ ~ , with equality when ~(~) ~ ~(~) . L i % t I e -
w o o d ' s C o n j e c % u r e asserts that this inequality is
true for coefficients of functions in ~0 . A proof of CONJECTURE I
could possibly tell us something new about how to attempt Littlewood's
conjecture, and this in turn might lead to fresh ideas about how to
prove (the stronger) Bieberbach's conjecture.
THEOREM I is easily deduced from a decomposition theorem obtained
by combining results of Helson and Szeg8 ~2~ and Hunt, Mucken_houp%,
and Wheeden [3]. Suppose ~e ~I(T ) and ~ real valued. Consi- ~~ der the zero-free analytic function ~ defined by~(~)=6xp(~(Z)t ~(~)),
• ~ ~ , where ~(~) denotes the harmonic extension of ~(~0)
and ~ the conjugate of ~ . Also, let ~(~) denote the set of all
334
functions obtained by "hyperbolically translating"
malizing,
and then nor-
and let ~ P denote the usual Hardy space. Part of Theorem I of [3]
can be phrased in the following w~7.
THEOREM 2. For ~G It(T) the following are equivalent.
(i~ ~ = ~ . ~ where ~ ~L~(?~ and LI~ll~ < x
(2) ~(P) u ~(~I~) is a bounde~ subse~ Of •
THEOREM I follows, since ~%/~satisfies (2)when ~ ~@ and O<A.<'t . •
THEOREM 2 may be regarded is a sharpened form of the theorem of
Fefferman and Stein [4], which asserts that ~ ~ %~ t ~ for
s o m e pair of bounded functions if and only if # is of bounded
mean oscillation.
To obtain CONJECTURE I in the same fashion as THEOREM I, we
need a result like THEOREM 2 in which the < ~[/~ of (I) is replaced
~y ~ / ~ Consideration of ~C~)----- 4+~ " 1--Z leads %o %he following
guess.
CONJECTURE 2. POF Bz 1,4(q) She followin~ are equiva,l,e,,nt.
(2 Y) ~(~)U ~(~/~) i~ a bounded subset of weak H ~ .
S%atement (2/) means %he following: There is a cons%an% C
such %hat for every~$~ , and every 6,6~(~) U ~(~/~)
~{ ~: 1 ~(~)I > ~ } -< 0~ -~ 1% is not ~r~ %0 prove, using subordim~%ion, %ha% (I / ) implies
(2/). If %he implica%ion (2 f) ~ (I f) is tr~e, %hen so is CONJECTURE ~.
Condi%ion (2 f ) can be resta%ed in a number of equivalent ways.
We men%ion one which is closely rela%ed %0 the subharmonic maximal
type function used by the au%hor in [5] and elsewhere,
335
(27[ ) There is a constant C such that
for ever~ meas~ble set ~, ~c~ , and ever~ ~, ae D ,
For ~¢ ~o , Theorem 6 of ~5~ asserts that (2 W) holds with
C-~0 .
In both the ?efferman-Stein and Helson-Szeg~ theorems the split-
ti~g ~ =IA~ t ~ is accomplished via duality and pure existence
proofs from functional analysis. It would be of considerable interest
if, given ~ , ~E~0 , one could show how to actually c o n s-
t r u c t the bounded functions ~I and ~ . We remark that if
~B~0 then some constant multiple of ~ satisfies (2F).
I can prove that (2f/) ' ~ (I/) provided we assume also that
i s m o n o t o n e on T , i . e . , there ex is t@t < ~ < ~ t + Z~ s u c h that
By composing with a suitable M6bius transformation, we may assume
~1 ~ 0 , ~ ~ Then, when O~ 0 , t~ can be constructed as foll-
ows. Let $6(0 ~) and x~(-1,1) be related by(1+X)(~-X
= [ t ÷ $ ~ B I J ~ - 6 ~ $ 1 - ' . T,et V be the harmon ic function in O with boundary values V(£~8) = ~(x) , 0< ~< ~ , and V(6"£g)=V(6 £8) .
Then it turns out that lV[~ ~/~ and ~-V~ 0(I) , so that ~-V
gives us (I[).
It follows that CONJECTURE I is true for functions ~, ~¢ ~o ,
which map ~ o n t o the complement of a "monotone slit".
REFERENCES
I. B a e r n s t e i n A. II. Univalence and bounded mean oscillati-
on. -~Lich.Math.J., 1976, 23, 217-223.
2. H e 1 s o n H., S z e g ~ G. A problem in prediction theory.
- Ann.Uat.Pura Appl., 1960, 51, (4), 107-138.
336
3. H u n t R., M u c k e n h o u p t B., W h e e d e n Ro
Weighted norm inequalities for the conjugate function and Hilbert
transform. - Trans.Amer.Math.Soc.~ 1973, 176, 227-251.
4. P e f f e r m a n C., S t e i n E.M. H P s~ces of several
variables. - Acta ~th.~ 1972, 129, 137-193.
5. B a e r n s t e i n A. II. Integral means, univalent functions
and circular s~mmetrization. - Acta Math.2 1974, 133, 139-169.
ALBERT BAERNSTEIN Washington University
St.Louis, Missouri 63130
USA
COMMENTARY
Conjecture 2 has been disproved by T.Wolff (see ref~ ~9~after the
Commentary to Problem 6.9)
337
6.11. BLASCHKE PRODUCTS IN~ o old
The class ~o consists of those functions ~ that are holomor-
phic in D and satisfy ~ (I-I$I)I~)~=0. It can be described al-
ternatively as the class of functions in ~ that are derivatives of
holomorphic functions having boundary values in the Zygmund class
(the class of uniformly smooth functions) KI , p.263S. It is a sub-
class of the class ~ of Bloch functions (those holomorphic ~ in f
satisfying ~(~-I~)I~(~)I ~ ) ; see, for example, K2]. It , . . . . I ~ 1 ~ 1
c o n t a i n s Y~U~ , the c l a s s o f ho lomoz~h ic f u n c t i o n s i n ~ whose boundary values have vanishing mean oscillation D~. The class~oNH ~
has an interesting interpretation: it consists of those functions in
~ that are constant on each Gleason part of H~ .
It is not too hard to come up with an example to show that the
inclusion V~0~ c ~'~o is proper. Indeed, it is known that ~ con-
tains functions that are not of bounded variation ~, p.48S. If
is the Poisson integral of such a function and ~ is its harmonic
conjugate, then the derivative of~, ~ will be such an example.
In connection with a problem in prediction theory mentioned in ~],
I was interested in having an example of a b o u n d e d function
in ~o which is not in V~0~ , and that seems somewhat more diffi-
cult to obtain. Eventually I realized one can produce such an example
on the basis of a result of H.S.Shapiro K5~ and J.-P.Kmhane ~6]. They
showed, by rather complicated constructions, that there exist posi-
tive singular measures on ~ whose indefinite integrals are in ~..
It is easy to check that the singular inner function associated with
such a measure is in ~0 . That does it, because the only inner func-
tions in V~0~ are the finite Blaschke products.
If ~ is an inner function in ~o and I ~ l < 1 then ~-C ,
is also an inner function in ~o , and it is a Blaschke product for
"most" values of G . Thus, ~@ contains infinite Blaschke products.
I should like to propose THE PROBLEM of ~ha~cterizing the Blaschke
products in ~o by means of the distribution of their zeros. One
has the feeling that the zeros of a Blaschke product in~ o must, in
some sense, be "spread smoothly" in ~ . A natural first step in
trying to find the correct condition would be to try to give a direct
construction of an infinite Blasc~e product in ~@ . The only infor-
mation I can offer on the problem is very meagre: A Blaschke product
in ~0 cannot have an isolated singularity on ~ . The proof, un-
fortunately, is too involved to indicate here. As A TEST QUESTION
338
one might ask whether a Blaschke product in ~o can have a singular
set which meets some subarc of ~ in a nonempty set of measure zero.
ANOTHER QUESTION, admittedly vague, concerns the abundance of
Blaschke products in ~o " Per instance, a Blaschke product should
be in ~o if its zeros are evenly spread throughout ~ . Ome is led
to suspect that, in some sense, a Blaschke product with random zeros
will be almost surely in ~o •
RE~RENCES
I. Z y g m u n d A. Trigonometric series, vol.I. Cambridge,
Cambridge Univ.Press. 1959.
2. A n d e r s o n J.M., C 1 u n i e J., P o m m e r e n -
k e Ch. On Bloch functions and normal functions. - J.Reine
Angew.~ath. 1974, 270, 12-37.
3. P o m m e r e n k e Ch. On univalent functions, Bloch functions
and VMOA. - Nath.Ann., 1978, 236, N 3, 199-208.
4. S a r a s o n D. ~hanctions of vanishing mean oscillation. -
Trans.Amer.Math.Soc. 1975, 207, 391-405.
5. S h a p i r o H.S. Monotonic singular functions of high smooth-
ness. -NLichigan ~th.J. 1968, 15, 265-275.
6. K a h a n e J.-P. Trois notes sur les ensembles parfaits llne-
aires.- Enseignement Math. 1969, (2), 15, 185-192.
DONALD SARASON University of California, Dept.Nath.,
Berkeley, California, 94720, USA
CON~NTARY BY THE AUTHOR
The problem is still open. ToH.Wclff has pointed out that the
measures constructed by Kahane and Shapiro can be taken with sup-
ports of Lebesgue measure 0 , so there do exist infinite Blaschke
products in ~o whose singularities form a set of measure 0 . (The
author was remiss in failing to notice this.) Wolff (unpublished)
has shown that the set of singularities on the unit circle of an in-
ner function in ~o meets each open subarc either in the empty set
or in a set cf positive logarithmic capacity. He conjectures that
"positive logarithmic capacity" can be replaced by "Hausdorff dimen-
sion 1."
3S9
6.12. ALGEBRAS CONTAINED WITHIN ~ co old
Let A---{~:~ analytic in ~ , ~ continuous in C~0~D=~U~ }.
Then A is an algebra contained within ~ , but there are two in-
termediate algebras that present some interest. First we require
some notation.
Let ~ denote the Banach space of functions ~ , analytic in
for which the norm
I~1,: 1 is finite. This is called the B 1 o c h
Bo=
space. We also define
t ~: ~ B , ~ Q ~ = o ( ' I - I ~1~)-1, I~l-,-'f }
For a survey of these spaces see ~]. The following facts are easily
established:
a H%B b H Bo c H'n o X is a subalgebra of
Similarly we define B~OA (an a 1 y t i c f u n c t i o n s
o f b o u n d e d m e a n o s c i 1 1 a t i o n) to be the
space of those functions ~ , analytic in ~ for which the norm
12;1< I is finite. Here II'~ is the ordinary H ¢ norm and
Similarly
The space VMOA cons is ts of those a n a l y t i c func t ions in ~ whose boun-
dary values on ~ have vanishing mean oscillation (see [2], p.591).
It is also easy to see that
d) ~B~OA , e) ~'4V~O~ ,f)~%VMO~de~is a subalgebra of~
340
it is not difficult to establish the following relation (see
eg E3]
.
The algebra X has already been studied. It was shown by Behrens,
unpublished, that ~ consists precisely of those ~ , ~6~ °@ , A
whose Gelfand transform ~ is constant on all the non-trivial Glea-
H son parts of the maximal ideal space of . It is also known ~]
that X does not possess the ~ -property or K -property in the
sense of Havin [4].
IT WOULD BE NICE TO HAVE A SIMILAR STUDY ~IADE OF ~ . The space
cannot contain any inner functions D], other than finite Blasch-
ke products, in contrast to X . But ~ does, of course, contain func-
tions having an inner factor - for example the function of [5], p.29
belongs to A .
REFERENCES
I. A n d e r s o n J.M., C 1 u n i e J., P o m m e r e n k e
Ch. On Bloch functions and normal functions. - J.Reine Angew.
Math., 1974, 270, 12-37.
2. P o m m e r e n k e Ch. Schlichte Funktionen und analytische
Funktionen von beschr~nkter mittlerer Oszillation. - Comment.~ath.
Helv., 1977, 52, 591-602.
3. A n d e r s o n J.M. On division by Ironer Factors. - Comment.
Math.Helv., 1979, 54, N 2, 309-317.
4. X a B ~ H B.H. 0 ~TOpHsa~Hz a~a~HT~ec~x ~y~En~, rxs~E~x
Bn~O¢~ ~0 rpaH~n~. - 3an.~ayqH.ceM~H.~0MM, 1871, 22, 202-205.
5. r y p a p H ~ B.H. 0 ~¢opHsak~ a0co~Ho cxo~n~xc~ p~OB
Ts~opa ~ HHTerpa~oB *yp~e. - 8aH.Hay~4.CeM~H.~0MH, I872, 30,
15-32.
J. M.ANDER SON Department of Mathematics,
University College, London
Londom WCIE 6BT
England
6.13.
341
ANALYTIC FUNOTIONS WITH ~INITE DIRICHLET INTEGRAL
I f #
_____(lj.l#~)l b e the Dirichlet integral of
the ~ollowing theorem is proved.
THEOREM. There is a constant Co <~ , such that if
tic on ~ , I(0)=0 and 3(~)~4 then
is an analytic function defined on D , let ~ (~)
, In B],
for all ~< I . O
is a=a!~'-
It would be interesting to know the size of C 0 and also the ex-
tremal functions (if exist) which correspond to the sharp constant C 0.
Actually, the above theorem is only a part of results similar to Mo-
ser's sharp form of the Trudinger inequality (see [2]). It would ac-
tually be interesting to see if there is a general form of extremal
functions which correspond to ~oser's sharp inequalities.
REFERENCES
I. C h a n g StY. A., M a r s h a I I D. A sharp inequalit# concer-
nin@ the Dirichlet integral. 1982, preprint.
2. M o s e r J. A sharp form of an inequality by N.Trudinger. - Ind.
Univ.Math.J,, 1971, 20, 1077-1092.
SUN-YUNG A. CHANG University of Maryland Math.Dept., College Park,
Maryland 20742
USA
6.14.
342
SUBALGEBRAS OF Lo°(~ ~) CONTAINING ~o@(?~).
Let H~(T) denote the Hardy space of boundary values of bounded
analytic functions defined on ~ . There has been a systematic study
of the subalgebras (called the Douglas algebras) between L"(T) and
H~(T) in the past 10 years. (For a survey article, see ~]). In par-
ticular, it has been noticed there is a parallel relationship between
subalgebras of L~) containing H~) to subspaces of B.H.O. (fun-
ctions of bounded mean oscillations) which contain V.M.O. (functions
of vanishing mean oscillations). For example, based on the fact that
on T , B.M.O.~ H(~) , where H denotes the Hilbert transform,
one can deduce that each Douglas algebra can be written as ~+someC~
algebra. There are some indications that relations of this type may
still hold on the hi-disc ~ (with distinguished bo~ry~ r ). For
example, if one views B.L0. (~) as L~T~)+ ~(~) + H~(L ~; +
+ ~i H~(L~) where the H$ ~,$=~,~ are Hilbert transforms acting on
~% variables independently with (~ ,~) e ~ and ~ ~ is the
composition of H 4 with H~ , one can ask the question whether each
subalgebra of ~(T ~) containing H~(~ ~ } has the structure of H~(~ z) .
some other three C*-algebras. It seems this problem can be studied
independently of the maximal ideal structure of H~(~ ~) . So far the
only case which has been worked out is the su~algebra of~(~ ~) gene-
rated by H~(~ ~) and C(T ~) (see [2]).
REFERENCES
I. S a r a s o n D. Algebras between L *Q and H ~ . - Lect.Notes in Math.Springer-Verlag, 1976, 512, 117-129.
2. C h a n g STY. A. Structure of some subalgebra of L ~ of the
toms. - Proc.Symposia in Pure ~sth., 1979, 35, Part I, 421-426.
SUN-YUNG A. CHANG University of Maryland
Nath.Dept., Collage Park,
Maryland 20742
USA
34S
6.~5. INNER FUNCTIONS WITH DERIVATIVE IN }~, O<p<~.
Let ~ be an inner function defined in the unit disc ~ .Por
~D let ~,(z)=W(z)-~)l(~-~(z) ) . ~ e t ~(~)~ d e n o t e
the zero set of ~A • From [I], theorem 6.2, we have:
THEOREM. Suppose that ~(~)=~ ~ is an inner function
and that ~/~ < ~ < ~ . Then the following are eguivalent:
i. ~'~.H ~
2, 7 I~I~ P<~
3. ~ (~-IZ~(~)I) ~-~ <oo for all AqD with the exoeDtio n
of a set of capacity zero. ,
For 0 < p <~/~ the situation is quite different. It is still
true that 1 implies 2 and 3. However, a~ is pointed out in [1], page
342, there is a Blaschke product ~(Z)=~ @~Z ~ such that 2 and ~z0
3 hold for ~ for all p , O<p<~/~ , but ~' is not a function
of bounded characteristic.
PROBLEM. Find a conditign ' on,,, the Tay%or coeff%cientsor on th~
distribution of values of an inner function ~ that is equivalent
to theoondition ~'aH P , 0<p<~/z .
REFERENCE
I. A h e r n P. The mean modulus and the d~rivative of an inner
function. - Indiana Univ.~th.J., 1979, 28, 2, 311-347.
PATRICK AHERN University of Wisconsin
Madison, Wisconsin, USA
EDITORS' NOTE
I.E.Verbitski~ has informed us about his result pertaining to
the Problem.
2) Y', i~,~l~ , f f < oo ;
344
then the followin~ ar e equivalent:
3) ~ (4-Im~C~)l) ~'SP <oo
for all & 6D with the exception of a set of capacity zero;
Here ~ (S) denotes the fractional derivative of ~ of order $
, B~ is the Besov class, i.e.
B ~ [I analytic in ~ : IIII(~)(z)IP(~-[~I) (~-~)~'{ ~<°° I,
being any integer >5~ E=~+~.
This theorem is implied by results of [I] when ~<~ , sp>~ . It is not valid when $-<~-4
gous result seems to be known i~ that case.
p~ ,0<~- < and no analo-
345
6.16. EQUIVALENT NORMS IN H p
Let H P denote the real variables Hardy space on o Let K~
be a Fourier multiplier operator whose symbol ~ is C~(~ W\ {0} )
and homogeneous of degree zero. For which families {~}~=q , i , s ' it
true that o
for all ~6~PG l, ~ ? This problem was solved for p=1 in [I~
and the results were extended in [2] to the case where p is only
sli@htly less than one. A subproblem is to decide whether the above
equivalence holds for all p< I when the family consists of the iden-
tity operator and the first order Riesz kernels~ See [3~ for related
results.
REFERENCES
I. U c h i y a m a A. A constructive proof of the Fefferman -
Stein decomposition of BMO (~) . - Acta Math.,1982, 148, 215-241+
2. U c h i y a m a A. The Pefferman - Stein decomposition of smooth
functions and its application to ~P(~) .- University of Chicago,
Ph.D. thesis, 1982. F
3. C a 1 d e r o n A.P., Z y g m u n d A. On higher gradients
of Ba~monio functions.-Studia Math?t964, 24, 211-226o
PETER W. JONES Institut Mittag-Leffler
Aurav~gen 17
S-182 62 Djursholm
Sweden
Usual Address:
Dept. of Mathematics
University of Chicago
Chicago, Illinois 60637
USA
346
6.17. DE2EITIO oP HP(R ). Suppose ~ is a distribution on ~n, ~ a compactly supported
C' -eunct io~ on , ~ - 4 , p > o . ~ t
T/(zl- sup{ IT(%,,)I &>0)
(the r a d i a i maximal function of T correspondin& to the mol-
lifier ~ ).
QUESTION. Does the inclusion T~eLP(R ~) imply Te HP(~)? The answer is YES if p >~ 4 or under the supplementary assumption
T E S ' ( ~ ~) [1 ] , r i f we replace T ~ by T ~ , the a n g u 1 a r maximal function (because then the inclusion t ~ S' (~) is easy
to prove). We were unable to answer the question following the patterns
of [ I ] .
REPERENCE
I. P e f f e r m a n C., S t e i n E.M. H P spaces of several
variables, -Acta Math,, 1972, 129, 137-193.
A. B. ALEKSANDROV
V. P. HAVIN
(B.II. XABI~H)
CCCP, 198904, .llsKam, pa.~, IIeTl~o.~ol~e~, ]~l~6ZmoTe*lNa8 2, .~e~ar'paz:oma2 rocy.~alOo~Bemm~ yHBBeI~TeT, MaTeMaTBEo-sexa-
347
6.18. HARDY CLASSES AND RIEMANN SURPACES OF PARREAU-WIDOM TYPE old
The theory of Hardy classes on the unit disk and its abstract
generalization have received considerable attention in recent years
(of. Hof~ [11, Helson [2], ~eZln [3])- The case of compact bor-
dered surfaces has also been studied in detail. It is thus natural
that we should try to increase our knowledge concerning the theory
of Hardy classes on infinitely connected Riemann surfaces. OUR BASIC
QUESTION is this: ~qr which class of R iemann surfaces can one get a
fruitful extension of the Hardly class theQry on the disk? A candidate
we believe most promising is the class of Riemann surfaces of Parreau
-Widom type, which is defined as follows:
DIPINITION. Let ~ be a hyperbolic Riemann surface, ~(~)
the Green function for R with pole at a point @6~ and B(~@)
the first Betti number of the region ~(@,~)~ lSe~: 6(~,~) > ~
with ~> 0 . We say that ~ is of P a r r e a u - W i d o m
o
We first sketch some relevant results showing that such sur-
faces are nice. In the following, R denotes a surface of Parreau-
Widom type, unless Stated otherwise.
(1) PARREAU [4]" (a) Every positive harmonic function on
has a limit along almost every Green line issuing from any fixed
point in ~ . (b) The Dirichlet problem on Green lines on ~ for any
bounded measurable boundary function has a unique solution, which
converges to the boundary data along almost all Green lines.
(2) WIDOM ~] : For a hyperbolic Riemann surface ~ , it is of
Parreau-Widom type if and only if the set ~@@(~,~) of all bounded
holomorphic sections of any given complex flat unitary line bundle
over ~ has nonzero elements.
(3) HASUMI [6] : (a) Every surface of Parreau-Widom type is ob-
tained by deleting a discrete subset from a surface of Parreau-Widom
type, ~ , which is regular in the sense that ~$e~: @(@,~)~8
is compact for any ~0 . (b) Brelot-Choquet's problem (cf. [7])
concerning the relation between Green lines and Martin's boundary
has a completely affirmative solution for any surface of Parreau-
Widom type. (c) The inverse Cauchy theorem holds for ~ .
In view of (3)-(a), w e a s s u m e i n what follows
that ~ i s a r e g u 1 a r s u r f a c e o f P a r-
r e a u - W i d o m t y p e. The Parreau-Widom condition stated
in the definition above is then equivalent to the inequality
348
~ ~(~,i6) : ~Z(@)}<~ , where Z(@) denotes the set of critical
points, repeated according to multiplicity, of the function ~ ~.
. we set
Moreover, let ~I be Martin's minimal boundary of ~ and ~& the
harmonic measure, carried by A I , at the point • . Look at the
following STATEMENT (DCT): Let ~ be a meromorphic function O n
such that 1'1,,,I h~s a harmonic majorant on ~ . Then A
= , where denotes the fine boun W A4
function for ~ . (Note: DOT stands for Direct Cauchy Theorem).
(4) HAYASHI [8]: (a) (DCT) is valid for all points ~ in
if it is valid for some @ . (b) (DCT) is valid if and only if each
-closed ideal of ~o@(~) is generated by some (multiple-valued)
inner function on ~ . (c) There exist surfaces of Parreau-Widom
type for which (DCT) fails.
We now mention SO~E PROBLEMS related to surfaces of Parreau-
Widom type. (i) Find simple sufficient conditions for a surface of
Parreau-Widom t,Tpe t0 Satisfy (DCT). Hayashi ~8~ has found a couple
of conditions equivalent to (DCT) including (4)-(b) above. But none
of them are easy enough to be used as practical tests. (ii) Is there ,
any criterion for a surface of Parreau-Widom t.ype to satisf,7 the
Corona Theorem? Known results: there exist surfaces of Parreau-Wi-
dom type for which the Corona Theorem is false; there exist surfaces
of Parreau-Widom type with infinite genus for which the Corona Theo-
rem is valid. Hayashi asks the following: (iii) Does ~,~) for
any ~ h@ve onl,y constant common inner factors? (iv) Is a generali-
zed P, and M.Riesz theorem true f o r measures on Wiener's harmonic
boundar,7 t which are o~ho~o~l to H~ ? Another problem: (v) Characte-
rize those surfaces ~ for which ~( ~, ~) for ever 7 ~ has an ele-
ment without zero~ This was once communicated from Widom and seems
to be still open. On the ether hand, plane domains of Parreau-Widom
type are not very well known: (vi) Characterize closed subsets
of the Riemann sphere ~ for which ~ \ E is of Parreau-Widom
(of. Voichick M, Dol).
Finally we note that interesting observations may be found in
349
work o f Pommerenke ~1~, Stanton D2~, Pranger ~13~ and others .
REPERENCES
I. H o f f m a n K. Banach Spaces of Analytic Functions. Prentice
-Hall, Englewood Cliffs, N.J., 1962.
2. H e 1 s o n H. Lectures on Invariant Subspaces. Academic
Press, New York, 1964 .
3. G a m e 1 i n T. Uniform Algebras, Pretice-Hall. Englewood
Cliffs, N.J., 1969.
4. P a r r e a u M. Th~or~me de Patou et probleme de Dirichlet
pour les lignes de Green de certaines surfaces de Riema~. -
Ann.Acad.Sci.Penn.Ser.A. I, 1958, no.250/25, 8 pp.
5. W i d o m H. ~p sections of vector bundles over Riemann sur-
faces. -Ann. of N~th., 1971, 94, 304-324.
6. H a s u m i ~. Invariant r~bspaces on open Riemann surfaces.
-Ann.Inst,Fourier, Grenoble,1974, 24, 4, 241-286; II, ibid.
1976, 26, 2, 273-299.
7- B r e 1 o t M. Topology of R.S. Martin and Green lines. Lec-
tures on Functions of a Complex Variable, pp.I05-121. Univ. of
Michigan Press, Ann Arbor, 1955.
8. H a y a s h i M. Invariant subspaces on Rieman~ surfaces of
Parreau-Widom type. Preprint (1980).
9. V o i c h i c k ~. Extreme points of bDunded -~alytic functi-
ons on infinitely connected regions. - Proc.Amer.Math.Soc., 1966,
17, 1366-I 369.
10. N e v i 1 1 e C. Imvariant subspaces of Hardy classes on infi-
nitely connected open surfaces. - Memoirs of the Amer.Nath.Soc..
1975, N 160.
11. P o m m e r e n k e Ch. On the Green's function of Fuchsian
groups. -~n.Acad.Sci.Fenn. Ser. A. I, 1976, 2, 408-427.
12. S t a n t o n C. Bounded analytic functions on a class of open
Riemann surfaces. - Pacific J.~ath., 1975, 59, 557-565.
13. P r a n g e r W. Riemann surfaces and bounded holomorphic func-
tions. -Trans.Amer.Nath.Soc., 1980, 259, 393-400.
MORISUKE HASUMI Ibaraki University,
Department of Mathematics,
Nito, Ibaraki, 310, Japan
35O
EDITORS' NOTE. A Parreau-Widom surface with a corona has been con-
structed in the paper
N a k a i M i t s u r u , Corona problem for Riemann surfaces of
Parreau-Widom type.- Pacif.J.Math., 1982, 103, N I, 103-109.
351
6.19. INTERPOLATING BLASCHKE PRODUCTS
I f &~m ~-Z
B:~ I~,r~l ~ -~ z
lation constant of ~ , denoted
is a Blaschke product, the interpo-
A well known result of L . Carlemon asserts that B is an interpola-
ting Blaschke product if and only if ~(~) >0 . It is also well
known that the following open prob-
lems are equivalent:
PROBLEM I. Can every inner function be uniforml,y approximated
b,y interpolating Blaschke products? i.e., Given an T Blaschke product
B and an ~ >0 is there an interpolating Blaschke product ~I
such that II B - B4 II < S ?
PROBLEM 2. Is there a function
Blaschke product ~ and a~y 6 > 0
product ~4 such that IIB-~III<
~(6) so that for ar4¥ finite
, there is a (finite~ B!aschke
These problems are stronger than Problem I posed by John Garnett in
"Some open problems concerning H~and BMO" in this problem book,
Problem 6. 9.
If these problems are eventually answered in the negative,then
the obvious question is to classify those inner functions which can
be so approximated. T.Trant and P.Casazza have observed (and this
may already be known) that changing convergence in norm to convergen-
ce uniform on compacta produces satisfactory classifications. Por
example,
PROPOSITION 3. Th e followin~ are equivalent for a function
Fell ~ :
a sequence {B~) of fini!eB!aschke products (1) There is
which conver~e to F uniforml~ on compacta for which ~ ~(B~)>
>0,
(2) F=BG where ~ is an interpolatin~ Blaschke product
and G is an oute r functio n satisfyin~
352
O< W IG(z)l.< ~ I (~(z)l.< ~. ~D ze~D
The proof that (2) ==~(1) follows by calculating the interpola-
tion constants of the approximating Blaschke products given in the
proof of Frostman's Theorem. By using some techniques developed in
[I], it is easily shown that (I)~(2).
I am particularly interested in the form of the function ~(~)
given in problem 2. A variation of this relates to a problem stated
in [I]. If K is a compact subset of the unit circle with Lebesgue
measure zero, let A~ denote the ideal in the disk algebra A
consisting of the functions which vanish on K . The most general
closed idea!s in A have the well ~ o ~ f o ~ Z F = { ~ . F : ~ A K } where m is an inner function continuous on the complement of K
in the closed disk. A sequence { Zm} in the open disk is called a
Carleson sequence if M-~{~(~-,z~l')I~(z~)I:#~H ~, ll~II.<~}<oo. n~4
In [1], the following problem appeared:
PROBLEM 4. l_~f {Z~} i@ a Carleson sequence and B th e Blasch-
ke product with zeroes {E~} continuous off K , d0es there exist
absolute constants ~ and A s O that
is a projectlcn onto}.<A&~M?
I have since discovered that the left hand inequality is true
(there does exist a universal constant ~ but the right hand ine-
quality is false (there does not exist a universal constant A ).
A new conjecture for the norm of the best projection onto am ideal
in A is needed. The calculations involved in computing this seem
to be related to those needed for problems (I) and (2) above.
REBmRENCE
I. C a s a z z a P.G.,P e n g r a R. and S u n d b e r g C. Comp-
lemented ideals in the Disk Algebra.lsrsel J.~th.,vol.37.Sol-2,
(198o), p.76-83. PETER G.CASAZZA Department of ~athematics University of
USA ~Lissouri-Columbia, Columbia, Nissouri 65211
CHAPTER 7
SPECTRAL ANALYSIS AND SYNTHESIS
Problems of Spectral Analysis - Synthesis arose long before
they were stated in a precise form. They stimulated, e.g., the deve-
lopment of Linear Algebra ("The Fundamental Theorem of Algebra",
Jordan Theory) and of basic ideas of PourierAnalysis The success
and the universal character of the last are the reasons why the
present theme was confined for a very long time to the sphere of in-
fluence of Harmonic Analysis. The well developed theory of trigono-
metrical series and integrals, group representations, Abstract Har-
monic Analysis - all these disciplines are directed at the same two-
fold problem: what are"the elementary harmonics" of an object (= a
function, an operator, ... ) which is undergoing the action of a
semi-group of transformations; what are the ways of reconstructing
the object, once its spectrum, i.e. the intensity of every harmonic,
is known? Another apparently different, but essentially identical
aspect stimulating the development of the theme has roots in Diffe-
rential Equations. The ritual of writing down the general solution of
~(~)~ = 0 using the z e r o s of the symbol ~ g e n e r a t e d numerous
investigations of differential-difference and more general convolu-
tion operators. The results always reflect the same routine: the ge-
neral solution is the limit of linear combinations of elementary so-
lutions ~ ~ corresponding to the zeros ~ of the symbol (Ritt,
354
Vallron, Delsarte). It was L.Schwartz who formulated the circle of
ideas in its real meaning and appropriate generality (in his classi-
cal paper in Ann Math , 1947, 48, N 4, 857-927) Now the Problem of
Spectral Analysis - Synthesis can be stated as follows: given a li-
near topological space X and a semi-group of its endomorphisms,
describe ~ -invariant closed subspaces, containlng non-tri-
vlal ~-invariant flnlte-dimensional parts ("Analysis"), and then
describe subspaces spanned topologically by the above parts
("Synthesis").
If ~ has a single generator then our problem actually deals
w i t h e i g e n - and r o o t - s u b s p a c e s o f the g e n e r a t o r and w i t h the subse -
quent r e c o v e r y o f a l l i t s i n v a r i a n t subspaces v i a t h e s e " e l e m e n t a r y "
ones . Systems o f d i f f e r e n t i a l and g e n e r a l c o n v o l u t i o n e q u a t i o n s l e a d
to f i n i t e l y - g e n e r a t e d ~ - i n v a r i a n t subspaces , ~ be ing the c o r r e s -
ponding group ( o r semi-group) o f t r a n s l a t i o n s ( i n ~ ~w T ~ e t c )
A n n i h i l a t o r s o f such subspaces become ( v i a F o u r i e r t r a n s f o r m ) modules
over the ring of trigonometrical(resp. "analytic" trigonometrical)
polynomials; the Analysis-Synthesis Problem converts into the well-
known problem of "localization of ideals". Roughly speaking the prin-
cipal role is played in this context by the concept of the divisor ofan
analytic function, and the Problem reduces to the description of di-
visorlal ideals (or submodules). After this reduction is accomplis-
hed, we may forget the origin of our problem and confine ourselves
exclusively to lhmction Theory. Namely, we are led to one of its key
questions, the interplay of local and global properties of analytic
functions. Thus, sts~ting with Analysis - Synthesis, we come to the
multipllcative structure of analytic functions (Weierstrass products
and their generalizations), the factorization theory of Nevanllnna-
Smirnov, uniqueness theorems characterizingnen-trivial divisors and
to many other accoutrements of Complex Anmlysis.
The problems of this chapter treat the above ideas in various
355
ways. Localization of ideals (submodules) in spaces of analytic func-
tions determined by growth conditions is discussed in Problems 7.1-
7.6, andin more special spaces in 7.7-7.11. These Problems overlap
essentially. We add to the references given in the text of Problems
books of L.Schwartz (Th~orie des distributions, Paris, Hermann, 1966),
L.Ehrenpreis (Fourleranalysis in several complex variables, N.Y.
1970) and J.-P.Perrier (Spectral Theory and Complex Analysis, N.Y.,
1973) (see also the bibliography in the survey D] cited in Problem
7.7). Analyzing spaces of holomorphlc functions defined by a family
of majorants requires a study of the intrinsic properties of maJorants
(see e.g.Problem 11.8 and §7.3 of Ferrier's book).
Problem 7.13 deals with an interestin~ question concerning fi-
nitely generated (algebraically) ideals in H ~ , a generalization
of corona Problem. And we mention once more Ferrier's book in connec-
tion with analogues for "~6rmander algebras" of that problem (inc-
ludlngmul%idimensional settings).
The more"rigid" is the topology of a space, the more profound
is the concept of divisor (and, as a rule, the more difficult it is
to prove that ~-invariant su~bpaces are divisorial). The series of
Problems 7.4-7.16 is very instructive in this respect. Another fea-
ture they have in common is that they aim at the well-known "secon-
dary" approx~maton problem of Analysis - Synthesis: to prove or to
disprove that any subspace with a trivial divisor is dense (of.
Wiener's Tauberian Theorem). This problem is implicit in arguments
of items 7.7-7.11, 7.14,ooncernlngweakly invertible (cyclic) func-
tions in corresponding subspaces.
Classical Harmonic Analysis has led to very delicate and diffi-
cult theorems in Spectral Synthesis and to a vast variety of problems-
from numerous generalizations of periodicity (which corresponds to
the simplest convolution equation (~-I~)* I~0 ) %o the theory of
resolvent sets of Malliavin - Varopoulos. This direction is repre-
356
sented by Problem 7.17-7.23 (see also Problem S.4).
Other problems related to Analysis - Synthesis are 4.9, 4.14,
6.11, 6.12, 8.1, 8.'~, 9.1, 9.3, 9.13, 10,1,10.3, 10.6, 10.,8.
We conclude by some articles connected with 7.1: B.H.r~u~aMO~OB,
~or~a~ AH CCCP, 1966, 168, N 6, 1251-1253; R.Narasimhan, Proc.Conf.
Univ.Maryland, 1970, Berlin, Springer, 1970, 141-150; H.Skoda, Ann.
Inst.Fo~rier, 1971, 21, 11-23. The references in 7-7 contain several
items concerning the localization of ideals ( for ~=I ) in the spi-
rit of 7.2. Man~ problems in 7.9 are discussed in t he book ~3~ cited
in 7.7.
357
7.~. ABOUT HOLO~ORPHIC FUNCTIONS WITH LI~ITED GROWTH old
Can one develop a theory of holomorphic functions satisfying
growth conditions analogous to the theory of holomorphic functions
on Stein manifolds?
Let ~ be a continuous non-negative function on ~ which
tends to zero at infinity; ~(~ will be the set of all holomorphic
functions ~ on the set ~:~0 such that ~N~ is bounded for N
large enough.
Research about the holomorphic functional calculus tl] led the
author to the consideration of the algebras ~(~ . The only rele-
vant algebras however were the algebras ~(~ where ~ is Lip-
schitz and I~I~(~) is bounded.
L.Hormander E2-4], has obtained results concerning algebras
that he called ~(~) , but ~(~)~(e -~) • His proofs used assumptions
about ~ which imply (up to equivalence) that ~-~ is Lipschitz and
I~ -~ is a bounded function of ~ .
He also assumed that ~ , i.e.-~O~ , is a plurisubharmonic
function. This is an expected hypothesis, it means that ~(~) be- ll
haves like the algebra of holomorphic functions on a domain of holo-
morphy. From the point of view of the holomorphic functional calculus,
the conditionU-~O~--- is p.s.h." is also significant,as I. Cnop
E5] showed (using Hormander's results). % J
The reason why L.Hormander and the author looked more specially
at the algebras ~(~) , ~ Lipschitz, were quite different. For Hor-
mander it appears that better estimates can be obtained when ~ is
Lipschitz. For the author, the only algebras relevant to a signifi-
cant application of the theory were the algebras ~(~) , with
Lipschitz. This coincidence suggests that the Lipschitz property is
an important property ~ has to possess if we want ~(~) to behave
somewhat like hclomorphic functions on an open set.
Unfortunately, it is not clear what should take the place of
this Lipschitz property when we investigate holomorphic functions on
manifolds. The Lipschitz property is expressed in global coordinates.
Manifolds only have local coordinate systems. An a~liaryRiemann
or K~hler metric could be defined on the manifold. Or one may notice
that ~(~) is nuclear when ~ is Lipschitzo
The plurisubharmonicity of - ~ involves the structure of
the complex manifold only. It generalizes the holomorphic convexity
of Stein manifolds.
L.Hormander has proved an analogue of Cartan's theorem B for
358
holomor~ic functions satisfying growth conditions. The full force
of the Oka-Cartan theory of ideals and modules of holomorphic func-
tions does not follow, until an analogue of Cartan's theorem on in-
vertible matrices has been proved, with bounds, and bounds have been
inserted in Oka's theorem on the coherence of the sheaf of relations.
We shall call B(~,8~(~)) the open ball with center G and ra-
dius ~(~) , and shall assume that ~ is small. This ensures that
~(~,g~(~)) ~ ~ , also that ~(~)/~(~) is bounded from above and
bounded away from below when~B(~,~(~)) , and that ~(~)/~(~)
is bounded from above and bounded away from below when B(~,6~(~))
and ~(~,50"(~)) have a non empty intersection (because ~ is lip-
s chit zian).
The following results should be a part of the theory.
CONJECTURE. Let ~,,...~ ~K be elements of (~(0~) ar . L eit
_ Se~(~) ~ be such that t~,~,...,%~K,~ can be found for each ~ , holo-
morphic on B (5, 6~)) , with
o.~n B(~,$~(¢)) , and I~,~(~)l < M~(~) -N for some M ,
and ~ , ~e ~ . Then ~ is in the submodule of O(1) @ generated
by ~I' .... ~K "
CONJECTURE. With the same conventions, assume tha t ~ is Riven
for each ~ , ~6 ~ , such that ~(~)II < M~(~)-N when ~
i_f M an~ N are large enough, Assume also that
set, and less than MI(~) -N
such that
wit h ~- ~ holomorphio on this open
. Then it is possible to find ~, M',
, an~
359
o_~n B(~,~'[(~)) , with V~,~ holomor~hic onB(~,$f~(~)) and IV$,~(~)I< . . f _N r
<M ~) when ~ ~B (~, ~rg(~)
A local description of the submodules of ~(~)~ would also be
welcome. Let ~ be a submodule of ~(~)% . Then, for each ~ ,
generates a submodule ~ of~(B(~,$~(~))) $ . When B(~,6~(~))
and B(~, ~(~)) intersect, M~ and ~$ generate the same submo-
dule of G(B(~(~))~B(~,%~(~))) ~ . Is it possible to find con-
ditions from functional analysis which ensure that a family of modu-
les ~ , which agree in the manner described, would be generated by
a submodule M of ~(~ ? J.-,.,e=ler [6J.[7] considers R~e's
theorem in the above context. Assuming ~ f to be two Lipschitz
functions he shows that the set of limits in ~(~f) of restrictions
of elements of ~(~) is - or can be identified with- some ~(~I)~
and ~ has some analogy with an " ~(~) -convex hull" of ~f .
However the limits that ~errier handles are bornological, not
topological. Ferrier cannot show that (~(~) is a closed subspace
of ~(~) . It might very well be that the limits of elements of
(~(~) would be elements of ~(~) with ~i~ 4 ~ ~f , etc. This spe-
cific problem is therefore open. So is the generalization of Fer-
rier's results to algebras of holomorphic functions satisfying
growth conditions on a Stein manifold ... once we know what is a
~ood analogue to the condition " ~ is Lipschitz".
The general problem described in this note is vaguer than the
editors of the series wish. It intrigued the author eighteen years
ago, when [I] was published, but the author could not make any head-
way and went on to other things. Hormander's breakthrough came later.
The author has not taken the time to investigate all of the conse-
quences of Hormander's results. Results have been obtained by several
authors, after H~rmander. They do not solve the problem as it is put.
But they indicate that significant progress at the boundary of com-
plex and functional analysis would follow from a good understanding
of the question.
REFERENCES
S. W a e I b r o e c k L. ~tude spectr~le des algebres completes.
- Acad.Royale Belg.M~m.C1.Sci.~ 1960, (2) 31.
2. H o r m a n d e r L. ~ -estimates and existence theorems for
the S-operator. - Acta Math.~1965, 113, p.85-152.
3. H o r m a n d e r L. An introduction to complex analysis in
360
several variables. New York, Van Nostrand. 1966.
4. H B r m a n d e r L. Generators for some rings of analytic
functions. - Bull.Amer.Math.Soc.,1967, 73, 943-949.
5. C n o p I. Spectral study of holomorphic functions with bounded
growth. - Ann.Inst.Fourier~1972, 22, 293-309.
6. F e r r i e r J.-P. Approximation des fonctions holomorphes de
plusieurs variables avec croissance. - Ann.Inst.Fourier, 1972, 22,
67-87.
7. F e r r i e r J.-P. Spectral theory and complex analysis.
North Holland ~th.Stud. 4. Amsterdam. North Holland. 1973.
L.WAELBROECK Univ.Libre de Bruxelles, D~p. de ~ath.
Campus Plaine. C.P.214 BRUXELLES
BELGIQUE
361
7.2. old
LOCALIZATION O~ POLYNOMIAL SUBMODLU~S IN SOME SPACES O~
HOLO~OP~HIC E~CTIONS AND SOLVABILITY O~ THE ~-EQUATION
Let K be a compact in . Consider its support function
For every positive integer ~ define a norm [I'II~, K on the space
of complex-valued functions in C ~ by
Let ~g be the space of all entire functions ~ in with
II~ U~, K < co for every ~ . This space can be considered as
a module over the algebra C~] of polynomials in C ~ with res-
pect to the pointwise multiplication. Therefore each ideal I of
C~ generates a submodule I ' ~k of ~K •
DEPINITION. A submodule I" ~K is called I o c a I if it
contains all functions ~ ~ satisfying the following condition:
for every ~J~C~ the Taylor series of # in t~
)'=J,!'" "J-! Y-. jI--V- '
belongs to the submodule I'T~ , where T~ is the C[~J -module
of all formal power series in ~-~=(~-~,"',~-~n).
CONJECTURE I. For any compact set K and for an~ ideal I i_~n
5~J the submodule I S K is local.
The CONJECTURE can be generalized to the case where~the ideal of
C K~S is replaced by an arbitrary submodule I of ~ ~ ~S
(the direct sum of ~ copies of ~ 5~S ). This more general
CONJECTURE is easily reduced to the case of the ideal I •
Since the support ftmction of a compact set coincides with that of
its convex hull, we can suppose~ to be a convex compact set. In this
case the space of the ~ourier transforms of SK coincides with the
space ~K of infinitely differentiable functions in ~ supported
on K . The validity of CONJECTURE I would lead, in view of this con-
nection, to some interesting consequences in the theory of differen-
tial equations with constan~ coefficients. Let us mention one of them.
362
COROLLARY. Let P ,b,9, a ($xS) matrix of differential opera-
tors in~ ~ with constant oqefficients. Then the system of equation,s S /
~=~ , ~=(~,...,~S) has a solution in the class ~K of
distributions on ~ for any ~ atisfyin~ the formal com r
patibility condition (i,e, ~#=0 f°r an2 matrix ~ of operators
with constant coefficients such that G~=0 ).
Conjecture I is induced by the following result.
THEOREM OF ~LGRANGE AND PALAMODOV ([I] ,[~ ). Let ~ be a con-
vex domain in ~ , ~ be the union of ~K over all compact sub-
sets M o_~f~ . Then for any ideal I of ~ [~] the submodule
I Sn is local.
The proof of this Theorem depends on the triviality of the V
Cech cohomologies for holomorphic cochains in C ~ with an estimation
of the growth at infinity or on the equivalent theorem on the solva-
bility of the ~ -equation in ~ with the estimation at infinity
as well. To use this way for the proof of CONJECTURE I one needs the
following assertion,
Let $~ be the space of ~ -differential forms
such that all derivatives ~ji~...~J$
every ~ .
CONJECTURE 2. Por ever~ ~C ~
that ~o~ = 0 there exists ~ i_~n
lj,,...,j, %^... ^
have finite norms t1"~$, K
~nd evel V ~ i_~n ~K
~ satisfyin~ ~@=o~
for
such
In this CONJECTURE ~he essential point is not the lecalproperties
of the coeffitients but their growth at infinity. We can assume them
to be locally square summable or even to be distributions. The opera-
tot ~ being elliptic the complexes corresponding to the different
local conditions are homotopic and therefore can satisfy CONJECTURE 2
only simultaneously.
The following result obtained for another purpose can be consi-
dered as an approach to CONJECTURE 2.
363
LE~ (~3S). Let ~ be a ball in ~n centered at the ori~in~
~+ be the intersection of ~ and a half-space of ~ . Then for
eve r~ ~ and for every ~ -closed form ~C~ there exists
a ~ - f o r m JB such tha t ( ) jB=~ and ll~(e')~,K.~G+ < oo
•
The following result was obtained recently.
THEOREM (Dufresnoy ~4~). Conjecture ~ is valid for any convex
cQmpact set with C z bo~dar~.
The proof is based on a well-known HBrmander's theorem on solva-
bility of the ~ -equation. A non-trivial point is the choice of an
appropriate weight ~ with plurisubharmonic ~ . It is here
where the smoothness of boundary is used.
REFERENCE S
1. M a I g r a n g e B. Sur les systemes differentiels a coeffi-
cients constants. Paris, Coil.Int. CNRS, 1963.
2. H a x a M o ~ o B B.H. ~zHe~H~e ~H~epemm~e onepaTop~ c nO-
CTOKH~ EOS~wu~eHTaMH. M., "HayEa", 1967.
8. H a ~ a M O ~ O B B.H. EoMnxeEc roao~op#H~x BO~H. - B EH. : Tpy-
ceM~Hapa HM.~.F.HeTpOBCEOrO, I975, ~ I, I77-210.
4. D u f r e s n o y A. Un r~sultat de ~l~-cohomologie; applica-
tions aux systemes differentiels a coefficients constants. - Ann.
Inst.Fourier 1977, 27, N 2, 125-143.
5. H o r m a n d e r L. Linear partial differential operators.
Springer-Verleg, Berlin-Gottingen - Heidelberg, 1963.
V. P. PAL~0DOV
(B.H.HAKAMO~OB) CCCP, 117284, MOCEBa
~eH~Hcz~e ropH,
MOCEOBCFm~ rocy~apcTBeHH~
yHHBepC~TeT, Mex.--MaT.~aEyJLBTeT
364
7.3. old
INVARIANT SUBSPACES AND THE SOLVABILITY
OF DIPEERENTIAL EQUATIONS
I. Let /l be a convex domain in C @ and let ~(.~ be the
space of all functions analytic in /I supplied with the natural to-
pology. L.Schwarz posed and solved (for/l=~ ) ~] the following
PROBLEM.
Does an~ closed subspace ~ ~(II) invari~nt under the operator
of differentiation contain exponential monomials, and if it does then
do such monomials span W ?
This problem is completely explored in [2] for ~=I . In case
>~ the problem has not been solved so far even for principal sub-
spaces sBch as
for examples
The positive answer to the question of L.Schwarz has been obtai-
ned only for special domains in ~ , namely for A~---~ ~ [3], [4],
for half-spaces in ~ 5 ~ for tube domains [6] and for d o m a i n s . ~
xn satisfying ~ ~ ~]. The proof in all listed
cases, besides the tube domains, exploits essentially the fact that
W is invariant D/Ider the translations. The condition ~+~c~
embraces a general class of domains with required invariance property,
As in the one-dimensional case the proof of the following co~ecture
Let~-*<6 ~,~
could be the key to the solution of the whole problem. ,I~/
be the generalized Laplace transform a adler E~ be the space of en-
tire functions coincidingwith the Laplace transform of continuous f
linear functionals on ~ (~) . T~e space [~ is endowed with the
natural t6pologyborrowed from ~(~).
CONJECTURE. Given q,~ ~ such that ~/~ is an entire
function there exists a sequence { P$}~)0 of polynomials satis-
The proof of this statement in case ~ hinges on the employ-
ment of canonical products and therefore cannot be directly transfer-
red to the case of several variables.
2. it is well known in the theory of differential equations that
365
~O~(/l) ~--- 0~(ll) for every differential operator~
with constant coefficients if and only if /I is a convex domain. A
natural complex analog of this statement can be formulated as follows.
CONJECTURE. Let ~I be a pseudo-convex domain in ~ . Then
~(/l) ------ H(/I) for ever2 differential o~erator
(~,"',~) with c0nstant coefficients if and onl2 if P= P /I
is strongly linearly convex (see 1.1~ of this volume for the defini-
tion).
The following facts are in favour of the conjecture. The property
of strong linear convexity is a sufficient condition [SJ. Conversely
if ~ is a pseudo-convex domain and ~(Jl) ~ ~(.O.) then
all slices of ~- by one-dimensional complex planes are simply con-
nected (the proof follows the lines of [9]). It is known ( ~ i n 113)
that this implies that ~ is strongly linearly convex provided all
slices of ~ are connected.
REFERENCES
I. S c h w a r z L. Th~orie g4n~rale des fonctions moyenne-p4rio-
diques.-Ann.Math,, 1947, 48, N 4, 857-925.
2. Np a c z~ ~ o B- T e p H o B c R ~ ~ H.~. MmmpzaHTm~e nO~-
upocTpSmc~Ba a~Tm~ec~zx ~yam~. I. CHe~TpaA~s~ CKHTes Ha B~-
nyi~m~x O6~aCT~X.- MaTeM.cd., I972, 87, ~ 4, 459-489; H - Ma~ex. cd.., 1972, 88, ~ I, 3-30
3~ M a I g r a n g e B. Existence et approximation des solution des
4quations aux deriv~es partielles et des ~quations de convolution-
Ann.lnst.Fourier, 1955, 6, 271-354.
4. E h r e n p r e i s L. ~.~ean periodic functions. - Amer. J.~ath.,
1955, 77, N 2, 293-328.
5. H a ~ a • ~ o B B.B. 0 uo~pocTpa~c~max saaawr~ec~zx ~ymm~,
w,~apza~T~x OTHOCITea~O c~m~a. - HsB.AH CCCP, cep.Ma~eM., 1972,
36, I269-I28I.
6. H a ~ a a ~ o B B.B. YpamHeKxe Tz~a cmepT~ ~ Tpy6~a~x o6~acTax
~. -M~B.AH CCCP, Cep.~aTeL, 1974, 38, 446-456.
7. T p y T ~ e B B.M. 0~ ypaB~em~ B cBepT~ax m B~mx o6aacT~x
npocTpaSc~Ba ~ . - B ~s. : Bonpoc~ MaTemT~. C6.~ay~s.~py~oB
510, TempesT, TrY, 1976, 148-150.
8. ~ a r t i n e a u A. Sur i~ notion d'ensemble fortement lin@-
366
ellement convexe. -Ann. Acad.Brasil., Ciens., 1968, 40, N 4, 427-
435.
9. ~ H H ~ y K C.H. 0 cy~eCTBOBaHzH rO2OMOIX~H~X nepBoo~paSHNX. -
~oK~.AH CCCP, I972, 204, ~ 2, 292-294.
V.M ~, TRUTNEV CCCP, 660075, EpacHo~pcK,
(B.M.TPYTHEB) KpacHo2pcEH~ Pocy~apCTBeHHN~
yHHBepcHTeT
CO~O~ENTARY
D.I.Gurevich proved in ~I0~ that in the space H(6 ~) there exist
closed non-trivial translation invariant subspaces without exponential
polynomials. The same holds in $(~), ~(~), ~f(~) too.
REPERENCES
1o. r ~ p e B a q ~.H. EosTpnpa~epu ~ n~odae~e Z.~BsI~a. - ~ .
aHaa~s ~ ero n-p~a., I975, 9, 2, 29-35.
367
7.4. LOCAL DESCRIPTION OP CLOSED SUBMODULES AND THE PROBLEM
old OF OVER-SATURATION
The space ~$ of all ~$-valued functions analytic in a do-
main ~ of the complex plane ~ becomes a module over the ring of
all polynomials ~ [~] under pointwise algebraic operations. Con-
sider a submodule P of ~$ endowed with the structure of a Haus-
derff locally convex space such that the multiplication operators by
polynomials are continuous. A great many problems in Analysis, such
as the problem of polynomial approximation [I], convolution equations
[2], mean periodic functions [3], the problem of spectral synthesis
[4],[5] etc., is connected with the problem of local description of
closed submodules lop . Such a submodule I defines a d i v i-
S 0 r (I) forms any point
of all germs at
% ,k . The m o d u l e
tai ng (I)
. The d i v i s o r is a mapping which trans-
~ ~ into a submodule I~ of the module ~k $
of ~$-valued analytic functions. The mapping
transforms every function in ~ into its ge~ at
IA is the smallest ~k-submodule of ~ con-
A submodule I is called a d i v i s o r i a I
dule if
I w a}
submo -
n
The module p =~ equipped with the topology of uniform conver-
gence on compact subsets of ~ provides an example of a module whose
all closed submodules are divisorial [6]. Many antipodal examples
can be found in ~], [4],[~.
The PROBLEM of localization consists in the characterization of
those conditions which ensure that ever 2 submodule of a Kiven modul e
The following concepts are useful for the solution of the prob-
lem of localization. Namely, these are the concepts of stability and
saturation, which separate the~algebraio and analytic difficulties
of the problem, Define =~ if ~Z6 .
DEFINITION I. A submodule [ ~9 ca lled stable if for eve ~ k~
368
It is natural to consider stable submodules for modules
possessing the property of the uniform stability. This property en-
sures a certain kind of "softness" of the topology in ~ .
DEFINITION 2. A module ~ is called uniforml,y stable if for
every neighbourhood V~ ~ of zero there exists a neighbourhood
[ ~ 9fi zero satisfyin6
v.
The following theorem explains the importance of the concept of
saturation which will be defined later.
THEOREM I. Let ~ be a uniformly stable module. Then the sub-
module I ~ ~ is divisorial iff it is stable and saturated.
The saturated suhmodules for ~ =~ can be described as fol-
lows. Let V be a neighbourhood in ~ and let ~ ~l(~g)
Set
~(~)" ~ F) z vl.
Suppose that for each ~I(~) and each ~
I~(~,)l..<C~,v(~), ~O ~ l~l~t (i)
Then I is called s a t u r a t e d. Note that (I) automatically
holds for ~ ~ I •
In general we proceed as follows. The dimension ~ l~ of
I~ over 0h (~ ~) is clearly not greater than ~ . Put
~4= ~ ~ I ~ Then it is easy to show, using standard keG
arguments with determinBnts, t ~t~IA =--~ in G o Moreover
t h e r e e x i s t s a ha~( ~n I suchthat ~(~('~,,~o, ~k ( t~(% family 1~ (0, . , D f o ~ s a ~ s i s o f I~ for every k ~ 8 . Set ~ I = ~
(the local ~ - ~ of I ). If ~ ( ~ ) ~ I x then (0+. (K)
: . , cj O , ( 2 )
369
and the germs ~j ~ in #$ the can be found as follows. Consicer ~ ortho-
gonal projection ~ onto the subspace spanned by el, ,. • •, ejK •
Here ~=(J~,...,jK) and {~j]~-4 is the standard basis in~$ . The
system of linear equations (with a (kx ~) -matrix) ~ ~ =
'
where the determinants are defined in accordance with
Kramer' s rule.
DEFINITION 3. A submodule I , ~----~ is called saturated
with respect to ~ ~ ~ if for every neighbourhood of zero V ~
the followin~ holds
> IcPI ¢st .
A submodule I is called s a t u r a t e d if it is sa-
turated with respect to every ~ ~ I (~) and I is called
o v e r - s a t u r a t e d provided it is saturated with res-
pect to every ~ ~ p .
The existence of suitable estimates for holomorphic ratios
5/@ (see ~1] ,E8],Eg] ) in many cases permits to prove that a
given submodule I is saturated. In particular the local descrip-
tion of ideals in algebras can be obtained in this way EI0-12]. If
P is an algebra then every ideal of P is stable (as a rule).
But if P is only a module, as for example in E4], then the role
of stability may turn out to be dominant.
THEOREM 2. Suppose that for ever~ collection ~ , ° . . ,
of elements of ~ the set
370
is contained in ~ and bounded. Then ever~J divisorial submodule
P , ~I = ~ is over-saturated.
The proofs of theorems I and 2 are to appear in Izvestia Acad.
Nauk SSSR ~).
It follows from Theorems I and 2 that for a uniformly stable mo-
dule ~ satisfying (3) every submodule of local rank I, containing
a submodu~e with the same properties, is divisorial. This shows how
important is it to extend ~heorem 2 to submodules of an arbitrary
local rank.
THE PROBLEM 0E SATURATION. Let ~ be a uniforml~ stable submo-
dule satisf~in~ (~). Is it true that every divisorial submodule I~
is over-s~turated? If not~ what are ~eneral conditions ensuring that
I is over-saturated?
The solution of the problem would clarify obsuure points in the
theory of the local description and in its own turn would lead to
solutions of some problems of real and complex analysis.
REFERENCES
I. H ~ E o ~ ~ c E H ~ H.E. Hs6ps2H~e sa~a~ BeOOBO~ ~OECHMa--
~ ~ cne~Tpaz~Horo aHa~sa. - Tp.MaT.~-Ta AH CCCP, 1974, 120.
2. Ep a c H ~E o B- T e p H OB C E ~ ~ H.~. 0~opo~Hoe ypaB--
HeHHe T~ua cBepTEH Ha B~uyz~X o6xacT~x. - ~oF~.AH CCCP, 1971,
197, ~ I, 29-31. 3. S c h w a r t z L. Th~orie g~n~rale des fonctions moyenne-
p~riodiqu~. -Ann.Nath°, 1947, 48, N 4, 857-929.
4. Ep a c ~ ~ E o B- T e p H 0 B C E H ~ H.~. HHBap~aK~e no~-
npocTpa~cTBa a~ax~TEecFmx SyH~ I. CneETpax~ C~HTeS Ha BH--
o6xacT~x. - MaTeM.C6., I972, 87, ~ 4, 459--488.
8~ Ep a c ~ E o B- T e p H 0 B C E X~ H.~. HHBap~aHT~e no~-
npoc~paHcTBa aHax~TEecE~x ~ ~. CneE~az~m~ c~Tes ~a B~--
od~aCT2X. - MaTeM.C6.,I972, 88, ~ I, 3-30.
6. ~ a r t a n H. Id~aux et modules de foncCions analytiques de
variables complexes.-Bull. Soc.Math. Erance, 1950,78,NI, 29-64.
7. K e 1 1 e h e r J.J., T a y 1 o r B.A. Closed ideals in
locally convex algebras of ar~alytic functions. - J.reine und
angew.Math., 1972, 225, 190-209.
*) Cf. HSB.AH CCCP, cep. MST., 1979, 43, N I, 44-46 sad N 2, 309-
341 - F_~L
371
8. Kp a c H ~ E o B- T e p H OB C E ~ ~ M.$. 0KeHEa cydrapMo-
HH~ecEofi pasHOCTE cyOrapMoHE~ecEEx ~y~ELm2 I. -- MaTeM.c6., 1977,
102, ~ 2, 216-247.
9. K p a c E ~ E o B - T e p H o B C E E ~ H,$. 0neH~a oydrapMo-
H~ecEo~ paSHOCTE cydrapMoHH~ecEl~X ~yHI<II~ II. - MaTeM.c6.,1977,
103, ~ I, 69--111.
I0. P a m e B C E E ~ H.E. 0 3a~HyTNX F~ea~I~Cg B O~HO~ CqeTHO- HOpM~oOBaHHO~ a~re6pe ne~x s2a~ETE~ecEmx ~yHEL~H~. -- ~IOF~.AH CCCP,
1965, 162, ~ 3, 513-515.
II. K p a c E ~ K o B E.G. 0 saME~y~u~ ~es~ax B xoEa~HO--B~
a~re6pax ne~x ~HE~Ua~ I, H. - HsB.AH CCCP,cep.MaT. I967, 3I,
$7-60; I968, 32, 1024-1082.
I2. M a n a e B B.M., M o r yx ~ c E E ~ E.3. TeopeMa~e~e-
~ aHa~T~ecEHx ~yHEL~ C ss~aH~o~ Ma~opa~TO~ E HeEowop~e
ee np~o~eH~x.- 3an.Hay~H.CeMHH.JlOMH, I976, 56.
I. P. KRASI CHKOV-TERNOVSKI I
(14. ~ .KPACI~IKOB-TEPHOBCI~ )
CCCP, 450057, Y~a
yx. TyEaeBa 50
OT~eJ~ (~3~E~ ~ MaTeMaTHEH
BsmE~pc~ ~a~ All CCCP
372
7.5- ON THE SPECtrAL SYNTHESIS IN SPACES OF ENTIRE FUNCTIONS
OF NORMAL TYPE
Let ~ be a positive real number and let H be a ~S~ -periodic
lower semi-continuous trigonometrically ~-convex function with va-
lues in (-CO, Oo~ . Denote bY ~ (H) the set of all trigonometri-
cally ~-convex functions ~ satisfying ~(~)< H(e) for every
~[0,~] . For ~c ~KL(~) let ~(~) be the Banach space of all
entire functions ~ with the norm N~IIC~ ~ I~(~e~)l ~J~(-~(e)$~).
The family I ~(~)I ~ E~t(M) is inductive with respect to natural
imbeddings and its inductive limit [~, H(@)) is a space of
)-type in the sense of J.Sebastiao-e-Sllva.
Multiplication by the independent variable is a continuous map-
ping of[#,H(eD into itself, so that [p,H(e)) is a topologi- cal module over the ring of polynomials and one may consider the la- ttice of the closed (invariant) submodules. The submodule I~,k =
: { ~ ~ [,P,H(e))" q(~) ..... ~(K)(a~= 0] defined by a~C
and k ~ + is of the simplest structure.
By a commonly accepted definition (cf. [11 ) a submodule I~[~, H(S))
a d m i t s t h e s p e c t r a 1 s y n t h e s i s (or is
1 o c a 1 i z e d ) if it coincides with the intersection of all
submodules IG, k containing it.
PROBLEM. Fin d necessar,y and sufficient conditions (on ~(~) )
for ever~j closed submodule I~[~, H(~)) to admit spectral s~nthesis.
In 1947 L.Schwartz [2] proved that every ideal is localized in
the algebra of all entire functions of exponential type ( ~= ~ ,
H(e) = + co ). The progress in the localization theory for spaces
invariant with respect to multiplication by the independent variable
in the weighted algebras and modules of entire functions is des-
cribed in the survey by N.K.Nikol'skii [1] ; there is an extensive
list of references there.
For ~J the discussed problem was solved by I.F.Krasi~kov:
every closed submodule I [ I, H(e)) is localized iff H is un- bounded [ 3] •
For an arbitrary 9 > 0 it was proved in [4] that if the
length of every interval, where H is finitegdoes not exceed ~/p
then every closed submodule I [9 H(e)) a~its localization
There is an indirect evidence that the last condition is not
only sufficient but necessary. Unfortunately, all my attempts to
prove its necessity failed.
373
REFERENCES
I. H H E o ~ B c E ~ ~ H.E. HEBapHaHTHHe no~npocTpaHcTBa B Teo-
pEE onepaTopoB E B Teop~ ~yHEn~. - B F~. : HTOr~ HayEE ~ TexH~--
F2. M.: B~TM, MaTeM.aHa~ES, I974, I2, I99-412.
2. S c h w a r t z L. Th6orie g@n~rale des fQnctio~ moyennes pet-
riodiques° -Ann.Math., 1947, 48, N 4, 857-929.
3. Ep a c ~ ~ ~ o B-- T e p H o B c E ~ ~ M.~. MHBap~aHT~e no~-
npocTpaHCTBa aHa~TEqecE~x ~y~E~. Iv -- MaTeM. c60pH~972, 87,
4, 459--489; H. -MaTeM.c60pH.,88, J~ I, 3-30.
4. T E a ~ e H E 0 B.A. 0 cneETp~HOM C~Tese B npocTpaHowBax
a~a~mT~ec~x (~yHELU~OHa~oB. -- ~oF~.AH CCCP, 1975, 223, ~ 2, 307-
-309.
V.A. TKACHENKO
(B.A. TEA~K0)
CCCP, 810164, Xap~KoB
np.J~eH~a 47,
$~Sm~O--TeXHH~ecEm~ ~HCTETyT
H~SE~X TeMnepaTyp AH YCCP
374
7.6. old
A PROBLEM IN SPECTRAL THEORY OP ORDINARY DIFFERENTIAL
OPERATORS IN THE COMPLEX DOMAIN
Let I~ be a domain in C and ~(~) be the space of func-
tions analytic in ~ supplied with the topology of uniform conver-
gence on compact subsets of ~ . Let ~k~ ~(~), ~=0,...~-~.
Consider a differential polynomial ~ , ~=(~/~E)~+~_~ (~/~E)~'~+...~@0.
Choose ~ linear functionals ~...,~
= < {, > - - 0 ,
def ines a l i n e a r operator ~ on ~f b y the =le o •
of solutions of the equation
on A(~) and set ~4=
The differential polynomial
that maps ©~ into A(~I) is a fundamental system
~-k~--0 (~)
normalized by equa l i t i es b~ (P) (O,k) =~K,p+4 ' k = ~ , - . . , ~ p= 0 , . . . , ~ - ~ , the spectrum of ~ coincides wi th the set of so lu-
t ions of the characteristic equation
A ( ~ ) = o , where ~(~)~(<~k(z,~),9i>)k~i={.
Since & is an entire function of ~ , the spectrum is, unless AmO,~
discrete set with possibly a unique limit point at infinity. In this
case a root subspace of finite dimension corresponds to each point
of the spcetrum.
The PROBLEM mentioned in the title consists in obtaining a des-
cription of the A(ll)-closure of the linear span of root vectors
of ~ . This problem is closely con2ectedwith completeness ques-
tio~fo=t~e ~yste~(Z,~)} of solutions of eq~tion (i) in A(ZI) , with the--O-construction of general solutions of differen-
tial equations of infinite order with respect to ~ , with the theory
of convolution equations and of mean periodic analytic functions. An
analogous problem for differential operators on the real line is
well-known.
CONJECTURE. I~f ~ is convex and A~O then the closure Ok
of the linear span of ~oPt vec$ors of ~ coincides with the domain
of all its powers I i.e ,. with th e subspace
375
0,,1,. . . ] .
The inclusion ~ C ~co follows immediately from the
~(~I) -continuity of $ and ~. ~ ~ . The inverse inclu-
sion is non-trivial and has been proved only in some particular
cases: by A.F.Leont'ev [1] in the problem of completeness of the sys-
tem { ~(2, ~i ) } ; by Yu.N.~rolov [2] in the problem of construct-
ing a general solution of equations of infinite order under some ad-
ditional restrictions on & (~) , see also the papers of the
same authors cited in [1~ and [2] ; by V.I.Matsaev [3] for a general
system { £~k} ~ k=~ but with ~ =C ; by the author [4] in some weighted sp~ces of entire functions. In the case of an arbitrary con-
vex domain ~ and ~= ~ the question under consideration is equi-
valent to that of the possibility of spectral synthesis in the space
of solutions of a homogeneous convolution equation; spectral synthe-
sis is really possible in this situation, this has been proved by
I.F.Krasickov-Ternovskii [ 5], [6], furSher generalizations can be
found in [7]. The results of [5], [6] imply that the convexity
condition imposed on ~ cannot be dropped. The question whether
the above conjecture is true for an arbitrary convex domain
and an arbitrary ~ remains open.
REFERENCES
I. ~ e o H T ~ e B A.$. E BOnpocy 0 noc~e~oBaTe~ocT~x ~e~x
al~eraTOB, oOpasoBaHHRX XS pemeH~ ~x~epeHnHaz~H~x ypaBHeH~. -- MaTeM.c6., I959, 48, ~ 2, I29-I36.
2; ~ p 0 X 0 B D.H. 06 O~OM MeTo~e peme~ onepaTop~o~o ypaB~e-
6ecEoHeqRoro nop~Ea. -~mTeM.c6., I972, 89, ~ 3, 461-474.
3. M a ~ a e B B.H. 0 pas~oxeH~ ne~x ~yHEn~ no COOCTBeH~M
I~ooe~HeHH~M ~JHEL~M O606~eHHO~ KpaeBo~ ss~a~o -- Teop.~ymc~.,
~s.aHa~s H ~x np~.~ 1972, 16, 198-206.
4. T E a q eHE O B.A. 0 pas~o~eH~ ne~o~ $ ~ Eo~e~o~o HO--
p~a no EopHeB~M ~ ~ 0~O~O ~z~x~epe~s~HO~O onepa~opa. -
~awe~.c6., I972, 89, ~ 4, 558--568. 5. Ep a c ~ ~ ~ o B- T e p H O B C ~ ~ ~ ~.~. 0~Hopo~e ypaB--
He~ T~a cBep~F~ Ha B ~ X o6~ac~x. - Ao~.AH CCCP, IOVI,
197, • I, 2~-3I.
376
6. Kp a c z ~ E o B- T e p H O B C E ~ ~ H.*. HEBap~aHTH~e noA- npocTpaHCTBa a~TE~ec~zx SyBZ~. H. CneETpaa~B C~TeS Ha BU--
o6aaCTSX. - MaTeM.c6., 1972, 88, ~ I, 8-30. 7. T E a ~ e H E O B.A. 0 oHeETp6UIBHOM c~Tese B npocTpa~cTBax
aHaxZT~xecE~x ~y~R~OHaXOB. - ~oEa.AH CCCP, 1975, 228, ~ 2, 307-
- 3 0 9 .
v.A. TKACHENKO
(B.A. TEACh, K0) CCCP, 810164, Xap~EoB npocneET ~e~Ha, 47,
~SHEO--TeXH~qecEm~ ~HCT~TyT H~SE~X TeMnepaTyp AH YCCP
COMMENTARY BY THE AUTHOR
S.G.Merzlyakov has discovered that my CONJECTURE IS FALSE.
Namely, pick up two entire even functions ~ and q of exponential type and of completely regular growth such that the zero-set of ~
is an ~ -set and all zeros are simple. For example
X4
f i t . Let {~k} and {~k} denote the zero-sets of q and ~ .
Then the functions
"~4 (~(Ak)C~'(~'k)(A'Ak) ' CA)=
are entire functions of exponential type and ~(~) + ~(~)A ~(A)--~q(~) define continuous linear functiomals on , , (~i ,
being the interior of the indicator diagram of ~ The operator ~ defined by (~/~£)W and these functionals is an ope-
rator with the void spectrum because A(~)=A(A)q(~) +~(~)~(~)~4. However, the domain ~09 of ~ contains a non-zero element, namely,
the holomorphic function defined in ~ by
377
S.G.Merzlyakov has communicated that ANALOGOUS COUNTER-EXAmPLES
EXIST FOR UNBOUNDED DONAINS AS WELL.
Nevertheless, to my knowledge THE GENERAL PROBLEM of describing
the closure of the family of roots vectors for an arbitrary operator
RENAINS UNSOLVED.
378
7,7~ o l d
TWO PROBL~S ON THE SPECTRAL SYNTHESIS
1 . Synthesis .is impossible. We are concerned with the synthesis of (closed) invariant subspaces of ~* , the a&Joint of the operator
of multiplication by the independent variable ~ on some space
of analytic functions. More precisely, let ~ be a Banach space of
functions defined in the unit disc ~ and analytic there, a~d suppo-
se that ~Xc X and the natural embedding X ~ ~(~) is con-
tinuous,~ ~ bei~ the space of all functions holomorphic in ~.
If #~ X then k~(~; denotes the multiplicity of zero of ~ at a
point ~ in ~ , and ~or any function k from ~ to nonnegative in-
tegers let
Xka. %:>-k A closed ~-invarlant subspace ~ of ~ is said to be DIVISO-
RiAL (or to have THE ~-PROPERTY) if E = X k for some
(necessarily k(~)~- ~EQ~) ~ ~ k~(~) , ~ ~)).
CONJECTURE I. In every space ~ as abov~ there exist non-divi-
sorial Z-invaria~t subspaces.
The dualized ~-property means that the spectral synthesis is possible~ To be more precise, let ~ be the space dual (or predual)
to X equipped with the weak topology 6"( ~, X) ( the duality
of ~ and V is determined by the Cauchy pairing, i.e. ~, $ ~ ~-- A
= E ~(~)~(~) for polynomials ~ ~ ). A ~-invariant sub-
space E of ~ is said to be SYNTHESABLE (or simply 6 -SPACE) if
(I)
with k~ kE~.. In other words ~ is an ~-space if it can be reco-
vered by the root vectors of ~* it contains.
All known results on Z-invariant subspaces (cf. KIS ) support
Conjecture I. The main hypothesis on X here is that X shoula be
a B a n a c h s p a c e . The problem becomes non-trivial if, e.g.
the set of polynomials ~A is contained and dense in X and
~. ~G~ :~A I P(~)I IIP~:X f < oo} ........... ~ . The existence of a sing-
le norm defining the topology should lead to some limit stable pecu-
liarities of the boundary behaviour of elements of X , and it is
379
these peculiarities that should be responsible for the presence of
non-divisorial ~-invariant subspaces. Spaces topologically contai-
ned in the Nevanlinna class provide leading examples. The aforementi-
oned boundary effect consists here in the presence of a non-trivial
inner factor (i.e. other than a Blaschke product) in the canonical
factorization. Analogues of ~nNer functions are discovered in classes
of functions defined by growth restrictions (~2], E3~, [4]); these
classes are even not necessarily Banach spaces but their topology is
still "sufficiently rigid" (i.e. the seminorms defining the topology
are of "comparable strength"). On the contrary, in spaces X with
a "soft" topology the invariant subspaces are usually divisorial. So-
metimes the "softness" of the topology can be expressed in purely quan-
titative terms (for example, under some regularity restrictions on
, a l l ideals in the algebra I ~ : ~HO$(~) , I ( ~ ( ~ ) 1
0(IC(~)), O ~---- C~ } are divisorial if and only if
0 ..... ~-- ~ ~ + co , [5], [I O] ). This viewpoint can be given
a metric character; it can be connected with the multiplicative struc-
ture of analytic functions, with some problems of weighted polynomial
approximation, with generalizations of the corona theorem, etc. (of. D,3,6]).
2, A~proximative s~thesis is possible. Let us read formula (I)
in the following manner: there is an increasing sequence { ~ } of
~W-invariant subspaces of finite dimension that approximates E :
E-~,~,,~ E~, ~'~'-/-~ { ~: ~ X , -R~ ~,~(÷, E~,) : o} . i,~ w.,
Removing one word from this sentence seems to lead to a universal des-
cription of ~-invariant subspaces.
CONJECTURE 2. Let ~ be a space from section I and E be a
~*-invariant subspace of ~ • Then there exist subspaces ~
with ~*E.cE., ~ E.<~ (~I~) so that E= {¢~ E~. There is a further extension of this Conjecture that still could
look probable. Namely, let ~ be a continuous linear operator on a
linear space Y and suppose that the system of root vectors of T
is complete in ~ • Is it true that T~c E----~E e~ E~ for
some sequence ~ with TE~c E~ , ~ $ ~ ~ ( ~ ) ?
But it is easy to see that without additional restrictions on T
the answer %o the last question is "no". A counterexample is provid-
380
ed by the left shift (i.e. still ~! ) (@0,~,...)~ ~ (~,~,...)
on ~P(~f~) with an appropriate weight { ~}~ 0 (decreasing
rapidly and irregularly). This operator posesses invariant subspaces
that cannot be approximated by root subspaces, [3]. In examples of
such kind it is essential that the spectrum of the operator reduces
to the single point 0~
A plenty of classical theorems on ~ -invariant subspaces (as
e~g., Beurling's theorem) not only support Conjecture 2, but also al-
low to describe ~cyclic vectors (that is, functions ~ with the
property span ( ~ : ~ 0) ~ ~ ) in terms of the approximati-
on by rational functions with bounded " X -capacities". If ~ is a
rational function with poles in C \ o~ ~, ~(~) ~ 0 then
o, capaci- ty of an arbitrary ~*-invariant subspace is defined similarly. If
~ (~)- ~w$~ $~ and ~p 6 @ ~ ~ then ~ is not cyc-
lic for m* ; analogously, ~ p c ~ p x E . ~ __> 45~ Ew==~ --- Y .
The last assertion can be converted, after a slight modification of
the notion of "capacity" [7,8]. Probably techniques of rational ap-
proximation should allow to prove Conjecture 2 avoiding estimates of
" X-capacities" of rational functions (that appears to be a more
difficult question; it is worth mentioning that this question is a
quantitative form of the uniqueness theorem for X ). The results on
this matter known up to now use, on the contrary, not only classical
uniqueness theorems but also the explicit description of ~ -invari-
ant subspaces in terms of the inner-outer factorisation.
RR~ERENCES
I. H H K o H b c K H ~ H.K. HHBapHaHTHNe HO~NpOCTpaHCTBa B TeopH~
oNepaTOpOB H TeopHH SyHE~M~. - B EH. : ~TOrM HayEH M TeXHHKH. Ma-
TeMaTHYecEH~ aHa2Hs, T.12, M., BHHET~, 1974, 199-412. 2. K p a c H y K o B - T e p H 0 B C K H ~ H.~. ~HBapHaHTHNe Ho~Hpo -
cTpaHCTBa aHa2HTHMeCKHX ~yHE~H~. If. CHeKTpa~bHN~ CHHTe3 Ha Bh~qyE-
HMX 06HaCT2X. - MaTeM. C6., 1972, 88, ~ I, 3-30. 3. H H E o ~ b c E H ~ H.K. Es6paHHMe s~a~m BeCOBO~ a~HpoKcMMaI~HH
H cneETpa~HOrO aHa~rHsa. - Tpy~M M~, 120, M.-2., HayKa, 1974.
4. K o r e n b I j u m B. A Beurling-type theorem. - Acta Math.,
1975, 135, 187-219.
8. A n p e c ~ H C.A. 0nHcaH~e aaredp aHaaHTHyecK~x ~yH~U~, ~onycKa-
mMx zo~aaHsa~ ~eaaoB. - 3an.Hay~H.ceMm~.~0~4, I977, 70, 267-
381
269
6. H H K o ~ b c E H ~ H.K. 0nHT Hcn0abs0Ba~Hs ~aKTop-onepaTopa ~
J~0K82LMSaILMM ~ -- MHBapMaHTHhKX H0~qp0cTpaHCTB. - ~oK.~.AH CCCP, I978 240, ~ I, 24-27
7. F p M 60 B M.B., H H K 0 ~ 5 O K H ~ H.I{. 14HBKpMaHTHMe I10/~-
~pooTps/4CTBa M pa~OHSwIBHaS a~poKcHMa~H~. - 3a~.H~.CeMMH.~0~4,
I979, 92, I03-II4.
8. H H K o a 5 c K H R H.K. ~e~HH 05 onepaTope cABHra I. - 3an.Ha-
y~H.CeMHH.~0~4, I974, 39, 59-93.
9. H i i d e n H.M., W a i 1 e n L.J. Some cyclic and non-cyclic
vectors of certain operators, - Indiana Univ.Math.J., 1974, 23, N 7,
557-565.
I0.~ a M 0 2 H ~.A. TeopeMs AeaeHH2 z saMEHy~Me ~eaJm B aape6pax
aHa~HTH~ecKHx ~yH~H~ C Ma~0paHToR EoHe~0~O poc~a. - MsB. AH Ap~.
CCP, MaTeMaTHKa, 1980, 15, ~ 4, 323-331,,
N. K • NIKOL ' SKII
(H.K.HMK0315C~)
CCCP, 191011, JleHHHrpa~
~0HTaHKa 27, H0~4
382
7.8. CYCLIC VECTORS IN SPACES OF ANALYTIC FUNCTIONS old
Let X be a Banach space of snalytic functions in ~ satis-
fying the following two conditions: (i) for each ~ , ~ ~ ,the
map~--*~(~) is a bounded linear functional on X , (il)~Xc~ .
It fellows from (ii), by means of the closed graph theorem, that
multiplication by ~ is a bounded linear transformation (more
briefly, an operator) on X . Finally, ~ X is said to be a
c y c 1 i o vector for the operator of multiplication by
if the finite linear combinations of the vectors ~ , $~ , $~$,...
are dense in X (when the constant function ~ is in ~ , one
also says that ~ is w e a k 1 y i n v e r t i b 1 e in X;
this terminology w~s first used ~ ~I~).
QUESTION I. Does strong invertibility imply weak invertibility?
(That Is, if ~,~, ~ are all in ~ , is ~ c~clic?)
Consider the special case when X is the Bergman space, that is
the set of s q ~ r ~ - i n t e ~ b l e analytic ~unctions-ll~]l¢=~ IS J z ~ ° ° "
CONJECTURE 1. I_~f ~ is in the Ber~man space and if I~(~)I>
>C(~-l~l) ~ for some O, @> 0 , then S is cyclic.
If correct this would imply an affirmative answer to QUESTION I
when X is the Bergman space. The conjecture is known to be correct
under mild additional assumptions (see [2], [3], [4~ ). In particular it
is correct when ~ is a singular ~ner function. In this case the
condition in the hypothesis of the conjecture is equivalent to the
condition that the singular measure associated with ~ has modulus
of continuity 0(~@~ ~/i) (see [1] ). CONJECTURE 2. A singular ~er function is c2clic in ~he Berg-
man ,space if and onl.T if its @ssociated sinRular meade puts no mass
on an~ ~rleson set. (For the definition of Carleson set see [5~,
pp. 326-327. )~)
For more discussion of the cyclicity of inner functions see §6
of ~6], pages 54-58, where the possibility of an '~nner-oute~' facto-
rization for inner functions is considered.
, i
u) or else 9.3 - Ed.
383
QUESTION 2. Does there exist a Banach space of anal,v~ic functi-
ons. satisfyin~ (i) and (ii). in which a function i is c~clic if
and onl~ i f ~t ~s no z eroj in ~ ?
N.F~Niko l sk i i has shown E7] t h a t no we igh ted sup-norm space o f a
c e r t a i n t y p e has t h i s p r o p e r t y . I f such a space X e x i s t e d t h e n t h e
o p e r a t o r o f m u l t i p l i c a t i o n by ~ on X would have t he p r o p e r t y t h a t
i t s s e t o f c y c l i c v e c t o r s i s non-empty, and i s a c l o s e d s u b s e t o f t he
space X \ {0} ( t h i s f o l l o w s s i n c e t h e l i m i t o f n o n - v a n i s h i n g a n a l y t i c
f u n c t i o n s i s e i t h e r n o n - v a n i s h i n g o r i d e n t i c a l l y z e r o ) . No example
o f an o p e r a t o r w i th t h i s p r o p e r t y i s known. (Th i s ~ y no l o n g e r be
c o r r e c t ; P e r En f lo has announced an example o f an o p e r a t o r on a Ba-
n~ch space w i th no i n v a r i a n t subspaces ; t h a t i s , e v e r y n o n - z e r o v e c -
t o r i s c y c l i c . The c o n s t r u c t i o n i s a p p a r e n t l y e x c e e d i n g l y d i f f i c u l t . )
H°S .Shap i ro has shown t h a t f o r any o p e r a t o r t h e s e t o f c y c l i c v e c -
t o r s i s a lways a ~ s e t ( s e e [8] , §11, P r o p o s i t i o n 40, p°110) . For a
d i s c u s s i o n o f some o f t h e s e q u e s t i o n s from t h e p o i n t o f view o f
weigh ted s h i f t o p e r a t o r s , see [ 8 ~ , ~ 1 1 , 12.
QUESTION 3. Let_ X be as before, and l e t ~, ~eX wish
c clic. # , clic?
This question has a trivial affirmative answer in spaces like
the Bergman space, since bounded analytic functions multiply the
space into itself. It is unknown for the Dirichlet space (that is,
the space of functions wi th ~l~'I~< eo ); the special case ~ = constant is established in [9] •
REFERENCES
1. S h a p i r o H a r o I d S. Weakly invertible elements in
certain function spaces, and generators in ~ . - Mich.Math.Je,
1964, 11, 161-165. 2. S h a p i r o H a r o i d S. Weighted polynomial approxlwa-
tion and boundary behaviour of holomorphic functions. - B ~. :
CoBpeMeH~e npo6xe~ Teop~ aRa~zT~ec~x ~m~, M., Hay~a, 1966, 326-335.
3. ~ a n ~ p o r . He~oTopHe saMe~zam~t,,~ o Becoso~ no,~z~o~,ma~z, Ho~ annpo~czMam~ rO~OMOp~HX ~ym~. -MaTeM.cS., 1967, 73, 320- -330.
4o A h a r o n o v D., S h a p i r o HoS., S h i e I d s A,L~
Weakly invertible elements in the space ef squ~re-summ~ble holo-
384
morphlc functions. - J.London Nath.Soc.~ 1974, 9, 183-192.
5. C a r I e s o n L. Sets of uniqueness for functions regular in
the unit circle. - Acta Nath.~ 1952, 87, 325-345.
6. D u r e n P.L., R 0 m b e r g B.W., S h i e I d s A.L.
Linear functionals on ~ ~ spaces with 0 < ~ ~ I . - J.fur reine
und angew.Math.~1969, 238, 32-60.
7. H ~ E o x ~ c ~ z ~ H.E. CneETpax~m~ C~mTes ~ sa~a~a mecoBo~
annpo~c~mmm B upocTpaRcTBaX asa~mTE~ec~x ~y~r~. - Hs~.AH Ap~.
CCP. Cep.MaTe~., I97I, 6, ~ 5, 345-867. 8. S h i e 1 d s A I I e n L. Weighted shift operators and analy-
tic function theory. - In: Topics in operator theory, ~ath.Surveys
N 13, 49-128; Providence, Amer.Math.Soc., 1974.
9, S h i e 1 d s A 1 I e n L. Cyclic vectors in some spaces of
analytic functions. - Proc.Royml Irish Acad.~1974, 74, Section A,
293-296.
AT,TRN L. SHIELDS Department of Nathematics
University of Michigan
Ann Arbor, Michigan 48109
U.S.A.
O0~ENTARY
QUESTION 1 has been answered in the negative by Shamoyan [10].
• ITZ~ l ~ D , 0<~<1 and l e t X~ d,eno¢e the spa,oe of ~11 func t ions ~ ana l y t i c i n the u n i t
=0 (0 ~(~)) f~r i~[<~, 5 ~ . Then polynomials are dense in X~ and
zor ÷ ~ ~ p (- r 1+z ~ ) we haw : ~,~ e X ~ but ÷ is not we- ~ - ~ s akl~ invert ible in X~.
CONJECTURE 2. The "only if" part can be found in E2] of 7.10, the
"if- part is proved in ,[_11] ° The same criterion of weak invertibility
of inner f~nction~ holds in all Bergman spaces ~P, ~ p <oo • in
spaces A Fd¢~ t ~ : ~ i s ana l y t i c i n 0 and l ÷(~) l~o((~-I ~1) -P ) , I~1--~ ~ anein pUo~----- p>U oAP.
Note, by the way, that 7.7 contains a conjecture in the spirit of
QUESTION 2, and that both QUESTION 2 and 3 (together with some 6there)
are discussed in ref. ~3] of 7,7.
385
REFERENCES
I0. m 8 M 0 S H ~.A. 0 C~1860~ O6paT~MOCT~ B HeKOTOIDHX I~OCT~HCTBSX
aHaaaT~.ec~x ~yHE~. - ~oEa.AH AI~.CCP, 1982, 74, ~ 4, 157-161.
II. K o r e n b i u m B. Cyclic elements in some spaces of analytic
functions. -Bull.Amer,Math.Soc., 1981, 5, N 3, 317-318.
386
7-9- old
WEAK INVERTIBILITY AND PACTORIZATION IN CERTAIN
SPACES O~ ANALYTIC PUNCTIONS
A measure ~ on ~ is called a s y m m e t r i c m • a -
s u r e i f ~ has the form ~ I ~ ( ~ , 0 ) = ( ~ ) - 1 ~ ( ~ ) ~ 0 , where T is a finite, p o s i t i v e Borel measure on [ 0 , ~ , hav ing no mass at 0 , and such that ~([~,~S) > 0 for all 0~ <~ . For any
function # analytic in 0 and any ~ , 0 ~ ~<eo , we define the
generalized mean
c~
0 ~ < 1 . The class EP(#) tic in ~ such t h a t
consists of all functions S analy-
~ p ; ~ ; j ~ < co (2)
In the special case where • is a single unit point mass at ~ ,
the means (I) reduce to the classical means, and the Er(~_)
classes to the standard Hardy classes on 9 o In all cases, EP(~)
is isometrically isomorphic to the Lr(~) -closure of the polynomi-
als. General properties of these classes are outlined in D,2,3~.
Numerous investiEatlons of special cases (e.g., the Ber~nan classes,
~-area measure) are scattered throughout the literature. A comp-
lete biblioEraphy would be quite extensive, and so references here
are restricted t o those which have had the most direct influence
upon the author's works
A function ~ , ~eEP(~) is said to be w e a k 1 y i n -
v e r t i b 1 e if there--is a sequence of polynomials ~w}
such t h a t p~ ~ '~' '~ i n t he m e t r i c o f EP(~) , Prom an o p e r a t o r - t h e o r e t i c p o i n t o f v i ew , such f u n c t i o n s a re s i g n i f i c a n t i n t h a t an
Er( ) is weakly invertible if and only if it is a element of w
c y c 1 i c V • c t o r for the operator of multiplication by
on EP(#) . (When ~-~ , this operator is unitarily equivalent
to a subnormal weighted shift. ) In the special case of the Hardy
classes, Beurling K4~ showed that a function is weakly invertible__ if
and only if it is cuter. In the more general context ef the Er(~)
classes, a complete characterization of the weakly invertible func-
tions awaits discovery. At this Juncture, however, it is not even
387
clear what Eeneral shape such a characterization might take. We know
of only a handful of scattered results which are applicable to these
spacial classes. The earliest of these can be found in three papers
by Shaplro [5, 6, 7] and in the survey article by Mergelyan [8].
More recent contributions have been made by the author [I, 2], Aha-
ronov, Shapiro and Shields [9] ; and Hedberg (see Shields [I0,p.112]).
Many of the known results on weakly invertible functions in the
EP(~) classes are essentially either multiplication or factoriza-
tion theorems. It is well known that the product of two outer func-
tions is outer, and that any factor of an outer function is outer.
Do these properties carry over to weakly inver%ible functions in
the EP(~) classes? We list a number of specific questions along
these lines.
(a) Suppose 5, 9' ~GEP(~ ) a~ ~=~ . l~f ~ and ~ are
we~kl,7 in vet%iDle, is # weakl~ inver%ible? Conversely. if # is
weakl~ invertible, are ~ and ~ weakly inver%ible?
(b) I f ~ ~P(~) is n0n~lqhin ~ and --~ ~ F~C~)-- ,~or s o m e
~, ~ 0 , i_~s ~ weakl,7 inve~ible.?
(c) If ~ EPC~) i,s,,,,weakly inver%ible and ~ > 0 , i~s ~
w~kly inver%ible in EP'-(~)'~ ?
(d) I.~f ~ EP(~) and ~ is w~Ikl,y invex~ible,,,,, ,,i n E~(~) fors~e ~ , ~<p , i_~ ~ .~inve~iblein E~(~) ?
(e) ~et ~ E P ( ~ , ~6E*(~) an__d ~ E * ( ~ , and let ~=%k . z_~ ~ and ~ dr,,, .~l,~ invertlble .in E'(~ and
Fs(~) , respectively, is ~ weakl,7 inver%ible i n E r(~)-- ? What
about %he converse?
Of course, all these things are trivially true in the special
case of the Hardy classes. Question (e) is the most general of the
list. The reader can easily convince himself that affirmative ans-
wers to (e) would imply affirmative asnwers to all the ethers. Con-
versely, affirmative answers to (a) and (d) together woul~ yield
affirmative answers to (e). The answers to questio~ (a) are known to be affirmstive if ~e~ e°
[2] )'). These results are insp~ ires by an ,arl~ier res' ult of Shapiro
~) See also [3] of 7.7 - Ed.
388
E5 , Lemma 2~. The question remains unanswered for unrestricted
and ~ •
Question (b) has a long history, and versions of it appear in
numerous sources. An affirmative answer may be obtained by imposing p+o
the additional condition ~ E (~) for some I , ~> 0 . The
legacy of results of this type seems to begin with the paper of Sha-
piro [6], and has been carried forth into a variety of different set-
tings in the separate researches of Breunan D1], Hedberg ~2], and
the author [.2]. A similar result with a different kind of side condi-
tion is to be found in the work of Aharonov, Shapiro and Shields K9 ~.
In its full generality, however, the question remains unanswered.
Question (d) seems in some sense to be the crucial question,
certainly in moving from the setting of question (a) to that of ques-
tion (e), but perhaps also in removing the side conditions from the
results cited above. Presently, however, there seems to be little
evidence either for or against an affirmative answer, nor can we of-
fer any tangible ideas on how to attack the problem. The key to its
solution in the special case of the Hardy classes rests upon the fact
that, there, weak invertibility can be accounted for in terms of be-
havior within the larger Nevanlinna class. Unfort~3~ately, in the more
of the EP(~) classes, none of the several diffe- general sett~Jlg
rent generalizations of the Nevanlinna theory discovered to date seems
to shed any light upon the matter. It may very well be that the ans-
wer to the question is negative. Clearly, a negative answer would in-
troduce complications which have no parallel in the Hardy classes~
However, in view of the negative results of Horowitz ~3~ concerning
the ~ero sets of functions in the Bergman classes, such complications
would not be too surprising, and perhaps not altogether unwelcome.
REFERENCES
I. P r a n k f u r t R. Subnormal weighted shifts and related func-
tion spaces. - J.Math.Anal.Appl.~ 1975, 52, 471-489.
2. F r a n k f u r t R. Subnormal weighted shifts and related func-
tion spaces. II. -J.Nath.Anal.Appl.,1976, 55, 1-17.
3. F r a n k f u r t R. Function spaces associated with radially
symmetric measures. - J.Nath.Anal.Appl. ~ 1977, 60, 502-541 •
4. B • u r 1 i n g A. On two problems concerning linear trans-
formations in Hilbert space. - Acta Math.~ 1949, 81, 239-255.
5. S h a p i r o H.S. Weakly invertible elements in certain func-
tion spaces, and generators of $I - - Mich.~ath.J.~1964, 11,
389
161-165.
6.S h a p i r o H.S. Weighted polynomial approxlmationand boun-
dary behavlour of analytic functions. - B ~H.: CoBpeMemme npo6-
xe~ Teop~ aHS~ZT~ec~x ~y~u~R. -- M., "Hsy~a", I966,326-335.
7. • a n ~ p o r. HeEoT0pHe 3aMeqaHEH o BeCOBO~ nOJLVLHO~a~BHO~
annpo~c~au~m rO~OMOp~X ~y~z~. - MaTeM.C6., I967, 73, ~ 3, 320-330.
8. M e p r e ~ ~ H C.H. 0 noJmoTe c~c~eM aHa~TH~ecEEx ~y~. --
YcnexH MaTeM.HayE, 1953, 8, ~ 4, 3--63. 9. A h a r o n o v D., S h a p i r o H.S., S h i • 1 d s A.L.
Weakly invertible elements in the space of square-smnmable hclo-
morphic functions. - J.Londen Math.Soc.~1974, 9, 183-192.
10. S h i e 1 d s A.L. Weighted shift operators and analytic func-
tion theory. - In: Topics in Operator Theory. Providence, R.I.,
Amer.Math.Soc., 1974, pp.49-128.
11. B r e n n a n J. Invariant subspaces and weighted polynomial
approximation. - Ark.Mat.,1973, 11, 167-189.
12. H e d b e r g L.I. Weighted mean approximation in Caratheodo-
ryregions. -~ath.Scand.~1968, 23, 113-122.
13. H o r o w i t z C. Zeros of functions in the Bergman spaces. -
Duke Nath.J.,1974, 41, 693-710.
RICHARD I~RANKPURT Dept.of~th., College of Arts
and Sciences.
University of Kentucky,
Lexington 40506, USA
EDITORS' NOTE. See also 7.8 and Commentary to 7.8.
390
7.10. WEAKLY INVERTIBLEELE~ENTS IN BERGNAN SPACES old
DEFINITION 1. ~5 is the Hilbert space of analytic functions
in ~ with the norm
(I)
DEFINITION 2 [1]. Let ~ be the set of all open, closed and
half-closed arcs l,lcT , including all single points, T and
o A function ~ ~ ~ ~ is called a p r e m • a s u r e
iff
DEFINITION 3. A closed set ~, ~c T is called B e u r -
I i n g - C a r i e s o n (B.-C. seT) iff
where I~ are the components of T\ P near Lebesgue measure.
mo~osiT~o~ 1 [1~2]. ~e~ ~
th~ followin~ properties hold:
(i) The limit
and I" I denotes the li-
and ~¢(2)=/=0 ( ~ D) . Then
exists for any arc I, I ¢ T
(s)
391
(il) The limit
exi,s,~,S f o r a~¥ sequence of closed arcs ( I , ) such that T4cT~ c . . .
and U I ~ I , I b e ~ a ~ v e~enarc;
(iii) ~(I) , defined b,y (~) for open arcs I ,I c ~ , ad_.-
mits a unique extension te a premeasur~;
(iv) for any B~-Oq set ~ , whose complementar~ arcs . are I~,
the series ~ ~ (I~) i s ~bsolutel.7 convergent. , ;
(v) if we define
(4)
for B,-O, sets ~ , then ~ admits a unlque @,xtension to a f~n!te
non-pgsitive Bergs!,, measure on every Bt-C t set.
DEPINITION 4. The measure ~ (defined on the set of all Betel
sets contained in a B.-C. set) is called % h e ~ - s i n -
g u 1 a r m e a s u r e a s s o c i a t • d w i t h $ , ~
(it is assumed that ~(~) =4= 0 in ~ ).
Proposition I follows immediately from the results of D,~'
since ~ implies
DEFINITION 5. An element I , ~ ~ , is called w • a k -
1 y i n v e r t i b 1 • ( o r c y o 1 i c ) iff clos{~:~¢H'}=
PROPOSITION 2. The followlng con~tions are nec,es,sar~ re,r ,an
element ~ , ~e ~ , to be weakl 7 inve~rt,ible:
(a) ~(z)=~ 0 (~G~) ; (6)
(b) ~ = 0 . (7)
392
This proposition follows easily from the main theorem in E2~
which gives a description of closed ideals in the topological algebra
A -~ of amalytic functions I satisfying
Z~c~) l ~ c~('I- I,~l) - ~
CONJECTURE 1. Cqndit$ons (6) and (7) are sufficient for ,an t ,
~e~ ~ , to be weakl 2 invertible.
CONJECTURE 2. The sam e conditions also describe wee~l~invertSb-
!e elements in any Berth space ~ P ( ~ p <~) of analytic functi-
ons ~ ~ ' t h t he norm
REPERENOES
1. K o r e n b I u m B. An extension of the Nevanlinn8 theory.
- Acta Math. 1975, 135, 187-219.
2. K o r e n b 1 u m B. A Beurllng-type theorem. - Acta Nath.
1977, 138, p.265-293.
BORIS KORENBLb~ Del~rtment of ~thematics,
State University of New York
at Albany
1400 Washington Avenue
Albany, New York, 12222, USA
C0~ENTARY
Both C o n j e c t u r e s a r e i n Commentary t o 7 . 8 .
supported by the results cited
393
7.11. INVARIANT SUBSPACES OF THE BACKWARD SHIFT OPERATOR
IN THE SMIR OV CLASS
Denote by N, the Smirnov class i.e. the space of all functions
holomorphic in the unitLdiScl D and such that[~+I~l]o<~<~
is uniformly integrable in = d( T, ~) . Here ~ #($~)~
is the normalized Lebesgue measure on the unit circle T . The
space N, can be indentified with the closure of the set of polynomi-
als (in ~ ) in ~ L , where ~ L is the space of all measurab-
le functions ~ on T such that U~(~+ ~l)~L ~ 6~L the distance p is introduced by p(~ ~)~ I @(~+~ ~I) %~
Let ~ denote the backward shift operator, T S*~ ~-~(0) =
N.
being a closed subset of T . In [2] and
are solved for the case of the Hardy spaces
case the real variable characterization of
portant role.
I. INVARIANT SUBSPACES AND RATIONAL APPROXIMATION
PROBLEM. Describe the invarlant subspaces of S : N,-~N, .
It should be noted that an analogous problem for the shift ope-
rator S" N, --~ N, , S~ = ~# , can be reduced easily to the fa-
mous Beurling theorem describing the invariant subspaces of- ~S:HL~H ~
(see [5] ).
THE PROBLEM is connected with the description of the clQsure in
{'} F of the linear span of the Cauchy kernels I-~ ~ F '
[ 3] analogous problems
H p (0<p< I) . In this
H P (see [7] ) plays an in-
CONJECTURE I. If F has no isolated points then the closure of
the linear span of {~I is the set of all functions :EF
to @\F •
The case F--T is considered in [5].
2. EXAMPLES OP ~-INVARIANT SUBSPACES. Let X c N, and let
I * be an inner function. Set I CX)~ I ~ ~X : ~I~ E N,I.
Denote by ~CI) the spectrum of I (see [I]). Let F be a clos-
ed subset of T , F -~QI)N T . we say that a function
k: F -~ N u i o o} is I-admissible if k(~)= co for all
E~(1) ~ T and for all non-isolated points ~E F . Denote by
394
I*(N., F, k) the set of aii functions # ~ I*(N.) ha~ing meromorphic continuation # to ~\F such that ~ is a pole of
~of order at most k(~) for all ~ F with k(~) ~ co . It is
easy to see that I*<N,,F, k) is an invsriant subspace of 5*: N~--~N~
Let E be an invariant subspace of S*: N~- , E=/=N, •
H ~ S* Then E n is an invmriant subspace of : H and En H~HT Hence E 0 H =l*CH )for some inner function I =I E (see [6]). Let
us construct a closed subset F= F E of T and an l-amdissible
function k = k E as follows
F =~{~T:C4-~z)L E}, k(~)~p{m~N:(~-~zf~E]
oO~OTm~E 2. E = I * (N,, F, k). I n c a s e F =~ t he c o n j e c t u r e is srme, s e e Oorollar~J 5.2.3 i n [5].
Results of the following section imply the inclusion E cl CN~).
3. CYCLIC VECTORS OF ~ . Let ~+ denote the Riesz projection.
The following proposition can be obtained from Remark I in [6].
PROPOSITION. Let <X~ Y) be a Smirnov dual Pair having proper-
t , ies 1 ° and 2 ° (see r 6 ] ) . Suppose t h a t X ~ m ~ Y~
T,e._~t E be an in~riant subspaoe of ~*: X-~X . Then if
"(Hh E/l H ~ I for some inner function T , then E~I (X) •
The p r o p o s i t i o n a i i o w s t o g e n e r a l i z e Theorem 5 . 2 . 4 i n [5] . Le t
be a function holomorphlc in D, t i~/ >/ ~. everywhere in D a~d
the PRO~gSITION for the Smirnov dual pair (~H ~, ,p+~) d~ I # ~4~ , we can prove the following
THEOREM. Let X be a Hausdorff topological vector space, X:-~,
that X has the following oroDerty:
~ N. 1 using P+ C.Q- L t )), (~
S*×c X • Suppose
Let g be an invariant subspace of S*: X--~X , E aH ~I (H)
for some inner f~otion I . ~he~ E=I~(X).
CORO~Y I. l.~f E is an invariant subs~aoe of S :N, --~ N,
E ~ N, , then E c I*(N.) , where I=I E
395
COROLLARY 2. The jclosure of the linear span of the famil[
[S does not coincide with N, a ~seu~ocant, i n ~ t , i o n (see [1] f o r the ee f~n i t i on ) .
4. DISTRIBUTION 0F VALUES 0F FUNCTIONS IN (N,) De~lote by ~W:(0,~)--~ the decreasing rearrangement of l~I ,
where ~ is a measurable function on ~ . In [4] (see also [5]) it
is proved that if ~ E ~(N,) and ~A~ ~*($)' ~ ~ 0 , then #=--0 •
an, d O ' t = O , i;-'O
then ~ ------ 0 •
Results of [4],[8] imply
t-~O t-~0
t-~-O
RE~ERENCES
I. H ~ E o a ~ c E ~ ~ H.K. ~e~m~H o6 onepaTope c~B~ea. M., Hsy~a,
I980.
2. A x e E c a H ~ p O B A.B. AnnpoEcm~ pan~oHax~m ~a~-
~ aHs~or Teope~ M.PHcca o conp~eEm~x ~yszT~x ~j~ npoc~a~cTB
I p c p~(0,1) . -~mTeM.c60pH.,I978, I07, ~ I, 3-I9.
3. A a e E c a H ~ p 0 B A.B. ~ap~a~T~e no~npocTpa~cTBa onepa-
Topa o6paTHoro c~Era B npocTpaHc~me HP(p¢(0, 0) . - 8an~c- E~ Ha~.ce~L~.~0~, I979, 92, 7-29.
4. A a e E c a H ~ p o B A.B. 06 ~-m~Terp~pyeMocmE rpa~r~x 3Ha-
~e~ £apMo~ecEzx ~y~. - ~TeM.sa~eTEE, I98I, 30, ~ 1,59-72.
5. A 1 e k s a n d r o v A.B. Essays on non locally convex Hardy
classes. -Lect.Notes Math., 1981, 864, 1-89.
6. A a e E c a H ~ p O B A.B. 2L~BapEa~T~e no%upocTpa~cTBa onepa-
TOpOB c2mvao A~C~OMaTzxec~ no~xoA. - 3an~cEH Hay~.ce~.~0~,
I98I, II3, 7-26.
7. c o i f m a n R.R. A real variable characterization of H P . -
Studia Math., 1974, 51, N 3, 269-274.
8. H r u s c e v S.V., V i n o g r a d o v S.A. Free interpolati-
on in the space of unifoz~ly convergent Taylor series. - Lect.Notes
Math. 1981, v.864, 171-213.
A. B. ALEKSANDROV CCCP, I98904, ~eHEH~p~, He~po~Bopen
(A.B.A~EECAH~POB) ~6X~oTe~a~ ~.2. ~aTeMaTEo-Mexa~m~ecEE~
~axy~TeT ~I~
396
7.12. DIVISIBILITY PROBLEMS IN A(~) AND H~(~).
We offer tWO problems on divisibility, one in the disc algebra A(D), the other in H~(D). The first is this. ~or which X C D t
~ , do we h a v e
Ax=N A {~}
where A× i s t h e r i n k o f f r ~ c t i o n s
{~/~ : I , ~ A ( ~ ) , ~vanishes nowhere in X}
and ~ is the local ,tin6 of,,,,,, fractions
We might point out that (1) holds if ~((IT is closed. (PROOF.
Put Y =X~T , ~= ~ , and let ~ ~ the right side of (1) . Then
~.~N A~.
I t i s p roved i n [1] t h a t (1) ho ld s i f X i s c l o s e d i n ~ ; hence
T-- F/G where F,G ~ A~ ) ~nd G ~ n i ~ e , ~owhere i n ¥ . ~,et ~ A ; t h e n ~ = ~ / ~ where ~,~ ~ A ( ~ ) , ~(~)~0 . We have
F~ ={~ ; t h ~ o(~,~}.<o(F,~) where o(~,~} i~ the order of vanishing of ~ at ~ . This proves that the Blaschke product
made up of the zeros of ~ in A (counting multiplicities) dlvi-
Let ~ be an integral domain, F its field of fractions, and
the space of all prime ideals p in ~ , ~ . Let ~ be an
ideal in ~ ; then ~ is said %o be an ideal of denominators if
where ~ 6 F . suppose ~ i s such t h a t e v e r y i d e a l o f denomina to rs
i s p r i n c i p a l ; t hen i f ~ c ~ , X~ ~ , we have
397
where ~X is the ring of fractions
and ~p the local r ing of f ract ions
This i s easy to prove. Now i n H~(~) every i dea l o f d e n o m i ~ t o ~ is principal. (Easy to prove, but not obvious.) Thus if X is any non .p ry set in the m~ximal ideal space of H ' (D) , then
where
oo
H x = H: ~;eX "
~ = { ~ / ~ ; f '~ 6 HOO, ~ vanishes nowhere in ~} co c o /%
This suggests that (1) might hold fo r every nonempty subset of ~ , although unl ike Hco (~) not every ideal of denomizu~tors in A(~) is principal. E.g. if ~ 6~ and
then P~ is an ideal of denominators, but it is not principal.
Here is our secon~ PROBId~. Let Q be a prime ideal in ~oo(~),
Q#0, H~(~)) o SUppose Q is finit~l,y ~enerated, Do,,we then have
= -- .ol
where ~6~ and ~>/0 ? (Yes i f ~ is maximal [2].) In A(~)
we have the following. ~e~ P and Q bsp~meid~lsin A C91 ~h P~Q and
PC Q C ~ where ~ eT ( P~ = the right side of (2)). Then the A(D) -moaule O/P is not finitely generated, i.e.
Q ~ P a finitely generated ideal in A (D).
398
This is a corollary to Nakayama's lemma. It suggests that the
answer to Problem 2 is yes, but then the proof would have to be dif-
ferent,
RE~RENCES
I. ~ o r e I i i F. A note on ideals in the disc algebra. - Proc.
Amer.Math.Soc. 1982, 84, 389-392.
2. T o m a s s i n i G. A remark on the Banach algebra ~ H~[~ N)'- ~
-Boll.Un.Mat.Ital. 1969, 2, 202-204.
FRANK FOREI~I Department of Mathematics
University of Wisconsin-Madison
Madison, Wisconsin 53706
USA
399
7.13. A REFINEMENT O~ THE CORONA THEOREM
The usual methods for proving corona type theorems [I ,2] use
existence of solutions with bounded radial limits for equations
~=~ when I~IiI~I~,~ U , or something similar to I~II~, U , is
a Carleson measure. Our problem is a variant of the corona theorem
for which apparently no Carleson measure is in sight.
~ o ~ . ~,pose { , I ~ , t~ H ~ with I.¢c")l~lI~Cz'~i+lT~cz)l ~or ~ Z ~ t3 ~ ~ho~e ~o ~,,~ H ~ -- {~
The answer is known to be yes if the exponent 2 is replaced by 3
(or 2 +~ if I has no zeroes) and no if 2 is replaced by I (or 2
- ~ ). See [2]. The answer is also yes if ~,~ are only required
to be in H i . This is by a ~ argument using the estimate
~TJ <oo
,.--~.1 ~,l(z)I,
which comes from the Ahlfors-Shimizu form of the first fumdamental
(% = characteristic theorem. If ,l
function of were always a Carleson
measure our problem could be answered affirmatively by arguments in
[2]; but in general H is not Carleson.
REFERENCES
I. C a r 1 e s o n L. The corona theorem. - Proc. 15th Scandinavian
Congress, Springer-Verlag, Lect.Notes in Math, 1970, 118, 121-132
2. Garnet t
Press, 1981.
T. WOLFF
J. Bounded Analytic Functions. New York, Academic
Department of Mathematics 253-37
California Institute of Technology
Pasadena, CA 91125, USA
400
EDITORS' NOTE. The results similar to the ones mentioned in the
Problem were obtained by Y.A.Tolokonnikov [3] in the following slight-
ly ~ifferent setting: for which increasing functions oCdoes the in-
equality
; H ~ I~l_<o~((. ~ 141") ~/') o~,.~,~
imply ~= E ~KSK See also [.4] .~)I
3 . T o j l o K o H H g K 0 B
~ea~u a ~ e ~ H,~sa~a~a JIOMH, I98I, II3, I78-I98.
Ke a ~ e a , ~ a.~,eci~ H "~ .
I~-20I.
with ~K~H ~, ~p ~ I$. I~ ~ D K~I
B.A. 0~eH~ B Teo~Me ~ e c o ~ m o ~ o ~ . e . Ce~e~a~Bg-Hs~a. - 3a~c~ say~s.OeMgH.
B.A. HHTepno~a[~o~e nioogsBe.~e~ B~sm- - 8sn~o~ Hay~.oeMHH.~G~H, I983, I26,
401
7.14. old
INVARIANT SUBSPACES OP THE SHIFT OPERATOR IN SOME
SPACES OF ANALYTIC FUNCTIONS
I. Let X be a Banach algebra of functions analytic in the
unitthat diSkcOx :- ~) (withand polntwiSelet ~% ~-~ ~additi°n{ MS and: X ~multlplicatl°n)'cA "~) ~ < oo. Assume
Let ~(n)C~) = {~ : ~ : : T ; ~(J)(~)=O , O ~ j ~ J .
DEFINITION. A closed subset E of the unit circle T is called
D. set for the a~geb= X if for any function ~ ~X with E(~(;)~- = E there is a sequence { ̂ ;K "JK~4 of functi-
ons in X , satisfying the following conditions:
( i ) I ~ ( ~ ) I~ C ~ [ ~ s ~ ( ~ , E ) ] ~+~, ~ O ,"
k
I t follows from [1 ] that every Beurling-Oarleson set ~) is a D ~ - -set for a number of standard algebras of analytic functions. In par-
t iotLlar, i t is true for the algebra HI+, ={~: ~("+°~H~, 4 < P < oo.
QUESTION I. Is eve~ Beurlin$-Oarleson set a D~ -set for
the algebr a H~+4 , I~ 0 ?
REMARK. I f E = 0 E , where mE K K-I ~ =0 , K= I,..,~ , and the lengths of complementary intervals of each E K tend to zero ex?onentially fast then (as proved in [2] ) E is a D~ -set for
H~+ and the sequence { ~ ) = . can be chosen not depen- .~ I D n , ,4 K K~I
dlng on ~ " _ ; ~ + 4 " 2. Let AP==~(D ) be the Bergman^space in the unit disc D
(i.e. the space of all functions in ~ <~) analytic in D ).
QUESTION 2. Is there a closed subspace ~ , ~A P , G~A P
invariant ~ d e r the shift o~e~tor s , ~(9<~ ~ ~ ~ (~) ,and
not finitel E ~enerated?
QUESTION 3. Let ~ , ~cA P be an invariant subsP~ce of ~ •
Assume that for eve~ ~, ~ D there is a function ~ , ~G
H) See the definition in 9.3~- Ed
402
C~) =~ 0 . Is it true that G is generated by one function ~ ,
i,e. G=V(~: I'1,>~0) ?
3 arise i~ the following way. Let REMARK. Questions 2 ~d and H Pc~ be the Hardy space in
b~the unit polydisk in
(see[3]). Denote by D~ the "diagonal operator" in HPC~) ,
i.e. D~(~) = ~(~, ~) , ~ D , and denote by ~ the
operator
5~C~)C~'~,t:,)= ~(~ ,t4) , ~ D , J~HeCD~), i,--~,~.
zt is kno~ that D~QM = A 0 < p < ~ (see [4] [5]} and there exist ~,~ -invariant and non-finitely generated sub-
spaces ([3], p.67). Moreover ~here is an invariant subspace generat-
ed by two functions in H~CD) ~thout common zeros in D s and
this subspace cannot be generated by any of its elements (see[6]).
REB~LENCES
I. m a M o ~ H $.A. CTpyETypa 3aMEHyTHX ~ea~oB B HeEoTopMx a~-
re6pax ~rHE~H~, ~T~qecENx B Kpyre ~ ras~Enx B~JIOTB ~O ero
rpa~m~. -~oE~.AH ApM.CCP, 1975, 60, ~ 3, 138-186.
2. ~ a M 0 H H ~.A. HOCTpoeH~e O~HO~ cne~s~HoH H0c~e~oBaTed~-
HOCTn ~ cTpyETypa saMEHyTHX ~ea~oB B HeEoTop~x a~re6psx aHa-
~T~eCE~X ~yH~L~. -- 'AsB.AH ApM.CCP, ~TeMaT~Ea, 1972, 2)[, ~ 6,
440--470.
3. R u d i n W. l~mction Theory in Polydiscs. Benjamin, New York,
1969.
4. H o r o w i t z C., ~ 0 b e r 1 i n D. Restrictions of func-
tions to diagonal of D . - Indiana Univ.Math.J. 1975, 24, N 7,
767-772. 5. m a M 0 ~ H ~.A. TeopeMa BaOXeH~ B npocTpSacTBaX n-rapMo-
m~qecz~x ~JHE~ ~ HeEoTop~e np~o~eHz~. -~oEa.AH ApM.CCP,1976,
X~, ~ I, 10-14. 6. J a o e w i c z Ch.A1. A nonprinmipal invariant subspace of
the Hardy space on the torus. - Proc.Amer.Math.Soc. 1972, 31,
127-129.
t~.A.SHAMOY~ CCCP, 3?5200, h'peBa~, 19, (~.A.NAM0~H) y~. Kape~a~yTaH, 240
~CT~TyT MaTeMaT~ AH ApM.CCP
403
7-15. ~LASCHKE PRODUCTS AND IDEALS IN Gf. old
Let A be the space of functions analytic in the open unit disc
and cont inuous i n ~ ; and let O ~ ' = { ~ A : ~ ( ~ ' ~ A , ~ = 0 , 1 , -.. }. Although the sets of uniqueness for C 7 have been described [I],
[2], [3], [4], and the closed ideal structure of C~ is known [5], there
are still some open questions concerning the relationship of Blaschke @@
products with closed ideals in C A . I pose two problems. Let I , CO
I c OA , denote a closed ideal and let ~ denote a Blaschke product
which divides some non-zero ~A function.
(I) For which ~ is it true that
(2) If ~ is the ~.c.d. (~Teatest common divisor) of the Bla~ch-
ke factors of the non-zero functions in I , when is (~)I £~
={~E ~:B ~eI } a closed ideal in C~ ?
Note that the corresponding problems for singular inner functions
are easier and are solved in section 4 of [5].
To discuss the problems for Blaschke products we need some no-
tation. Let
~ I
and let
Z'(1)= n Zhl) , ~,~-0
CO
Z(I)= { Z~(I) } ~=o
cO
I f ~ ( ~ ( I ) ) deno te s t h e c l o s e d i d e a l o f a l l ~ , ~ C A , w i t h
~(~)(~) ~ 0 for ZE~(1) ~ ~= 0,1, ..., then the closed ideal struc-
ture theorem says I= ~I (~(I)) , where ~ is the g.c.d, of the
singular inner factors of the non-zero functions in I .
DEFINITION. A sequence{~ c ~ has f i n i t e d e g -
r e • o f c o n t a c t at E , ~c ~ , if there exist ~ ,
404
k > o , and ~ , ~ > o , such t ~ t ~ - l ~ l ~ f ( ~ / I ~ l , E ) k
for all ~ . (Here ] denotes the Euclidean metric. )
The following unpublished theorem of B.A.Taylor and the author
provides solutions to problems (I) and (2) in a special case.
m m o ~ , , (a) As,s~eZ ±)=Z ( I ) . z~ o r d e r tm~t Bi it is
necessar 2 and sufficient that the z ergs of B have finite degree of
c o n t a c t a t Z~(D. z__~ Bl=C~ ,then multiplication ,b,y B ,iS oontinu-
ous,,on i, 5i is closed,and the , inv~.rse operation is...continuous.
(b) Ass~e ~l)n 8~-~ ~I). ~et E ~e t.ae., g,c.d,, of the
Blaschke factors of the non-zero ' ~ctions in I • In order that(~/E)l
b.e closed i,t,, ,,i,s, ~ecessary and suffioient that t.he. zeros of ~ have
~inite degr¢,~,,of contact at ~ql) •
THE PROOF of sufficiency in (a) is primarily a computation of
the growth of the derivatives of ~ near ~(~) . The computation
has also been done by James Wells [61 . The proof of necessity in (a)
requires the construction of outer functions. (One can assume without
loss of generality that the g.c.d, of the singular inner factors of
the non-zero functions in I is ~ . ) In section 3 of [5] it is de-
monstrated that there is an outer function ~ , ~e OA , vanishing
to infinite order precisely on ~oo(l) , and such that ~I ~(~@)[~
~--00(-~?(~)), where ~ is continuous, const.~(~%)~ ~(~) <~
~const.~(6 ~%) , and 0O is a positive increasing infinitely differen-
tiable function on ~ which can be chosen so that ~ ~-~(~)=+oo ~-++00
as slowly as desired. An appeal to the closed ideal structure theorem
places P("~l~ for a l l I~ . NOW, s i n c e Bg and B~ 'f are assure-
reed to belong to C~ , B~Z=~BP~iBP~ C~ Thus
,~lB~4~ I= - ~1 ~M~I-,- 0(~)= ~c-~.~c~) + Oc'f)
for all choices of ~o . Hence, for some ~>0 ,
or[~{O)l: O(~(6~t ~I) ~k ) • A computation shows that this implies
that the zeros of B have finite degree of contact at ~°°(I) .
405
The last assertion follows from the closed graph theorem.
To prove sufficiency in(b), let
O0
J={5 C, A : }
Y
let
Then ~(~) = ~(J) = ~(_I) and (~/~)I C J o Applying (a) to
, one concludes (4/B) I is closed. To prove necessity in (b),
( Again, one can ignore singular inner factors. ) Then ~ ~)~-
=~(K)=~(1)n SD ~nd by the closed ideal structure theorem o@
(~/B)I~ K. Thus BKcl~ O A ; and so, applying (a) to ~ ,the
zeros of B have finite degree of contact at ~K)=~I) n SD. •
Let us consider problem (I) in the more general case where
~ I ) caD but ~°C)=~ ~ ( I ) in t h e l i g h t o f t h e above r e s u l t s .
From t h e c o m p u t a t i o n r e f e r r e d t o i n t h e p r o o f o f s u f f i c i e n c y i n
THEOREM (a), it is clear that if the zeros of B have finite degree
of contact at ~(I) , thenBlcC? ; however, it is not difficult
t o construct e wamples t o show that this condition is not necessary.
On the other hand, THEOREM (a) along with the closed ideal structure
theorem implies that a necessary condition for BlC O A is that the
zeros of~ have finite degree of contact with ~o(i) ; however,
this condition is clearly not sufficient. It appears that the sets
~(I) , O< ~<oo , play a role in determining whether or not
Blc C~ Similar remarks apply to problem (2). That is, if the zeros of
B have finite degree of contact at ~'(I) , %hen(~/S) I is
closed; and, if (I/~) I is closed, then the zeros of ~ have finite
degree of contact at ~(I) N ~ ~ .
In regard to problem (2), it is not always the case that ~)I
is closed. In fact, it is possible to construct a closed ideal I
where the zeros of B , the g.c.d, of the Blaschke factors of the
non-zero functions in I
~°(1) naD and, hence,
We note that if B cO
zero C A function, then
, do not have finite degree of contact at
(4/2) I is not closed.
is a Blaschke product which divides a non-
there is a Carleson set E , Ec a~ ,
406
such that the zeros of B have finite degree of contact at E . In
f a c ~ one can t ~ e E = ~ | ~ I I ~ I : B ( z ) = O 1 ; s e e ~ e o r e m 1 . 2 o f [ 3 ] .
REFERENCES
I. K o p e H 6 x m M B.M. 0 ~yH~n~x ro~oMop~x B ~pyre ~ rxa~-
z~x B~OT~ ~O ere rpa~L~.-~o~.AH CCCP,19VI,200,~ I, 24-27.
2. c a u g h r a n J.G. Zeros of analytic function with infini-
tely differentiable boundary values. - Proc.Amer.~ath. Soc. 1970,
24, 700-704.
3. N e 1 s o n D. A characterlzation of zero sets for G~ . -
Nich.Math.J. 1971, 18, 141-147.
4. T a y 1 o r B.A., W i 1 1 i a m s D.L. Zeros of Lipschitz
functions analytic in the unit disc. - Mich, Nath.J. 1971, 18,
1~9-139.
5. T a y I o r B.A., W i 1 1 i a m s D.L. Ideals in rings of
analytic functions with smooth Boundary values. - Can.J, Nath.
1970, 22, 1266-1283.
6. W e 1 1 s J. On the zeros of functions with derivatives in H I
and H °° . -Can.J.Nath. 1970, 22, 342-347.
DAVID L.WILLIAMS Department of Mathematics
Syracuse University,
Syracuse, New York, 132 I0,
USA
407
7.16. CLOSED IDEALS IN THE ANALYTIC GEVREY CLASS
Let ~ denote the open unit disc in C . The (analytic) Gev-
rey class of order ~ is the class of holomorphic functions
in D such that nA
The class ~@ is endowed with a natural topology under which it be-
comes a topological algebra. So it is natural to ask for the struc-
ture of the closed ideals of ~& . For ~ ~ the class ~ is
quasianalytic and so this question is trivial in this case.
For 0 <~< ~ , this characterization should be along the lines
of previous works on this topic ([2] and [3]) concerning the classes K
, ~ 0,~,...,oo. Namely, for a closed ideal I one considers for
k= 0,4~,.,
K Z CI)= {~ ~: ~(~)=0 for all # inI, j=O,..,,k]
and
= greatest common divisor of the inner parts
of functions in I
The precise QUESTION is then stated as follows:
ideal I cha~cterized by the sequence zk(I) is every closed
a~d 8I in the sense
I-{~G -SI divides ~ and Z(~)c::Zk[I)} .~ VV~
Hruscev's paper [1] is basic in this context for it contains
the characterization of the sets of uniqueness for G~ , i.e.
the sets E which can appear as kz°°CI ) . The imitation of the
proof of [2] or [3] for the A -case just gives the result in spe-
cial cases which include the restriction ~ <4/~ °
RE FEREN CE S
1. H r u s ~ e v S.V. Sets of uniqueness for the Gevrey classes.
-Ark.for Mat.~ 1977, 15, 235-304.
408
2. h O p e H 6 x ~ M B.I~. 3amm~yTme ~eax~ Eo~ha A . - ~ .
aHax. ~ ero np~., 1972, 6, ~ 3, 38-52.
3. T a y I o r B.A. and W i I I i a m s D.L. Ideals in rings
of analytic functions with smooth boundary values, Canad.J.Math.~
1970, 22, 1266-1283.
J. BRUNA Universitat aut~noma
de Barcelona
Secci~ matematiques
Bellaterra (Barcelona)
Espa~a
409
7.17. old
COMPLETENESS OP TRANSLATES OF A GIVEN FUNCTION IN
A WEIGHTED SPACE
w i ~ h ~ I~ ~ (~) Here the weight
be the Banach space of measurable functions
, t h e no= defined
is a measurable function satisfying
(~)
The multiplication being defined by
the space becomes a Banach algebra w i thout u n i t ; tha t fo l l ows f rom( l ) ,
Condi t ion (1) ensures the ex is tence of f i n i t e l i m i t s
~±eo ~ = o~± and so for ~ in L~ the Fourier trans-
--~ turns out to be continuous in the strip ~= { ~ : ~ _ ~ <~ ~} and analytic in ~ ~ . The mammal
ideal space for L~ is homeomorphic to ~ ( [I] ), and any maximal
ideal ~ (~e) ( ~ ,S ['I ) has the form:
Let us consider together with ~4
~ . ~ = { ~t,%c ~): ~c~-~ o,
: ~ c ~ . ~ ~ - o }
its closed subalgebra
is homeomorphic to the half- The maximal ideal space of h~(~+)
plane I~ <~ ~+ , and the Tourier transform of the function [ ,
~ hl (~+) turns out to be continuous in this half-plane and analytic in its interior.
THE PROBLEM we treat here is the following: let ~ be a famil~
of functions in ~(~) ( ~ ma~ consist of a sing~le element) and
let 11~ be the space sp~nued b~ all translates of functions from~.
What are the conditions on ~ for T~ to coincide with ~(R),
i,e, when ever~y" func,ti,o n in [,~(~) can be approximated in ~(~)
410
by linear combinations of translates of functions in ~ ? 4
The problem for T~---I (tL~^(I~)~Lt(~,)) was stated by N.Wiener
[2], he proved that I~ = ~ (~) if and only if the Fourier trans-
forms of ~ have no common zero on ~ . Since I~ is the smallest
closed ideal of [/ containing ~/~ ~ Wiener's theorem means that a
closed ideal of h~(~) is contained in no maximal ideal if and only
if i% is equal to 11(~) .
A.Beurling [3] discovered the validity of Wiener's theorem for
the space [i (~) if the weight @ satisfies condition (I) and
I ~ t e r i t t ~ . e d out ( [-1] ) tha t simple'rJ3a~B.ch a lgebra" argmnents prove both Wiener's and Beurlin~s theorem and work in a more general
case of any regular Banaoh algebra. The regularity of ~ is en- sured by (~).
The Wiener-type theorem for non-regular Banach algebras were
obtained by B.Ny~an [4~, who ~roved the following theorem for the
case (~($)~ 8 ~lSf , ~>O:I ----~ (~)iff ~ourier transforms of ~ have
no common zero in the strip I I~l~ ~ and
E-J • ,~ ~:e~ ~-.--~o ~'P ~ (3)
we can see that in the algebra 4L~(]~) Scru t in i z ing condi t ions (S) there are closed proper ideals contained in no maximal ideal. Such
ideals will be called p r i m e i d e a I s c o r r e s p o n d-
ing t o infinity p o int s of the strip
Independently Nyman's result was rediscovered by B.Korenblum,
who described completely the prime ideals corresponding %o infinity
points of the strip I I~4 ~I~ ~ . The question concerning Wiener-
type theorems in algebras ~i (~) , where the weight ~ satisfies (I) and
R oo , (4)
411
~+-~-~_ ~ 0 , is still open.
The methods used earlier don't work in this case. The author
does not know a necessary and sufficient condition for the validity
of approximation theorem even under very restrictive conditions of
regularity of the weight ~ (for example ~(~)~ C~p (4+I$Ii) )" For a weight ~ , satisfying (I), (4). one can find chains of
prime ideals corresponding to infinity points (see, for example, [6]).
The reason for the existence of prime ideals of this sort is that
by (4) %here are the functions in L~(~) with the greatest possib-
le rate of decrease of Pourier transform (see [7], [8] ). It remains
~own whether all prime ideals are of this sort.
There is a similar question for the algebra I.~(~+) . We
have the following theorem [9] : Let ~ satisfy (I), (2), and let
I~ be the_ , _cl°sure of the linear span of all right translates of~.
Then I~%~'~(~) if and only if the following two conditions
are fulfilled:
I ) there is no interval I adjacent to the origin and such that
all functions in ~ vanish a.e. on I ,
2) Fourier transforms of functions in ~ have no common zero
i n l ~ 0 .
It is worthwhile to note, that this case is simpler than the
case of h'~(~) because there are no prime idemls of the
above type. (The case when ~= ~$ doesn't differ from the case
when ~= ~ ). We conjecture that the previous assertion remains
true also for the case when ~ satisfies condition (4). In particu-
lar the following conjecture can be stated: cO
If the Fourier transform ~ doesn't va sh in 0
and if ~ ~0~I~({~)I ~_~_~ ~ = 0 , then the ~ equali%,7
j ~(~)@(~+~)~ = 0 VT> 0 implies ~(~)= 0 .... once. a,e,
There are some reasons %o believe the conjecture is plausible. We
have no possibility %o describe them in detail here, but we can note
that condition (2) is much stronger than the condition
412
I which is a well-known condition of "non-quasianalyticly" of 4~(~÷).
A more detailed motivation of the above problem and a list of relat-
ed problems of harmonic analysis can be found in DO].
In conclusion we should like to call attention to a question on
the density of right translates in ~(~) . Let ~ ~(R ) , let
be the closure in ~I(~) of linear span of all right translates
of ~ . It is easy to prove that I I ---- implies that
and that ~is nowhere zero. However these conditions are not suffi-
cient. There are some sufficient conditions but unfortunately they
are far from being necessary. I think that it deserves attention to
find necessary and sufficient conditions.
REFERENCES
l. re~B~aH~ 14.M., P a~ E o B ~.A., ~ I~ Jl o B r.E.
KO~yTaTHBHHe HOpMHpOBaHHHe EOXB~a, M., $.-M., 1960.
2. B E H e p H. HHTeI~a~ $ypBe H HeEoTopNe ero Hp~o~e~, M.,
~.-,M., 1963.
3. B e u r i i n g A. Sur les integrales de Fourier absolument con-
vergentes et leur application a une transformation fonctionelle.
Congres des N~th. Scand., Helsingfors, 1938.
4. N y m a n B. On the one-dimentional translations group and semi-
group in certain function spaces. Thesis, Uppsala, 1950. 5. K o p e H 6 a D M B.H. 06odmeHEe Tay6epoBo~ TeOpeM~ B~Hepa
rapMoHH~ecE~ aHaxHs OHcTpopacTym~x ~yHE~. -- Tpy~H MOCE.MaTeM.
o6-~a, I958, 7, I2I-I48.
6. V r e t b 1 a d A. Spectral analysis in weighted ~-spaces on
•- Ark.Math., 1973, 11, 109-138. 7. ~ • p 6 a m a H M.M. TeopeM~ e~EHCTBem~ocT~ ~ npeoSpasoBa-
H~ ~ypBe H ~ 6eCEOHe~HO ~E~epeHsEpyeM~x ~yHIg/~. -- HSB.~
ApM.CCP, cep.~.-M., I957, I0, ~ 6, 7-24.
8. r a 6 e H E 0 K.M. 0 HeEoTop~x F~accax npocTpSHCTB 6ecEoHe~Ho
~H~epeHnEpye~x #yH~. -~oEx.AH CCCP, I960, I32, ~ 6, I23I-
-I234.
9. F y p ap ~ ~ B.H., Z e B E H B.H. 0 HOXHOTe C~CTm~
C~B~eE B npocTpaHcTBe ~(0,~) C BecoM. -- 8an.MeX.-MaT.~-Ta
XY7 E XM0, I964, 30, cep.4, I78-I85.
413
I0. r y p a p H ~ B.H. rapMoH~ecE~ a~s B npoc~ps2cTBaX C
BecoM. -- Tpy~ MOOE.MaTeM.O6--Ba, 1976, 35, 21--76.
¥. P. GURAR!I
(B.H. IVPAPM~)
CCCP, 142432, EepHoroxoBEa,
MOCEOBOEa~ 06~aCTB, ~ES~
E-Ta X~ecE0~3~LEE AH CCCP
414
7-18. TWO PROBLEMS OF HARMONIC ANALYSIS IN ~rEIGHTED SPACES
We consider the space ~ (~] of measurable functions on with the norm II~II = ~ ~ I~(~)I/q(~) , The weight ~ is supposed to be measurable and to satisfy the conditions
A ssign to each function ~ g ~ e (~) the smallest ~*-clos- ed subspace of L~ (~) (denoted b; ~ ) invariant under all tran- slations and containing ~ . The set
is called the spectrum of ~ .
Denote
for {<0 ].
For each
owg:
~+ in ~ (~+) a spectrum
A +. I
~÷ is defined as fell-
Here ~ is the smallest ~J*-closed subspace of h~ (~÷] invariant under translations to the left and con-
taining ~ . A~ A~ closed subset of the The spectrum (resp. ~+ ) is a
real line (resp. of the lower halfplane). The spectrum ~÷ (or ~ ) is said to be "simple" if
the only functions in B~÷ ( resp. ~ ) that have one-point spect- rum, are exponentials times constants.
415
PROBLE~ 1. Describe the subsets ~ of ~ with the followir~
prop err.y: ever~ ~ function ~+ ~ (~÷) with a "simple" spectrum
admits an extension ~ to the whole of ~ so that
and
+
I f ~ [ ~ ) ~ the set Z of all integers is an example of
such set. Indeed, if 9+ ~ L~[~+) and ~%÷ c Z then the
theorem on spectral synthesis in ~[~,) proved in [1] implies
that ~, lies in the %E*-closure of the trigonometric polyno-
mials with frequencies in ~ . Thus ~+ admits a ~@ -periodic
extension ~ to the whole of ~ , and clearly (I) holds for this
. There also exist more refined examples.
~?nen treating the spectral synthesis in L~ [~Q the
following problem might be useful.
tk£L%(~+) for 0 g k ( ~ ( ~ is a positive integer or
the symbol oo ). Suppose also that
0
(2)
and
0
(3)
wh,e,re 0 is a constant. Describe wei~t,s ~ such that, (,,2,) and (~)
imply C= 0 .
~or the weight ~, ~(~)~I this implication has been proved in [I]. If ~(~) =I+~| , a proof has been proposed by E.L.Suris.
Some considerations concerning Problems I and 2 are implicit
in [2].
416
REI~ERENCE S
1. r y p a p ~ ~ B.II. CHeETpa~H~U~ c~Tes OI~paH~eHHNX ~yHEL~
Ha HOJ~yOOE. -- ~yHE~.aHa~. ~I ero np~., 1969, 3, B~.4,84-48.
2. r y p a p ~ ~ B.H. rapMOH~ecEH~ 8aa~s B npocTpaHCTBaX 0 Be-
COM. --Tpy~J~ MOCE.MaTeM.o6--Ba, I976, 85, 21--76.
V. P. GURARII
(B. II. rYPAP~0 CCCP, I42432, qep~orozoBza, MOCEOBCEa~ 06~., 0T~e~eH~e
~H-Ta X~M~ecEo~ ~S~E~ AH CCCP
417
7.19. A CLOSURE PROBLEM FOR FUNCTIONS ON ~+. old
A w e i g h t f u n c t i o n ~/ is here a positive,
bounded, decreasing function on ~+ , satisfying ~-I~ ~(0o)-~ -oo,
as ~ ~ co • ~t~ is the Banach space of functions ~ on ~+ with
~e ~(~+). Tor every ~, ~e R+U {0} the translation T~ , de-
finea by
[ is a contraction in ~'~r * Aur is the set of all ~ , ~e~u~ ,
which do not vanish almost everywhere near 0 , and ~ is the set
of cyclic
the translates ~@
elements in I~ , i.e. elements ~ such that
, ~ ~ 0 , span a dense subspace. Obviously
Some light is thrown on this problem by the corresponding prob-
lem on 7+U~0} . A w e i g h t s e q u e n c e is a posi-
tive decreasing sequence ~(~)~0 , satisfying ~-I~_~_ co
as ~ ~ co • ~r is the Banach space of sequences C~ (C~)~ o
w i t h Ot ,~= (C~t~I~)R~Oe¢4(Z.U{0}) stud t h e t ~ l a t i o ] ~ l Tfl, t, ,
are defined as above, giving contractions of ~t~ - A~/ is the set
of o, ceSur , with Co=~ 0 , ~r is the set of cyclic elements.
~urcA~r, and we ask whether ~t~A~. This time results are easier
to obtain. Let us say that ~ is of submultiplicative type if
~ + ~ C~T~ , ~,~e ~+ , for some constant C . In that
case, ~r is a unital Banach algebra under convolution, with~\~
as its only maximal ideal, and ~f=~z follows from elementary Ba-
nach algebra theory. It should be observed that the submulitplicati-
vity condition is an assumption on the regularity of I~ , not a res-
triction of its growth at co . If this condition is not fulfilled,
there are cases, when~-~-A~ [1], and other cases when~=~=A~[2~,
rq. I n an a n a l o g o u s way we say t h a t a w e i g h t f u n c t i o n ~ i s o f s u b -
=ultiplicative t e, if ,
for some constant C . Using the results of Nikolskii and Styf it
is easy to produce weight functions ~f, of non-submultiplicative
type, for which ~ ~r • But if we from now on restrict the atten-
tion to weight functions of submulitplicative type, we can in no
418
single case answer the question whether ~-----~w - It is tempting
to conjecture, in analogy to the discrete case, that the answer is affirmative for every ~/ . Now againwe have a convolution Banach
algebra, bu% the absence of a unil prevents us from carrying over
the arguments from the discrete case. A vague indication that the
answer perhaps is yes, at least if ~ tends to zero rapidly at in-
finity, is given by the circumstance that ~ur=A~ if the corres-
ponding problem is formulated in the limiting case when ~ is non-
negative and vanishing for large X . (This follows from Titch-
marsh's theorem).
It is a direct consequence of Hahn-Banach's theorem that~
if and only if the convolution equation
CO
o
has the zero functions as only solution with ~ / ~ o o ( ~ , ) . Thus
3~= A~ if and only if there exists a ~e ~ such that the equa-
tion has a non-zero solution. ~aybe function theory, in particular
the theory of special functions, can provide an example showing that
~w~__.A~.. for at least some ~ .
Here are some s u f f i c i e n t c o n d i t i o n s
for S~ B~
o
(This follows directly from the results in ~] or [4], and is valid
also for ~ of non-submultiplicative type.)
2. Suppowe--~ ~/ is convex and~-¢~1~ "---~--°o, a s x ---~-°° .
Let ~ and suppose that for some S t ~ S t~ (~+), with oo~-
pact ~uppor% and coinciding with S near O,
eo
0 o
for large ~ , ~ ~÷ , where C is a constant and ¢~-t denotes the
inverse of ~/ . Then S ~ . (In par~icular,~(x)~-x P , p>~,
yields the right hand member ~ ¢0cp I-C~ ~ } , for some C , where
419
~--~oo . Let ~eLw and suppose that ~ is of bounded variation
near zero with I~0)~=0 . Then I~B~ .
REPERENCES
1, S t y f B. Closed translation invariant subspaces in a Banach
space of sequences, summable with weights Uppsala University,
Dept. of Math., Report 1977:3
2. H H E o 2 b c K H ~ H.E. 06 MHBapEaHTH~X no~HpocTpaMCTBaX B3Be-
meHHMX 0~epaTopoB C~BEra. - MaTeM.cd., 1967, 74, ~ 2, 171-190.
3. N y m a n B, On the one-dimensional translation group and semi-
group in certain function spaces Uppsala, 1950
4, F y p a p H ~ B.~. CneKTpa~H~ CHHTe3 oPpsH~eHH~XSyHrd~ Ha
no~yocH. - ~yHE~.a~a~. H ePo npH2., 1969, 3, ~ 4, 34-48.
YNGVE DOMAR Uppsala Universitet
Matematiska Institutionen
Sysslomansgatan 8
75223 Uppsala, Sweden
COMMENTARY
The proofs of Propositions I-3 can be found in E5~ . In an impor-
taut paper E6] it is shown that A~=B~if~x~$ ~ is eventually con-
rem is proved under the same hypotheses for the spaces l,~, ~ p < ~ .
These results are derived from a general theorem on convolution equa-
tions in Lt-spaces which is a strong form of the famous Titch-
marsh theorem.
Concerning all these and many other problems on translation in-
variant subspaces and ideals in~t(~,Lt(~) see also ~7] , a very
informative book.
420
REFERENCES
5. D o m a r Y. Cyclic elements under translation in weighted
spaces on ~. - Ark.mat. 1981, 19, N 1, 137-144.
6. D o m a r Y. Extensions of the Titchmarsh convolution theorem
with applications in the theory of invariant subspaces - Proc
London Math.Soc.(3), 1983, 46, 288-300°
7. Radical Banach Algebras and Automatic Continuity, Proceedings,
Long Beach 1981, Ed. by J.M.Bachar, W.G.Bade, P.C.Curtic Jr.,
H.G.Dales, and M.P. Thomas.- Lect.Notes.in Math., 1983, 975,
421
7.20. TRANSLATES OP IrffNCTIONS OF TWO VARIABLES old
i~ D] ~ d [21 the rollo~ng theore~ is p~ved= if~(g+) and $(X)~-0 for m < 0 , then the system of functions{I{~-~): ~eR,}
is dense in ~'(~) if and only if the following conditions are ful-
filled:
I. The function O0
o
d o e s n ' t vanish in I ~ 0 .
2. There is no ~>0 such that ~(X)=O a.e. on ~0,~)
i n 2 ~ \ Pt . P lnd ne,cessa,r~ an d s u f f , i o i e n t c o n d i t i o n s f o r t h e sZs -
tern ~ (~-(×-~/~,-~,) : ~>~0, k~ 0 } to be dense in ~(P,) .
REI~ERENCES
I. N y m a n B. On the one-dimensional translation group and semi-
group in certain function spaces. Uppsala, 1950.
2. ryp ap ~ ~ B.H., Z e B ~ H B.H. 0 nOXHOTe C~CTe~ C~B~ez B
npocTpaHCTBe ~(O, oo) C BecoM. -- 3an.Xap~z.MaTeM.o-Ba, I960, 30,
cep.4.
B. Ya. LEVIN
(~.~.~H) CCCP, 310164, Xap~zOB
np.ZeH~Ha 47
• ~SEEO--TeXH~ecz~ ~HCT~TyT
HHSZ~X TeMnepaTyp AH YCCP
422
7.21. ALGEBRA AND IDEAL GENERATION IN CERTAIN RADICAL
BANACH ALGEBRAS
Let C[[ Z]] denote the algebra of formal power series over C . We say that a sequence of positive reals {~/(~)~ is a r a d i -
c a 1 a 1 g e b r a w e i g h t provided the following hold:
(1) W(O)='~ and O<W(~)~<'~ for a l l I't,(~Z+. (2 ) f o r a l l
If these conditions hold it is ~outine to check that
is both a subalgebra o f C[[~] ] and a r a d i c a l Banach algebra with
identity adjoined. The norm is defined in the natural way: I~I~--
~-~I~(~)IW(~) . The multiplication is given by the usual convolu-
t~oOn of formal power series. We shall generally refer to ~ (W (~))
as a radical Banach algebra and { W(~)} as simply a "weight".
Let in all the fonowing. Besides A itself, there
are obvious proper closed ideals in A :
o@
for ~= ~,~.-. ~ and, of course, the zero ideal. Such closed ideals
are referred to as s t a n d a r d i d e a 1 s. Any ether closed
ideals are denoted n o n- s t %n d a ~:~) i d e a 1 s . Note
that the unique maximal ideal in is = .
We first discuss the problem of polynomial generation. Let @@
~ = ~ ( ~ ) ~ be an element of A with ~( l )~O. One says
that ~$ g e n e r a t e s a n o n- s t a n d a r d c 1 o s -
e d s u b a 1 g e b r a if the smallest closed sub~lgebra con-
taining S~ is properly contained in M . Since this algebra is the
423
closed linear span of polynomials in ~ , we could equivalently
say
is properly contained in M (4).
The requirement that ~(~)~0 is necessary, otherwise (4) is vacuous.
If the weight is very well behaved there are positive results which
show that non-standard closed subalgebras are not present [3]. On the
other hand, it was shown [5, Theorem 3,11] that, for certain star-
shaped weights, non-standard~._#losedl subalgebras exist ( A weight
W is star-shaped if tW[~) t~ is non-increasing). Hence one
problem is the following.
PROBLEM I. Characterize the radigal al~ebra weights ~ such
that ~4 (W(~ 9 ~.,.....non_standa ~ clos.9..d su~l~ebras.
We next consider the problem of ideal generation. Whether I _ _ ~ ( W ( ~ has only standard closed ideals or not is the problem
whether each non-zero element ~ generates a standard closed ideal
or not. If we let T be the operator of right translation on A
we could equivalently say [I, Lemma 4.5]
(5) contains a power of Z~
for each non-zero X in A . If ~W is a concave function it is
well known [I, Theorem 4.1] **) that all closed ideals are standard.
More generally, it can be shown [4,Corollary 3.6] that if W is star-
. l 4 ) shaped and W( ) is 0 -~ for some ~>0 then all closed
ideals are standard. This is in contrast to the fact that star-shaped
weights can support non-standard closed subalgebras. Apparently ~ilov
first posed the problem whethe{r or not there exists any radical al-
gebra weight W such that ~ (W(~)) contains a non-standard ide:
al. The answer is affirmative [6, Theorem] for certain seml-multipli
., i, i i i . L
*) This is also a part of Lemma I in HXEox~c~ H.K.,H3BeCT~ AH
CCCP, cep~ ~aTe~., I968, 32, II23-I137. - Ed.
**) This is also a part of Theorem2in HxI~o~c~ H.K., BeCTHN~ cep~ NaTeH.Nex. ~ aCTpOH., 1988, ~ 7, 68-77. - Ed.
424
cative weights [5, Definition 2.1]. These are weights where W(M%~)
actually equals W(~)W(~) for many vslues of ~, ~ in Z+ •
Hence we propose the following
PROBLEM 2. Characteriz,e,,,,,,,,,,,,,the radical al~ebra weiKhts W su~ch
that - ~(W("~ has non-s,t,andard ideals.
Even substantial necessary conditions on the weight W for the
existence of a non-standard ideal would be welcome.
Finally we remark that one can consider related radical algebras
~t(Q÷, W) built upon Q+ rather than E+ (we again Tequire (I)-
(3) for ~,~ in ~). Define for ~ non-zero in ~I(Q+, W)
Also define
~4
We pose the final problem.
PROBLEM 3- Does there exist some ~ (Q+, W) containir~ am
element ~ ~(~)=0 , such,,, that the closed ' ,ideal ~enerated b[ 0~ ins
properly contained in M ?
Preliminary results on this problem can be found in [2].
RE~RENCES
I. G r a b i n e r S. Weighted shifts and Banach algebras of
power series. -American J.Math., 1975, 97, 16-42.
2. G r o n b a e k N. Weighted discrete convolution algebras.
"Radical Banach Algebras and Automatic Continuity", Proceedings,
Long Beach, 1981, Lect.Notes Math., N 975.
3. S ~ d e r b e r g D. Generators in radical weighted ^ ~ ,
Uppsala University Department of Nathematics Report 1981:9.
4. T h omt ~a s M.P. Approximation in the radical algebra ^~4(W~)
when ~W~ is star-shaped. "Radical Banach Algebras and Automa-
tic Continuity", Proceedings, Long Beach, Lect.Notes in Math.,
N 975.
425
5. T h o m a s M.P. A non-standard closed subalgebra of a radical
Banach algebra of power series. - J.London Math.Soc., to appear.
6. T h o m a s M.P. A non-standard closed ideal of a radical Ba-
nach algebra of power series, submitted to Bull.Amer.Math,So¢.
MARC THOMAS Mathematics Department
California State College
at Bakersfield
9001 Stockdale Hwy.
Bakersfield, CA 93309
USA
426
7.22. HARMONIC SYNTHESIS AND COMPOSITIONS old
Let~ ~ be the algebra of all absolutely convergent Fourier
series on the circle ~ :
We say ~ admits t h e h a r m o n i c s y n t h e s i s
(~ -h .~ ) i f t h e r e i s a sequence I~,}c~' such t h a t n~-~I{¢ , 0
and I~(O)C I~$ ~(0), ~=J,~, .... The algebra ~CI contains
functions not admitting h.s. though every sufficiently smooth func-
tion admits h.s.
QUESTION I. Let ~ admit h~s~ Is it ~ossible to choose functi-
ons ~ in the ~bove derlnit~on so that ~= ~o ~ , ~. bein~ so~e
Zunctlons on ~-1,1] ?
Denote by [~] the set of all functions ~ on [-1,4]such that
~°~6~' . This set is a BAn~ch algebra with the no~U~ {{[~ ] =
={{~@ ~, . It contains the identity function ~(X)~ X . NOW
we can reformulate our question,
QUESTION 2. Let ~ admit hls , Is it possible to approximate
~(X) ~ X in the algebra [~ ] b~ functions vanlshin~ ~a r the
po±nt, × = 0 ?
I f I f ] c C I [ - ~ , J ] ( t h i s embedding has to be con t inuous by the Banach theorem) , then the f u n c t i o n a l ~[1 ~ )' ~ ~ O) s e p a - r a t e s X from f u n c t i o n s i n [ ~3 v a n i s h i n g a t a v i c i n i t y of ze ro . So
the following question is a particular case of our problem.
QuEstioN 3. ~et ~ admit h,s. Is it ~ossible that [~]cC'[-tI] ?
so
we have a further specialization.
QUESTION 4. Is it possible to construct a function ~G~ f
admittin~ h~ s. with
427
Now (1983) very little is known about the structure of the ring
[~]. The theorems of Wlener-Le~ type (~]ah.Vl [2]) give some suf-
ficient conditions for the inclusion ~c [~] , but these conditi-
ons are much stronger than C1-smoothness of ~ . On the other hand,
let the function $--~(~) be even on ~,~] and strictly mono-
tone on [0,~S . Thus any even function on [-~,~] has a form~@#
and our Question 1 has the affirmative answer. Hence If] ~ 0!
and all known theorems of Wiener-Levy type are a priori too mough
for this ~ . Kahane [3] has constructed examples of functions
with [~]~ O~4,J ] . Thus, the ring [~] is quite mysterious.
A possible way to answer our questions is the following. If
C then the functional ~ : F ~ ~ O) is well-de-
fined on [~] and generates a functional I~) on the subalgeb-
r a [ [ # ] ] = ~ ~o~ : ~ [ # ] } c ~¢' I f there is an exten-
sion of this functional to ~ with < I~), ~ > = 0 for
~(0) c I~$ ~-' (0) , then I cannot admit h.s. It is in this way
that Malliavin's lemma on the absence of h.s. has been proved. Namely,
@O
then Malliavin's functional
gives the desired extension of ~t(~)
The author thanks professor Y.Domar for a helpful discussion
in 1978 in Leningrad.
REFERENCES
1. K a h a n e J.-P. S~ries de Pourier absolument convergentes,
Springer, Berlin , 1970.
2. ~ w H ~ E ~ H E.M. TeopeMH T~na B~epa-~eB~ ~ o~eEE~ ~ onepaTo-
pOB B~epa-Xon~o - MaTeMaTH~ec~e ~ccae~oBs2~, 1973, 8, ~ 3, 14-25.
428
3. K a h a n e J.-P. Une nouvelle reclproque du theoreme de
Wiener - L~vy. - C.R.Aead°Soi.Paris, 1967, 264, 104-106.
E.MoDYN'K!N CCCP, 197022, ~eR~rps~
yx. npo~.HonoBa, 5
~eHEH2pa~cE~ SxeE~poTexH~ecE~
~HCTETyT HM.B.H.Y~B~HOBa (~eH~a)
429
7.23. DEUX PROBT,~RS CONCERNANT LES S~RIES TRIGONO~TRIQUEH old
I. Soit ~ ~ $ une s~rie trigon~m~trique dont les coeffi-
cients tendent vers 0 et dont lee sommes partielles tendent vers 0
sur un ensemble ferm6 ~cT :
N - ~ ~=-~
~ ~---- 0
Soit ~ ~+(~) une mesure positive por%ee par Y ,
^ ~
telle que
°
A-t~on nec,essazrement
Une r~ponse positive (dont je doute) donnerait une nouvelle preuve
de l'existence d'ensembles ~(5) de Zygmund de mesure pleine.
, , ^ ~, ~,~'~ 2. solt S=LPcT) ~~ ~ ~(~ . Pout-on =~roo~er t
dans LP(~) par d~s pol,ynomes tri~onom~triques 9=~ ~(~)~
A A
t e l e que ~ ( 1 4 ¢ ) : ~(~)---> P (~ I~ ) : P(~) ?
La question a et~ posse par W.Rudin [ I ] pour ~ : 1 (la r~pon- f . # #
se est alors negatzve [ 2 ] ) . Pour p:~ , la reponse positive est evi- dents. Pour p~- oo , la question n'a d'int~r~t que si on suppose
continue (la r~ponse est negative). La question est ouverte pour~<~
BIBLIOGRAPHIE
I. R u d i n W. Fourier analysis on groups. N.Y., Interscien- oe, 1962.
430
2. K a h a n e J.-P. Idempotents and closed subalgebras of ~(~)
- In: l~nct, algebras, 198-207, ed.T.Birtel, Proc.Intern.Symp.
Tulane Univ., 1965, Chicago, Scott-Forestmann, 1966.
J.-P.~AWANE Universit~ de Paris-Sud,
Nathematique, B~timent 425, Centre
d'0rsay 91405, 0rsay Cedex, France
COMMENTAIRE
La reponse au second probl~me est negative (voir [3] pour
et 4,5] pour ~<p<O0 ).
BIBLIOGRAPHIE
3. R i d e r D. Closed subalgebras of C(T) . - Duke ~th.J.,
1969, 36, N I, 105-115.
4. 0 b e r I i n D.M. An approximation problem in ~P[ 0,~S •
< p<oo i - Studia Math., 1981, 70, N 3, 221-224. 5. B a c h • i s G.F., G i I b • r t J.E Banach algebras with
Rider subalgebras. Preprint, 1979.
CHAPTER 8
APPROXIMATION AND CAPACITIES
M o s t problems of our Collection may be viewed as approxima-
tion problems. That is why selection principles in this Chapter are
even more vague and conventional than in the others. Problems collec-
ted under the above title illustrate, nevertheless, some important
tendencies of modern Approximation Theory.
Some Problems below are closely related to the ideas of the pre-
ceding Chapter. This is, of course, not a mere coincidence, the app-
roxiMation heing really the core of spectral analysis-synthesis. An
attentive reader will not be deceived by the seemingly scattered con-
tents of items 8.1, 8~3, 8.4, 8.8, which can be given a unlfied in-
terpretation from the (broadly conceived) "spectral" point of view.
What really matters is, after all, not w h a t or by w h a t
m • a n s we approximate (by rational functions or by exponentials
with prescribed frequencies, by weighted polynoBials or by ~-mea-
sures within a spectral subspace), but the intrinsic sense, the aim,
and the motive impelling to the approximation, i.e. singling out ele-
mentalTharmonlcs (with respect to an action) and subsequent recove-
ring of the o b j e c t t h e y a r e g e n e r a t e d by,
The v a r i a n t o f s p e c t r a l s y n t h e s i s ment ioned i n Problem 8.1 i s
aimed at ~ -approximation by solutions of elliptic differential
equations (in particular, by analytic and harmonic functions). The
432
same can be said about Problems 8.8-8. I 0 Problem 8 . 9 deals also with
some estimates of the derivative of a conformal mapping. Such estima-
tes are useful in connection with "the weak invertibility" (see Chap-
ter 7 again) and especially with the "crescent effect" discovered
by M.V. Keldysh.
Problems 8.5-8.7 are interesting variants of the classical uni-
form approximation (in the spirit of ~ergelyan - Vitushkin - Arake-
Zyan).
Pad~ approximations, an intensively growing branch of rational
approximations, is presented in Problems 8.11 and 8.12 (this direc-
tion seems to be promising in connection with some operator-theore-
tic aspects.See Problem 4.9).
The best approximation ~ la Tchebysheff, the eternal theme of
Approximation Theory, emerges in Problem 8.13 (as in Problem 5. I
amid Hankel operators and ~-numbers).
Problem 8.14 concerns some ideas arising in Complex Analysis
under the influence of the Theory of Banach Algebras.
But all this explains only the first half of the title. As to
the second, it is a manifestation of close connections of many modern
approximation problems with potential theory. Items 8. I, 8.9,
8. I0~8.15 -8.18 make an extensive use of various kinds of capacities
though all of them have in mind (or are inspired by) some approxima-
tion theoretic problems.
"The capacitary ideology" appears here also in connection with
other themes, namely with the solvability of boundary value problems
for elliptic equations (see the "old" Problem 8.20, its Commentary
being a new problem article), with metric es~imstee of capacities
(8.15-8.19, 8.21) and with removable siD~ularities of analytic func-
tions (8.15-8.19).
Sobolev spaces play an essential role in many approximation
problems of this Chapter. In Problem 8.22 they are considered in
433
their own right.
Five problems (8.1,5-8.19) dealing with removable singularities
o f b o u n d • d a n a l y t i c f u n c t i o n s ( o r , wha t i s t h e s a m e , w i t h
analytic capacity) formed a separate chapter in the first edi-
tion. Here we reproduce the translation of some fragments from its
preface.
"Analysts became interested in sets of removable singularities
of bounded analytic functions in the eighties of the last century,
attracted by the very possibility to formulate problems in the new
set-theoretic lan@~age. This interest being still alive today (as
it is witnessed by this Chapter whose five sections have a non-void
and even a fairly large intersection), modified its spirit many times
during the past century. Now conneoted with the classification of
Riemann surfaces then with extremal problems of Function Theory it
was born again in early sixties after Vitushkin's works on rational
approximations ~..~.
The problem of relations between analytic capacity and length
was the theme of active debates during the Yerevan Conference on
Analytic Punctions (September 1960) when L.D.Ivanov pointed out in
his conference talk the role which irregular plane sets (in the Besi-
covioh sense) are likely to play in the theory of removable sungula-
rifles of bounded analytic functions. But an essential new progress
(namel~ the proof of the Denjoy conjecture*) became possible in the
last (1977) year only, after the remarkable achievement by A. Calde-
r6n, namely, after his proof of the ~ -continuity of the Cauchy
sinaTular integral operator on a smooth curve. The whole Chapter is
written under the influence of the Calder~n theorem. May be, thanks
to it the time is near when the geometric nature of singularities
of bounded analytic functions will be completely u n d e r s t o o d ~
The h i s t o r i c a l i n f o r m a t i o n c o n c e r n i n g a n a l y t i c c a p a c i t y i s g i v e n
See Problem 8.15.
434
in 8.15 and 8 .16 . We should like to add the article Uryson P.S. Sur
une fonction analytique partout continue. - Pund.N~th. 1922, 4, 144-
150 (the Russian translation in the book YpHCOH H.C. Tpy/~ go TONO~O--
~ ~ ~py~ od~aCT~ ~aTe~aT~m, T.I,, M.~.I~TT~, I95I, 3-I00).
Capacitary motives can be heard also in other Chapters. The clas-
sical logarithmic capacity appears (rather unexpectedly) in the item
1.10 devoted to the isomorphic classification of spaces of analytic
functions . The analytic capacity (the main subject of Problems 8.15-
8.19 influenced by the recent progress in singular integrals, see
Chapter 6) takes part in the purely operator-theoretic item 436. The
use of capacities in the Operator Theory is not at all a novelty or a
surprise. Spectral capacities describing the sets carrying non-trivi-
al spectral subspaces, the exquisite classification of the uniqueness
sets for various classes of trigonometrical series (the particular
case corresponding to the shift ~-~Z£ ), metric characteristics of
spectra in the classification of operators (transfinite diameters et
al.), all these are the everyday tools of Spectral Theory and the
corresponding connections are well illustrated, e.g., by Chapter 4.
435
8. I. SPECTRAL SYNTHESIS IN SOBOLEV SPACES old
Let X be a Banach space of functions (function classes) on
~ . We have in mind the Sobolev spaces W~ , ~.<2<oo ,~e~+
or the spaces obtained from Sobolev spaces by interpolation (Bessel
potential spaces hP~ , 4> 0 , and Besov Spaces ~'~ , ~ > 0 ).
Then the dual space X t is a space of distributions. We say that a
closed set ~ in ~ admits X-s p e c t r a I s y n t h • -
s i s if every T in X [ that has support in ~ can be approxi-
mated arbitrarily closely in X / by linear combinations of measures
and derivatives of order < ~ of measures with support in ~
PROBLEM. Do all closed sets admit X -spectral synthesis for
~]%e above s~ces?
The PROBLEM c a n a l s o be g i v e n a d u a l f o r m u l a t i o n . I f • i s a
m e a s u r e w i t h s u p p o r t i n ~ s u c h t h a t a p a r t i a l d e r i v a t i v e ~ k
belongs to X [ , then one can define S~ ~ for all ~ in X .
Then ~ admits X -spectral synthesis if every ~ such that
~k~ ~.~ ~ 0 for all such ~ and all such multiindices ~ can
be approximated arbitrarily closely in X by test functions that va-
nish on some neighborhood of ~ .
The PROBLEM is of course analogous t o the famous spectral syn-
thesis problem of Beurling, but in the case of ~ this terminology
was introduced by Fug!ede. He also observed that the so called fine
Dirlchle% problem in a domain ~ for an elliptic partial differenti-
al equation of order ~5 always ~s a unique solution if and only if
the complement of ~ admits W~-spectral synthesis° See [I ; IX,
5. I~.
In the case of ~ the PROBLEM appeared and was solved in the
work o f V°P.Havin [2] and T.Bagby [3] in c o = e c t i o n w i t h the prob lem o f a p p r o x i m a t i o n i n l, P by a n a l y t i c functions. For W~ t h e s o l u t i o n
a p p e a r s a l r e a d y i n t h e work o f B e u r l i n g a n d Deny ~4]. I n f a c t , i n
t h e s e s p a c e s a l l c l o s e d s e t s h a v e t h e s ~ e c t r a l s y n t h e s i s p r o p e r t y °
T h i s r e s u l t , w h i c h c a n be e x t e n d e d t o ~ , $ ~ ~ < ~ , d e p e n d s
mainly on the fact that these spaces are closed under truncations°
When 5 > ~ this is no longer true, and the PROBT.RM is more com-
plicated. Using potential theoretic methods the author [5] has given
sufficient conditions for sets %o admit spectral synthesis in ~(~)~
~6~, . These conditions are so weak that they are satisfied for
all closed sets if ~>W~IX(~/~, ~-4/~) , thus in particular if
436
e-~ and ~-~ or 3. There are also some still unpublished results
for ~ and B~ P showing for example that sets tha~ satisfy a cone
condition have the spectral synthesis property.
Otherwise, for general spaces the author is only aware of the
work of H.Triebel ~,where he proved, extending earlier results of
t, i o n s and Nagenes, that the b o u n d a r y of a ~ domain admits spect-
ral synthesis for and .
REI~ERENCES
I. S c h u I z e B,-W., W i I d e n h a i n G. Methoden der
Potamtialtheorle fur e~liptische Differautialgleichungen beliebi-
ger 0rdnm~g. Berlin, Akad~m~e-Verlag, 1977.
2. X a B ~ H B.H. A~O~Cm~Sn~ B cpe~HeM aHax~T~ecm~M~ ~ym~-
M~. - ~o~.AH CCCP, I968, Iva, I025-I028. 3. B a g b y T. Quasi topologies and ratioual approximation. - J.
l~amct.Anal.,1972, 10, 259-268.
4. B e u r 1 i n g A., D e n y J. Dirichlet spaces. - Proc.Nat.
Acad°Soi., 1959, 45, 208-215.
5. H e d b e r g L.I. Two approximation problems in function spa-
ces. - Ark.ma%.~1978, 16, 51-81.
6. T r i e b e 1 H. Boundary values for Sobolev-spaces with
weights. Density of ~ (~i) . - A~.Sc.Norm.Sup.Pisa,1973, 3, 27,
73-96.
LARS INGE HEDBERG Department of Mathematics
University of Stockholm
Box 6701
S-11385 Stockholm, Sweden
CO~S~ENTARY BY THE AUTHOR P
For the Sobolev spaces W S , ~ < P <oo, 5~Z+ , the problem
has been solved. In fact, all closed sets admit spectral symthesis
for these spaces. See L.I.Hedberg [7S, L.I.Hedberg and T.H.Wolff [8S,
and concerning the Dirichlet problem also T.Kolsrud E9S.
REFERENCES
7. H e d b e r g L.I. Spectral synthesis in Sobolev spaces, and
uniqueness of solutions of the Dirichlet problem. - ActaMath.,
1981, 147, 237-264.
437
8. H e d b e r g L.I., W o i f f T.H. Thin sets in nonlinear
potential theory. - Ann.Inst.Fourier (Grenoble), 1983, 33, N 4
(to appear).
9. K o 1 s r u d T. A uniqueness theorem for higher order elliptic
partial differential equations. -~ath.Scand., 1982, 51, 323-332.
EDITORS' NOTE. I) The works [7] and [8] are of importance not
only in connection with the Problem but in a much wider context rep-
resenting an essential breakthrough in the general nonlinear poten-
tial theory.
2) When ~=~ some details concerning the problem of synthesis
in W P(~) , W P(T) are contained in the following papers" J.-P. S . 5
Kahane, Semi~aire N.Bourbaki, 1966, Nov. ; Akutowicz E.G., C.R.Acad.
Sci., 1963, 256, N 25, 5268-5270; Ann.Scient.~cole Norm.Sup., 1965,
82, N 3, 297-325; Ill.J.~th., 1970, 14, N 2, 198-204; 0ca~ H,M.
Y~p.~aT.z., I974, 26, ~ 8, 669-670.
3) If XcC(~) then the spectral synthesis holds: every ideal
of X is divisorial, i.e. is the intersection of priory ideals. Ho-
wever, the identification of divisors generating closed ideals is a
non trivial task. This problem is the theme of articles by L.G.Hanin
(Z.r.Xa~H): reoMeTp~ec~a~ ~cc~Kanz~ ~xea~oB ~ a~redpax ~epeH-
n~pye~x ~y~ ~yx nepe~e~x, in the book "~CC~e~oBaHNH nO Teopg~
~/HKI~ ~0rZX Bemec~eR~x nepeMe~x", Hpoc~aBa~, z3~-Bo STY, I982,
122-144; "FeoMeTp~ec~ss Kaacc~KSa~ ~eaaoB B aaredpax ~peH-
I~pye~x ~yHm~", ~IOKa. AH CCCP, I980, 254, ~ 2, 303-307.
438
8.2. APPROXIMATION BY SMOOTH FNNCTIONS IN SOBOLEV SPACES
Le t ~ c ~ be a bounded domain whose b o u n d a r y i s a J o r d a n c u r v e .
Pu t
WK'P~G.-)--- ~ { ~ : ~"~L,P(.C:,-~ , 0~1o~1~ K }
T h i s i s t h e u s u a l S o b o l e v s p a c e d e f i n e d on ~ . I~S G°°(R¢)[ G dense i n WK'P(G), ' ~ K , p~oo ?
(The corresponding question for a disc minus a slit has a negati-
ve answer). The only thing I know is that this can be verified when
K= 4 and p=~ (Use con_formal mapping). To the best of my knowledge
%his question was firs% raised by C.Amick.
PETER W.JONES Institut Mittag-Leffler
Auravagen 17
S-182 62 Djursholm
Sweden
Usual Address :
Dept , of M a t h e m a t i c s
University of Chicago
Chicago, Illinois 60637
USA
439
H ~ 8 . 3 . sP~z~r~G ~D Bo~D~z BE~VZ~ ~ Om~ ~A~S old
Let ~ be a finite Borel measure with compact support in C •
Even for very special choices of ~ the structure of H~(~) , the
~(~) -closure of the polynomials, can be mysterious. We consider
measures~ ~ - $ + ~ , where ~ is carried by 9 and W is in
h'(~) . If '~O0/W ls in ~ ( ~ ) , H~(~,) Is well understood and
behaves like the classical Hardy space ~ (~4) [I]. We assume that
is circularly s~etric, having the simple form~$~-~(~)%~,
where ~ > 0 on ~,I]. Hastings [2] gave an example of such a measu-
re with W> 0 ~ ~.e. such t~t ~%~=H~(~) • L~(Wd~ we say then that ~(~) s p I i t s. A modification of this
example will show that given a n y W with ~ ~ W ~------oo~
c~n be chosen to be positive and non-increasing on [0,1] such
that ~(~) splits. Suppose G is smooth and there exist C , G>O~
and ~ 0<~, so that
@(~)¢
for 0~$< ~ . Suppose further that for some 6 , 6> 0 ,
0
(2)
THEOREM I. [3S Let ~ satisfy (I) and (2). SuDDose that
u~ (~tl~ < co f o r some,,,,,,,,arc r o f T , and t h a t W : O on a, set P
of positive measure in ~\ ~ . Then - ~(~) splits of and onl~ if
t -or ~- oo (3)
for small ~ .
This theorem settles the question of splitting only when
is well-behaved. Conditions similar to (3) were introduced by Keldy~
and Dzrba~an, and have been used by several authors in the study of
440
other olos~e prob~e=s (cf. ~ and [5]). QUESTION I. ,Can W be found such #hat spllttln~ occurs when
the integral in (~) is finite, or even when ~ ?
For ~ in ~ the point evaluation p .~ p(~) is a bounded li-
near functional on H~j~) (at l eas t for thos~ ~ that we are con-
sidering) ; let ~J~(~) denote its norm. If ~ (A) is analogously
defined, then E~(~)~ ~(~) (an upper bound for E ~# ). It is easy
to show that H~ splits i~ and only if E~<A)=E~(A) for ~ll in ~ . At the orther extreme, there is always an asymptotic
lower bound for E "t~ [_6]: ~'~ (4--%~)E~ei@)~>/ 4/W(~) ~ -..e. ~-+'I
Sometimes equality holds a.e. on an arc ~ of ~ :
~ - - 4 WCQ)
e.g. if $~ ~ is in [i(~) , then (4) holds with P~T . One
verifies tha% if W does not vanish a.e. on ~ and if (4) holds, then H~(~) cannot split.
• ~o~= 2. D] Suppose that ~ ~ W A~ ~.-~ ~,a r,
4
0 <t~ (5)
~hen (4) holds, every ~ in - H~(j~) has boundary values
~,e, on f ~=~ ~,e,~P and ~ ~I~I~ -~
whenever ~ is a closed arc i~t@rior ~o ? and £ ~ 0 i_~n ~"(~) .
Every zero set for H~(~) wi~h nq l!mlt points outside of Y is a
Blaschke sequence.
The hypothesis on W is weaker than that in Theorem I, (5) is
stronger than finiteness of the integral in (3) and the conclusion is
stronger than the "only if" conclusion of Theorem I by an unknown
amount. Eq.(4) can fail if the hypothesis on W is removed. Fix ~, 0<~ < ~, and let
441
sat isf ies (5). Define A(@, ~) : ( [ / ~ ) M# I ~: ~ ~ [@-~ @ ÷ ~]m
W(~.<,mp (-~)} and note that 0 .</I~ i .
THEOREM 3. ([3]). l.~f ~ is a s in .(6), there exist co~,tants
~:, o with J , < ~ e ~ ~p(~ ..0-~,~-~,~ ) ~o~ , .......... ( ~-~3~
all ~ in D •
I f S .~W~i~¢ ;~ - -~ , then O(O,#)~-O(# " ) as ¢--*- 0 ~ - -a.e. onr- , r and Theorem 3 yields no information near ~ . On the
other hand, for any ~, ~> ~ , one can construct W , W >0 aoe°
with ~,I~ ><~(-~¢>-~ for ~ =~ll ~nd all ~ [3]. ~hus
(4) can fail even if (5) holds.
QUESTION 2. Assm~e that %he inte~r~l in (~) is finite! or even
that & is ~iven b x (6). o,r that ~---~ . Is ~here a measurable
se~ E , E c T ,:',.~.h
where %h e first summand consists of "~ly%ic" functions? ~i~h% such
an E C,o~%ain an,y arc on which ~( ~, ~) (or a suitable an%lo~ue)
tends %o zero sufficien%l y slowly as I--~ 0 ? If ,~here is no such
F with ME > 0 , ~x~ctl~ how ca n the variou s conclusions of Theo-
rem 2 fa,,ilt, if indeed ~he,y can?
QUESTION 3. Let W(~) be smooth with a single zero at ~-----0 .
Assumin~ the integral in ~) is f tni%¢, describe the ,invarian% sub-
spaces of the operator "multiplication b~ • " on H~(~) in te~
of the rates of decrease of W(~) near 0 and ~(~) near I.
Perhaps more complete results can be obtained than in the simi-
lar situation discussed in [8].
Finally we mention that the study of other special classes may
be fruitful. Recently A.L.Volberg has communicated interesting re-
lated results for measures 9 t W~ ~ where ~ is supported on a
radial line segment. (See [I0], [13] in the reference list after Commentary. - Edo )
442
REPERENCES
I . C I a r y S. Quasi-similarity and subnormal operators. - Doct.
Thesis, Univ.Michigan, 1973.
2. H a s t i n g s W. A construction of Hilbert spaces of analytic
functions. - Proc.Amer.r~ath.Soc., 1979, 74, N 2,2295-298.
3. K r i e t e T. On the structure of certain H (~) spaces. -
Indiana Univ.Math.J., 1979, 28, N 5, 757-773.
4. B r e n n a n J.E. Approximation in the mean by polynomials on
non-Caratheodory domains. - Ark.Nat. 1977, 15, 117-168.
5. M e p r e x ~ H C.H. 0 no~oTe CzCTeM a~a~zm~ecEHxSyH~. -
Ycnex~ MaTeM.RayK, I958, 8, ~ 4, 3--68.
6. K r i e t e T., T r e n t T. Growth near the boundary in
M~) spaces. - Proc.Amer.Math.Soc. 1977, 62, 83-88.
7. T r e n t T. ~(~) spaces and bounded evaluations. Doer.
Thesis, Univ.Virginia, 1977.
8. K r i e t e T., T r u t t D. On the Cesaro operator. -
Indiana Univ.Nath.J. 1974, 24, 197-214.
THOMAS KRIETE Department of Math.
University of Virginia
Charlottesville, Virginia
22903, USA
COMMENTARY
THEOREM (A.L.VoI'Berg) There exists W , W>0 a,e, on T
such that m2(~) splits even for ~ ~ ~ .
The theorem gives an affirmative answer to QUESTION I. It may be
seen from the proof that ~ (~,S) tends to zero rather rapidly for
every ~ . The proof follows an idea of N.K.Nikolskii [9], p.243.
PROOF. It is sufficient to construct a function W , ~ >0 a.e.
on T and a sequence of polynomials {P~ } ~4 such that 2
(P.IT, ID) =C0, in the Hilbert space IZ(WcI,I'II,)~I~L(~ I$
Let {~}~4 be any sequence of positive numbers satisfying
443
Z S. <4 ' S~O , and let
Pick any smooth outer function ~
= ~ on T\~n ( ~ 0 ) and such that ~(0) =
condition implies the existence of an integer N~
O
Consider now the set e~={r~T: ~N"~r. }
fTI. e. ~. and therefore t~(~.4 U e~)= 0 the inoreasing family of Bets ~g = T \ U ~k
exhausts the unit circle:
to define the weight W
with the constant modulus I~
• The last
such that
(7)
. It is clear that
This implies that
- A Txe K almost
~A~MS ~K= I . Now we are in a position K
W(O: , ~ 5 K \ S K _ I , K - - ~ , , . . ,
where C~ stands for ~oe
set H~(~)~ ~.(N.) because ~nE T\e~ . Clearly
and note that IH~l=sn on S~
C~ = + co . These imply
2
T S~ The last inequality together with (7) yields obviously the desired
conclusion. •
THEOREM 1 in the text of the problem can be strengthened. Sup-
pose that the function G satisfies some regularity conditions and
splits iff 4
444
T
The new point here is that we do not require for W to be identical-
ly zero on a set of the positive length. See ~ for the proof.
QUESTION 2 can be also answered affirmatively. Recall that a
closed subset E of T satisfies (by definition) the Carleson con- 4 dition i f ~ ~ ( ~ ) ~ ~ < + oo. Here {~} stands for the fa-
Y mily of all complementary intervals of E . Let ~ be the family
of all closed E , tY~E>O which do not contain subsets of positive
length satisfying the Carleson condition.
Suppose again that G ~ .
THEOREM (S.V.Hru~v). Let E ~ . Then there exisItISLI a ~iosi -
tive wei~t ~J such that
= H(v+ T,E W& )eL(%E
where the first s ummand does not split.
It has been shown in ~1~ that such sets E do exist. For
example, any set of Cantor type having positive Lebesgue measure and
not satisfying the Carleson condition does the job.
PROOF. Pick a closed set E in ~ and consider an auxiliary
region ~ having the smooth boundary as it is shown on the figure
The region ~ abuts on T precisely at the points of E
and its boundary F has at these points the second order of tangency.
Let ~ be a conformal mapping of ~ onto D . Then ~ does not
distort the Euclidean distance by the Kellog's theorem (see ~2],
445
p.411). It follows that ~CE)~% • By theorem 4.1 in [11] there
exists a sequence of polynomials ( P~} fl,~4 satisfying
P~ ----" 0 uniformly on ~'CE) ;
P. = 4 uniformly on compaot ~ b s e t s of 0 1t, ¢o~st,
IP. (~)1 ~ 0 _ l ~ l ) V ~ "
Using the Kel log 's theorem a ~ i n , we see that the seq~enoe {j~.}.~.~ ,
j ~ C~) = P. ( ~ ( ~ ) ) s a t i s f i e s the fo l lowing
~ ~ = 0 uniformly on E ; (8)
n uniformly on compact subsets of ~ (9)
co~st ~ c~os O. (lo)
Define
wCt)= f 4, t~E (~f,}~, tee , being a complementary
in%erval of E.
The function ~C~) = ~$~C-~,~) ~" ~ C4¢L~')/~"/%~ i s evidently ~ummable on T ~nd dominates l ~ l ~" with (9) this implies
. Together
T',E
by the Lebesgue theorem on dominated convergence. Besides, (@) and
(40) yield
446
(see [/1] ). ~ i ~ l l y , ~ L a( ,~,1~ I ~ = 0 , see (8). 15
The space H~(,~+ ~ F W ~) does not split because TXE
~g ~J(t) = W~(~)~I >0 for every complementary interval ~ .
This can be deduced either from theorem 2 cited in the text of the
problem or from theorem 3.1. of [I I]. @
Note that an appropriate choice of E provides the additional
property of the weight ~ in the theorem: P
T
for every p, p < ~ . Pick a Cantor type set E in ~ satisfying
+ for
The construction of the theorem can be extended for other
weights G satisfying (5). Such a splitting cannot occur if
j ~ ~ > - o o and E~ ~E >0 satisfies the Carleson condition E (see theorem 3.1 in [I I] ).
REFERENCES
9. H ~ ~ o x ~ c ~ H ~ H.E. Hs6paH2~e 8s~aH~ BeCOBO~ annpo~cHMa-
n~ ~ cne~Tpa~HOrO aHaz~sa. - Tpy~ Ma~.~H--Ta ~M.B.A.CTeF~oBa
AH CCCP, 1974, 120.
I0. B o a ~ 6 e p r A.~. ~orap~M HOqTH--aHaJL~THqecEo~ ~yHl~
cy~pyeM. -~oF~.AH CCCP, 1982, 265, ~ 6, c.1297-1302.
II. X p y ~ "e" B C.B. Hpo6xeMa O~HOBpeMeRHo~ annpoEczMa~H~ ~ cr~pa-
~e OcO6eHHOCTe~ ~HTerpa~oB THHa EO~. -- Tpy~ MaT.~H--Ta ~M.B.A. CTeF~oBa AH CCCP, I978, I30, c.I24-I95.
I2. r o ~ y 3 E H roM. reoMeTpE~ecEa~ Teop~H ~yHE~ EOM~eECHO--
ro nepeMeHHo~o. M., "HayEa", 1966.
13. B o a ~ 6 e p r A.$. 0~HoBpeMeHHa~ aHnpoEc~MaU~ no~Ho~a~
Ha oKpy~HocT~ ~ BHyTpH Kpy~a. -- 8an.Hay~H.CeM~H.~0ME, I978, 92,
60--84.
447
8.4. oi~
on the line ~ , let ZCA)= I,~(~, ~A) and let ZT(A)
the closure in ~(A) of finite trigonometric sums ~C~e $~J
I~I~T . It is readily checked that ZT4(A)cZT~(A) for~T~
z' U (A) is dense in Z(A) . Let T~ 0
with the understanding that To(A)= oo if the equalityZT(A) =~(A)
is never attained.The following 3 examples indicate the possibili-
ties:
(I) if A(~)~I(~I) -I~ then To ~-~ ; 0
ON THE SPAN OF TRIGONOMETRIC SUMS IN V~IGHTED~ SPACES
Let A=A(~) be an odd non-decreasing bounded function of
denote
with
and that
(2) if A(~)=I 6-I~I ~ then To=0 ; O
(3) if A is a step function with jumps of height ~/(~+~)
at every integer ~ , then To= ~ .
PROBLEM. Find formulas for To , or at least bounds on To ,
in terms of A •
DISCUSSION. Let A ! denote the Radon-Nikodym derivative of
with respect to Lebesgue measure. It then follows from a well known
theorem of Krein D~ thatTo= oo as in example (I) if
--0@
A partial converse due to Levinson-McKean implies that if A is ab-
solutely continuous and if Af[~) is a decreasing function of I~I
and ~.~K!~} ~ =-oo (as in example (2)), thenTo=0. A proof
of the latter and a discussion of example (3) may be found in Secti-
on 4.8 of [2]. However, apart from some analogues for the case in
which A is a step function with jumps at the integers, these two
theorems seem to be the only general results available for computing
T o directly from A • (There is an explicit formula for T o in terms
of the solution to an inverse spectral problem, but this is of
448
little practical value because the computations involved are typi-
cally not manageable.)
The problem of finding To can also be fox~ulated in the lan-
guage of Fourier transforms since ZT(A) is a proper subspace of~(A)
if and only if there exists a non-zero function ~ Z (A) such that @@
* o r . T h u s
for Il;1 in Z (A) } .
Special cases of the problem in this formulation have been studied by
Levinson [3] and Mandelbrojt [4] and a host of later authors. For an
uptodate survey of related results in the special case that A is a
step function see [5]. The basic problem can also be formulated in
~P(~,~A) for 1~e~ . A number of results for the case ~eo
have been obtained by Koosis ~],~7] and [8].
REFERENCES
]. K p e 2 a M.F. 06 o~o~ 9NCTpa~o~s~HOHRO~ npo6~eMe A.H.Eo~Moro-
poBa. - ~o~.AH CCCP, 1945, 46, 306-309.
2. D y m H., M c K e a n H.P. Gaussian Processes, Function
Theory and the Inverse Spectral Problem, New York, Academic Press,
1976.
3- L e v i n s o n N. Gap and Density Theorems. Colloquium Publ.,
26, New York, Amer.Math.Soc., 1940. #o 4. M a n d e 1 b r o j t S. Serles de Fourier et Classes Quasi-
analytiques. Paris, Gauthier-Villars, 1935.
5. R e d h e f f e r E.M. Completeness of sets of complex exponen-
tials. - Adv.Math. 1977, 24, 1-62.
6. K o o s i s P. Sur l'approximation pond~r~e par des polyn~mes
et par des sommes d'exponentielles imaginaires. - Ann.Sci.Ec.Norm.
Sup., 1964, 81, 387-408.
7. K o o s i s P. Weighted polynomial approximation on arithmetric
progressions of intervals or points. - Acta Math., 1966,116,
223-277.
8. K o o s i s P. Solution du probl~me de Bernstein sur les en-
tiers. - C.R.Acad.Sci.Paris,Ser.A 1966, 262, 1100-1102.
HARRY DYM Department of ~thematics The Wei~annInstitute of
Science Rehovot, Israel
449
8.5. DECOMPOSITION OF APPROXI~[BLE FUNCTIONS
Let ~(~0) be the space of all analytic functions in some open
subset ~0 of the extended complex plane C . Let ~* denote the
one point compactification of~0 .
If F is relatively closed subset of ~ , A©(F) is the func-
tions on F being uniform limits on F by sequences from H( O) The problem of characterizing A (F) ~s raised by N.U.Arakelyan
some years ago ~] . A closely related question was raised in [2 ] . Recently we obtained the following characterization of A~(~)
for a large class of sets ~0 :
Am(F) = U]) (F))+ H (9)
.here is the space of analytic functions on
FU~(E) with a continuous extension to the Riemann sphere, and where
Z)(F) is the smallest open subset of ~[)\F such that~O~\(FU~(F~ is arcwise connected. For details see [4].
PROBLEM 1: Obtain a de compositi£A ' like (I 7 for any proper nonemp-
ty open subset of the Riemann sphere.
PROBLEM 2: Obtain decompositions ' like (1) when ~ is the unit
disc {I~I <~I and is replaced by other function spaces
i_~n~ .
REMARK: A positive answer to Problem I, will immediately give a
solution to problem 9.6 in [0] in light of the results about g~a(F)
in [3].
REFERENCES
O. A n d e r s o n J.M., B a r t h K.F., B r a n n a n b.A.
Research Problems in Complex Analysis. - Bull.London Math.Soc.,
1977, 9, 152.
1. A r a k e I j a n N.U. Approximation complex, et propri~t~s
des fonctions analytiques. - Acres Congr~s intern.Math,, 1970,
2, Gauthier-Villars / Paris, 595-600.
2. B r o w n L., S h i e 1 d s A.L. Approximation by analytic
functions uniformly continuous on a set. - Duke Math.Journal, 1975,
42, 71-81.
450
3. S t r a y A. Uniform and asymptotic approximation. - Math.Ann.,
1978, 234, 61- 68.
4. S t r a y A. Decomposition of approximable functions.
ARNE STRAY Agder Distriktshogskole
Postboks 607,
N-4601 Kristiansand
Norway
451
8.6. A PROBLEM OF UNIFORM APPROXIMATION BY PUNOTIONS ADMITTING QUASICONFORNAL CONTINUATION
The following subalgebras of the B~nach space C(~) of all
continuous functions on a compact set ~ ~ K cC , are important in
the theory of rational approximation. These are the algebra A(~)
of all functions in ~(~) holomorphic in the interior of K and
the algebra ~(K) consisting of uniform limits of rational func-
tions continuous on K .
For ~>0 let K~ ~:K+D(~), D(s)~-~{zeC.Izl<a}. Consider the Beltrami equation in K~
~ ---~(Z)IZ ' (1)
being a measurable function such that
K~
A c o n t i n u o u s f u n c t i o n I i s s a i d t o be a g e n e r a l i z e d s o l u t i o n o f (1) i f i t s g e n e r a l i z e d d e r i v a t i v e s ( i n t h e s e n s e o f t h e d i s t r i b u - t i o n t h e o r y ) be long t o ~ l o c a l l y and s a t i s f y (1) a . e , on K6 o
Clear ly { I K s C ( K ) f o r such a s o l u t i o n and i t i s kno ,~ t ~ t IIK~A(K), provlded ~ 0 on K [1].
Fix k ¢ ~ and consider a set ~8(K) of all restrictions
~IK , where I ranges over the family of generalized soluti- ons of (I) in ~8 with j~-0 on K~ • Let ~(K) be the clo-
sure of ~0 ~ ( ~)- in C(K) . Then clearly
R(K) c B(K)cA(K). PROBLEM I . Is there K such that R(K)¢B(K) ?
An affirmative answer to the question would entail the following problem,
PROBLEM 2. Find necessary and sufficient conditions on K for
a) B(K)-C(K) and for
.1
b) B(K)= A(K).
452
Suppose k= 0 . Then a complete solution of problem 2 is given
by Vitushkin's theorem [2],[3]. The case k>0 corresponds to the
problem of approximation by functions admitting a quasi-conformal
continuation.
One of possible ways to solve problem I consists in the const-
ruction of a "Swiss cheese" satisfying ~(K)~C(K)~ ~(K) =0(K),
These problems were posed for the first time at the Internatio-
nal Conference on Approximation Theory (Varna, 1981).
REFERENCES
I. L e h t o 0., V i r t a n e n K.I. Quasiconformal Mappings
in the Plane, Springer-Verlag, Berlin. Heidelberg. New-York,1973.
2. B E T y m E E R A.r. ARaaET~ec~as ~OCT~ MRo~ecTB B 3a~s~ax
TeopzE npEdJm~eRE~. -Ycnexa MaTeM. sayE,1967,22,.~iS, 141--199.
3. Z a i c m a n L. Analytic Capacity and Rational Approximation.
Lect.Notes in Math., 1968, 50.
V. I. BELYI CCCP, 340048, ~osenH 48,
YHEBepcETeTCEa2 77,
14RCTMTyT llpEEaa~o~
MaTeMaT~E~ E MexsaHE~
453
8.7. TANGENTIAL APPROXIMATION
Let F be a closed subset of the complex plane C and let
and G be two spaces of functions on F . The set F is said to
boa set of tangential a p p r o x i m a t i o n
of functions in the class ~ by functions in the class ~ if for
each function ~ ~ ~ and each positive continuaus ~ on F , there
is a function ~ G with
Carleman's theorem [I] states that the real axis is a set of
tangential approximation of continu@us functions by entire functi-
ons Hence, tangential approximation is sometime called Carleman ap-
proximation
PROBLEM: For given classes of functions ~ and G , characte-
rize the sets of tangential approximation.
Of course, this problem is of interest only for certain classes
and ~ . We shall use the following notations:
H(~) : entire functions
MF(C) :meromorphic functions on C having no poles on F •
H(F) : functions holomorphic on (some neighbourhood of) F .
U(F) :uniform limits on F , of functions in H(F) .
A(F) : ~unctio~ continuous on F and holomorphlc on F ° .
C(F) :continuous complex-valued functions on F .
Each of these classes is included in the one below it. We con-
sider each problem of tangential approximation which results by
choosing ~ as one of the first three classes and choosing ~ as
one of the last three. Thus each square in the following table cor-
responds to a problem
H((:)
MF(£) H(F)
U(F) A(F)
[~]
C(F)
[9]
[3]
[3]
454
The blank squares correspond t o open problems. For partial re-
sults on the central square, see [5]. In [3], the conditions stated
characterize those sets of tangential approximation for the classes
~= C(F) and G=~F~C) . One easily checks that these conditions
are also necessary and sufficient for the case ~=C(F) and ~ =H(F)
The first column was suggested to us by T.W.Gamelin and T.J.Lyons.
One can formulate similar problems for harmonic approximation.
The most general harmonic function in a neighbourhood of an isolated
singularity at a point ~ ~n , ~$~ can be written in the form
=po. +
where
KCO ) = g-~
and Pk,~K are homogeneous harmonic polynomials of degree k ,
k>~O . The sin~alarity of ~ is said to be n o n - e s s e n -
t i a I if pk=O , k>~ k o . An e s s e n t i a 1 1 y h a r-
m o n i c f u n c t i o n on an open set ~c-~ n is a function
which is harmonic in ~ except possibly for non-essential singulari-
ties.
Let F be a closed set in , ~>~ . We introduce the fol-
lowing notations:
~(~) : functions harmonic on ~t
~F(~): essentially harmonic functions on ~ having no
singularities on F.
~(F) :functions harmonic on (some neighbourhood of) F.
~(F) : uniform limits on ~ , of functions in ~(F) . @
~(F) :functions continuous on F and harmonic on ~.
c~F) continuous real-valued functions on F .
As in the complex case, we have a table of problems.
455
mFc
,CF)
c(p)
[s]
RE~RENCES
I. C a r 1 e m a n T. Sur un th~oreme de Weierstrass. - Ark.Mat.
Astronem.Fys. 1927, 20B, 4, I-5.
2. K e a ~ ~ m M, B. ~ a B p e H T B e B M.A. 06 O~OR ssaa-e EapaeMaHa. - ~oEa.AH CCCP, I989, 28, • 8, 746-748.
M e p r e a a H C.H. PaBEoMepm~e np~Oa~zeHm~ ~ysxm~ EOMEae~cHo--
rO nepeMem~oro. - YcnexE MaTeM.HayE, I952, 7, B~n.2 (48), 31-123.
(English. Translations Amer.~ath.Soc. 1962, 3, 294-391).
A p a E e a a H H.Y. PaBHoMepm~e z ~acaTea~m~e np~OazxeH~a saa-
x~T~ec~mm ~z~m. - ~sB.AH ApM.CCP, cep.MaTeM., 1968, 3, J~ 4-5, 273-286.
3. H e p c e c a ~ A.A. 0 yaBHoMey~o~ ~ ~aca~ea~Ho~ aunpo~cm~a~
MepOMOp~ ~ . -- HsB.AH ApM.CCP, cep,MaTeM., I972, 7,
6, 405-412.
R o t h A. Meromorphe Approximationen. - Comment.Math.Helv.
1973, 48, 151-176.
R o t h A. Uniform and tangential approximations by meromorphic
functions on closed sets. - Canad.J.Math.1976, 28, 104-111.
4. H e p c e c a H A.A. 0 ~Ho~ecTBax Kapae~saa. - MsB.AH ApM.CCCP,
cep.MaTeM., 1971, 6, ~ 6, 465-471. 5. B o i v i n A. On Carleman approximation by meromorphic func-
tions. -Proceedings 8th Conference on Analytic ~unctions,Blaze-
jewko, August 1982, Ed.J.Lawrynowicz (to appear).
6. ~ a ~ ~ ~ a H A.A. 0 paBHoMepHo~ ~ EacaTex~HO~ ~apMo~m~ecEo~
annpoEc~Mmm~ Henpep~mm~ Sy2~m~ Ha ~po~sBOa~SHX COBOEyrK~OCT~X.
--MaTeM.ssaeTE~ 1971, 9, Bm~.2, ISI-142. (English: Mat.Notes
1971, 9, pp. 78-84).
7. G a u t h i e r P.M. Carleman approximation on unbounded sets
by harmonic functions with Newtonian singularities. - Proceedings
8th Conference on Analytic 2unctions, Blazejewko, August 1982,
Ed.J.Imwrynowicz (to appear).
456
8. L a b r ~ c h e M. De l'approximation harmonique uniforme. Doc- • . J toral Dissertation Unlverslte de Montr6al, 1982.
~D~ BOIV~
PAUL M. GAUTHIER
• • o Departement de Mathematlques et de Statistique o • • Unlverslte de Montreal
C.P. 6128, Succursale "A"
Montreal, Quebec
H3C 3J7
CANADA
457
8.8. THE INTEGRABILITY 0P THE DERIVATIVE 0F A CONPORNAL NAPPING old
Let ~ be a simply connected domain having at least two boun-
dary points in the extended complex plane and let ~ be a cenfermal
mapping of ~- onto the open unit disk ~ . In this" note we pose the
following QUESTION: Per which numbers p is
J'I ? /L
For p- -~ the integral is equal to the area of the disk and is
therefore finite. In general, it is known to converge for $/J<p<8
amd if .~I is the plane slit along the negative real ax~s then it ob-
viously diverges for p-----~/S and p=~- These facts are consequen-
ces of the Kcebe distortion theorem and were first discovered by
Gehrlng and Hayman (unpublished) for p< ~ and by Metzger EI3 for
P • ~ . Recently, the author has succeeded in proving that the upper
bound 3 can be increased. The following theorem summarizes the kmo~rm
results.
THEOREM I. There exists a number ~, ~ 0 , not de~end~ on
~l, such that
Jl I < #.
± f 4-/~ ~: p < 3 .+ ~ .
Per a wide class of regions, including "starlike" and "close-to-
convex" domains, p~ ~ is the correct upper bound (of. 12], ~eorem 2). Quite likely, IflgtlP~×~<.~ fer~/~ < p<÷ in all cases
but, unfort.~ately,-- U t h e argument im E2~ will net give this result.
Here is a SKETCH OF THE PR00P 0F THEORE~ I. We shall assume that
Xe~ ~ , ~ (X0)~ 0 and we shall denote by ~(~) the Euclidean
distance from the point ~ to ~- . It is easy to see, using polar
ceerdinates, that
where ~go¥ i s h.a_~o~Lo measure on the curve I ~ I -~-~ r e l a t i v e t o ~o . Moreover , i t f o l l o w s f rom the Koebe ~Lstoz@ion theorem t h a t
t~(z~[ "~ K 'l-lq~(~l &£. a,~d, co~equently, .... ~ , . ( ~ ) n e a r
458
if and only if
~ .(~)p,_~,,,,, ~,~, < . - ~ o
Thus, Theorem 1 is now an i.~ediate consequence of the following lem-
on the of the inte l 5 as
LE~A 1. There exists a constant ~ , ~> 0 , such that if
~ ~/~ then
Of course, if we could prove the lemma for all ~ , ~ ,
them we could prove Theorem I for 4/~ < p < $ . So far, however,this
has still not been done. The proof of the lemma is based on an idea
of Carleson [3], which he expressed in connection with another prob-
lem. The QUESTION is the following: On a Jordan curve is harmonic
measure absolutel,y ~ontinuQus with respect to ~-dlmensiemal Haus-
dorff measure for every ~, ~ <~ ? On the one hand, according to the
Beurling projection theorem (cf. [4], p.72), the question can be an-
swered affirmatively if ~ <~ 4/~ . On the other hand, Lavrent'ev [5],
McMillan and Piranlan [61 and Carleson [3] have shown by means of
counterexamples that absolute continuity does not always occur if
~=I . In addition, Carleson was able to show that the upper bound
I/2 in Beurling's theorem can be increased. It is interesting to spe-
culate on the extent to which it is possible to observe a similarity
between the two problems. For example, it is well known (of. [7~,p.44)
that harmonic measure is absolutely continuous with respect to 1-di-
menslonal Hausdorff measure if there are me points ~ on ~ for
w h i c h
~r~ ~p 0~ (~-~)=~o (I) ~ --,- ~ ~XL
~ ~,~ ~} (~_ ~)=-oo (2)
459
The Q~STION arises: if t h i s comet!on is s a t i s f i e d mus t J~I~IIP~X~<
< co for $/Z ~ p<~ ? At this time the answer is not known# Be-
fore proceedimg to the solution of %he gemer~l problem i% apparently
remaims to answer this more modest question,
To the best of my knowledge, the question about the integrabili-
ty of the derivative of a conformal mapping arose in connection with
several problems in approximation theory. We shall mention only one
of these and then imdicate an application of Theorem I. Our problem
was first posed by Keldy~ in 1939 (cf.[8] and [9], p.10) and he ob-
tained the first results in this direction. Fur%her progress has been
achieved in the works of D~rba~jan ~0], ~ginjan ~, Maz'ja and
Ha.vin D2], D3] and the author D4], ~5],[2]. A complete discussion of the results obtalnedup to 1975 can be found in the surveys of
Mergeljan [9], Mel'nikov and Sinanjan ~6].
Let us assume that ~ , ~ are two Jordan domains in the comp-
lex plane, ~cD , and let.~=~(P~) . We shall denote by
HP(~),~, the closure of the set of all polynomials in the space
hP(..~L,,~) and we s h a l l denote by hq (-.~-.) t h e subspace consisting
of %hose functions ~ , ~eh~(~) , which are analytic in ~- . Clear-
ly,~Fch~ . An imterestimg question concerns the possibility of
equality in this inclusion. It is well known that in order for H P and ~ to coincide the de%erminimg factor is the "%himmess" of the
region~l near multiple boundary points (i.e. near points of ~DN
N ~ ~ ). Here is a result which gives a quantitative description of
that dependence. The proof is based in part on Theorem I (cf.~] and
~, pp.143-148).
THEOREM 2. Let ~7(~) be the distance from Z to ~ \ D and
let ~ be harmo~c measure on ~ relativ~ to the domai m ~ .
There exists an abso!ute constant ~ ~ >0 , n£.t de~.ending on/[ ,
such that if
,then P
for a n p,
The QUESTION remains: i_~s ~ ~ the upper bomad or i~ the theg- rem true for all p, p~oo ?
14.
15.
18.
460
REFERENCES
I. M e t z g e r T.A. On polynomial appro~mation in A$(2) • -
Proc.Amer.Math.Soc.)1973, 37, 468-470.
2. B r e m n a n J. The integ1~bility of the derivative in con-
formal mapping. - J.London Math. Soc. ,1978, 18, 261-272.
3. C a r I e s o n L. On %he distortion of sets on a Jordan curve
under conformal mapping. - Duke Math.J., 1973, 40, 547-559.
4. M C M i 1 1 a n J.E. Boundary behavior under confromal map-
ping. - Prcc. of the N.R,L. Conference on classical function
theory, Washington D.C. 1970, 59-76.
5. ~ a B p e R T B e B M.A. 0 ReEoTop~x FpaRE~HHX 3a~a~ax B TeopEE
O~OAEOTH~X ~YH~%7~. -- ~aTeM.06., I963, ~ I, 815-844.
6. McM ~ i I a m J.E., P i r a n i a n G. Compression and
expansion of boundary sets, - Duke Math.J. ~1973, 40, 599-605.
7. MoM i 1 1 a n J.E. Boundary behavior of a conformal mapping.
- Acta Math.,1969, 123, 43-67.
8. K e ~ ~ H m M.B. Sur l'approximaticn en moyenne quadratlque
des fonotions analytiques. - MaTeM. cd.$939,47,~ 5, 391-402. 9. M e p r e a s ~ C.H. 0 nOaSOTe C~CTeM asax~T~ecm~x ~yRm~. -
Ycnex~ MaTeM. RayE, 1953, 8, ~ 4, 3-63. I0 . ~I X p 6 a m a R M.M. MeTp~'qec~i~e T e o p e ~ o nom~oTe ~ npe~cTaB~-
MOCT~ ssaa~T~,~ecr~x ~y~mm~. ~O~T.~ccepTsam~, Epe:sas, I948. II. m a r ~ H a s A.~. 06 O~0M np~sRa~e senom~oT~ C~CTe~ aHaa~-
T~qe0~zX ~yRm/~. - ~oE~.AH ApM.CCP, I946, Y, ~ 4, 97-I00.
I2. M a s ~ ~ B.F., X a B ~ R B.H. 06 annpo~c~sa~z B cpe~HeM
aHS/fgTRqeCEEM~ ~yRm/~aME. -- BeCTH.ZeH~HPp.yH--Ta, oep.MaTeM. ,MeX.
acTpoH., I968, ~ I3, 62-74.
13. ~ a s ~ ~ B.F., X a B i~ H B.H. llp~xo~eR~ (~,$) - e~OCT~
aec~o~E~M sa~ayaM Teop~ ~CI~Te~RI~X MRo~eoTB. -- MaTeM.C6.,
1973, 90, ~ 4, 558--591.
B r e m m a n J. Imvarlamt subspaces sad weighted polynomial
approximation.- Ark.Mat,,1973, 11, 167-189.
B r e n n a m J. Approx~%ion in %he mean by polynomials om
nom-Oaratheodory domains. - Ark.Mat. 71977, 15, 117-168.
M e n ~ ~ E E o B M.C., C ~ R a R a R C.0. Bonpoc~ Teop~
np~6m~eH~fi ~y~m/m~ o~moro ~oMa~eEcRoro nepeMeRRo~o. - B E~. :
CoBpeMeRsNe nqoo6J~eM~ ~aTeMaTE[4~, T.4, HTOI~ Ha~ E T~HEEI~,
MocEBa, B~{I4TH, 1975, 143-250.
J.~RE~A~ University of Kentucky Lexington, Kentucky 40506
USA
461
8.9. WEIGHTED POLYNOMIAL APPROXINATION
Let Jl be a bounded simply connected domain in the complex
plane $ , let dxdy denote two-dimensional Lebesgue measure and let
• /(~) > 0 be a bounded measurable function defined on ~L . For each
p , ~ p<oo , we shall consider two spaces of functions:
(i)~P(~l,~/~)~ the closure of the polynomials in1.P~f~)
the set of functions inL ( (ii)
which are analytic in ~ .
If t/ is bounded away from zero locally it is easy to see that
~P@ is a closed subspace of [~P and that ~c ~ • It is an
OLD PROBLEM to determine: for which regions ~. and weiF~ht s ~/ i_~s P
Li ? enever this happens the polynomials are said to be c o m p 1 e t e in ~ .
As the problem suggests, completeness depends both on the re-
gion ~ and on the weight %~ . In this article, however, we shall
be primarily concerned with the role of %I when no restrictions are
placed on/4 , save simple connectivity. The main difficulty then
stems from the fact that ~ may have a nonempty i n n e r b o u n-
d a r y; that is, there may be points ing~ which belong to the
interior of~ (vlz., a Jordan domain with a cut or incision in the
form of a simple arc from an interior point to a boundary point).
Roughly speaking,~a,%q~) ~ ~ if %9"(~) ; 0 sufficient-
ly rapidly at every point of the inner boundary. But, this is not
the only factor that must be considered and in order to avoid cer-
tain snags we shall make the additional sssumption that ~ is cons-
tant on the level lines of some conformal mapping ~ of ~ onto
t h e open unit disk O (i.e., i6(~) = W(4-1~(~)l ),where W(~) ~ 0
as ~ , 0 ). Put another way, %~ depends only on Green's function.
With this requirement the problem becomes conformally invariant and
every significant result going back to the early 1940's and the se-
minal work of Keldy~ [I] makes use of this or some equivalent fact.
Additional information and background material on the completeness
question can be found in the survey article of Hergel~an [2], in the
author's papers [3], [4~ and in the references cited therein.
In the ensuing discussion ~ will denote the unbounded compo-
nent of ~\ ~ and 0~ is harmonic measure on S~ relative to some
462
convenient point in Jl . For weights which depend only on Green's
function the author [4] has obtained the following result"
THEOREM le Suppose that W($) } 0 as ~ ~ 0 and that ~(~)>0 .
Then there exists a universal constant ~ > 0 such that
0
...... b P whens,re=" •
Since there are only two restrictions, one on ~ and one on
the ~P-class, TWO QUESTIONS arise:
(I) Can the assumption ~($~o0)> 0 be removed?
(2) I s the theorem true for all ~, ~p<oo ?
If W(~) } 0 in a sufficiently regular fashion then the answer to
both questions is yes. For example, if W(~) ~ o -~($~ and~(~)~+ 6°
as ~ ~ 0 the divergence of the integral ~.*~(~)~ is suffici- p v o
ant tO guarantee that Hr(..Q.,'MF~Q~I,J,)= LIo. ' for all p:, '~ p < o o , even ~ e n ~ ( ~ L ~ ) ~ O.
I n o rder to g ive an i n d i c a t i o n of hew the hypotheses are used, here i s a b r i e f o u t l i n e of the p roo f of Theorem 1. For each ~> ¢ we shall denote by 05 the capacity naturally associated with the
Sobolev space W I'$ and A~ will stand for ~-dimensional Haus-
dorff me&sure. A comparison of these two set functions together with
their f@rmal definitions can be found in the survey article of
~az' ja and Havin [5] • Let ~ be any function in ~$(O W~X~) , ~ p/(~-~), with
the property that ~ Q ~ , ~ # , ~ = 0 for all pol~onials Q and form the Cauchy integral
~-~ ~L
Evidently, ~ v~nishes identically in ~ and so, by "continuity",
~ 0 a.e. -C$ on ~lao . To establish the completeness of the po-
lynomials we have only to prove that 5-----0 a.e. -C$ on the rest
of the boundary as well, this approach having first been suggested
by Havin E6] (cf.also E3] and E4] ). The argument is then carried out
463
in two stages; one verifies that:
STEP 1. ~0 a.e. with respect to harmonic meas~e on ~l ;
STEP 2, ~--0 a.e. with respect to the capacity C~ on ~- .
Horeover, in the process it will be convenient to transfer the
problem from ~ to ~ by means of confoz~nal mapping. With ~= ~-~
let ~= ~(~ . ~or each ~ > 0 let A~={~ :I ~o¢~1 < 4 - ~. } and p u t
{~(~) = I ~(~ ~ # ~- r~ a n d ~6 ~-- "~6 (~) •
T h u s , ~ a n d ~a a r e b o t h d e f i n e d on ~ and ~8 i s a n a l y t i c n e a r
3 D . STEP 1. By c h o o s i n g ~ ~ 0 and s u f f i c i e n t l y s m a l l we c a n f i n d
a corresponding ~> 0 such that the following series of implicati-
ons are valid for any Borel set E ~- ~ll :
0 (1.1)
Here $~ p/(p-J) and p< ~+~ ° The first implication (i) is essen-
tially due to Frostman K7~- Although he considered only Ne~onian
capacity, his ar6ument readily extends to the nonlinear capacities
which enter into the completeness problem (cf.~az'ja-Havin [5~ ).
The second assertion (ii) is a consequence of a very deep theorem of
Carleson [8]. Because I= 0 a.e. -05 on ~oo and ~(~-oo) > 0
it follows that ~ 0 on some boundary set of positive harmonic mea-
s~LTe. Consequently, taking radial limits, ~-- 0 on a set of positive
are length on S~ .
We may now suppose that W (~)~ 6 -~($) , where ~(~)~ + co
as t , i o . Then, n s i ~ the f a c t t ~ t ~ l ~ ' l ~ & ~ < ~ f o r p < ~ + ~ (of. [9] and ElO~, the latter being reprinted in this collection, Prob- lem 8.8) it is an easy matter to check that
I~i=~
464
where o and ~ are constants independent of 5 . Because ~----- 0
on a set of positive ~ measure on SD and I@ ~(~)~=+oo ,
if follows that ~ 0 a.e. -~ . The argument here is based on a
modification of Beurling's ideas ~IS and i% states, in essence,that
those functions on ~ which can be sufficiently well approximated
by analytic functions retain the uniqueness property of the approxi-
mating family. As a general priciple, of course, this goes back to
Bernstein [_12]. The upshot is ~-~-0 a.e. -~0 on ~ and Step I
is complete.
STEP 2. At this point we are required to show that from £ = 0
a.e. -~c0 it can be concluded that # = 0 a.e. -6~ on ~4 which runs
counter to the known relationship (1.1) between harmonic measure and
capacity. We shall be content here to simply note that the reasoning
is based on an ar@~ent from the author's article ~3] and that es-
sential use is made of the fact that Stepl is valid for e v e r y
8nnlhilator ~ .
In light of what has now been said one IMPORTANT QUESTION re-
ma~n.: Is the dlver~ence of the Io~ lo~-integ~al necessary for comp-
leteness to . . . . . . . occ~...{, that is, if ~ ~ W--.~-~'~ < - e o o ' l -is
% , L% ° H.A,w~,x,~,,~,)==/= ? In case the inner boundary o f /1 conta inB
an isolated "smooth" arc and W($)$ 0 as $~ 0 the answer is yes
and the proof is a simple adaptation of an argument of Domar [I 4].
REI~,~LENCE S
I. K e I d y c h M. Sur l,approT~mation en moyenme par polyn~mes
des fonctions d'une variable complexe. - MaTeM.c6opH~K, 1945,
58, ~ I, 1-20. 2. M e p r e a a H C.H. 0 HO~mOTe CZOTeM aaam~T~ecE~x ~yHE~R. -
Ycnex~ MaTeM,HayE, 1953, 8, ~ 4, 8--63. 3. B r e n n a n J. Approximation in the mean by polynomials on
non-Caratheodery domains.- Ark.Mat.~1977, 15, 117-168.
4. B r e n n a n J. Weighted polynomial approximation, quasiana-
lyticity and amalytic continuation. - Preprint.
5. M a 8 H ~ B.r., X a B ~ H B.H. H~HHe~Ha~ Teop~ noTemmsaa.
- Ycnex~ MaTeM.HayE, I972, 27, ~ 6, 67--138.
6. X a B ~ H B.H. AnnpoEc~ saaazT~qec~ ~ ~ m B cpe~-
HeM. --~oEa.AH CCCP, 1968, 178,~ 5, 1025-1028.
7. F r o s t m a n 0. Potentiel d' ~ equilibre et capacite des en-
465
sembles.- Meddel.Lunds Univ.N~t.Sem., 1935, N 3, 1-118.
8. 0 a r I e s o n L. On the distortion of sets on a Jordan
curve under conformal mapping. - Duke Nath.J.j1973, 40, 547-559.
9. B r e n n a n J. The integrability of the derivative in con-
formal mapping. - J.London Math Sec., 1978, 18, 261-272.
10. B r e n n a n J. HHTeI~pyeMocT~ ~pO~SBO~HO~ KOH~p~oro OTO6--
pazeH~. -38n.H~.CeM~H.~0MM, 1978, 81, 173-176. 11. B e u r i i n g A. Quasianalyticity and general distributions.
Lecture Notes, Stanford Univ., 1961. J f
12. B e r n s t e i n S.N. "L~$ons sur los Proprietes Ext~males
et la Meilleure Approximation des Fonctions Analytiques d,une
Variable R~elle", G~uthier-Villars, Paris, 1926.
13. B r e n n a n J. Point evaluations, invariant subspaces and
approximation in the mean by polynomials, - J.Punctional Analy-
sis,1979, 34, 407-420.
14. D o m a r Y. On the existence of a largest subharmonic mine-
rant of a given function. - Ark.~at.~1958, 3, 429-440.
J. BREN AN University of Kentucky
Lexington, Kentucky, 40506
USA
466
8.10. APFROXINATION IN THE ME~ BY HAP~[ONIC In/NOTIONS
We discuss analogues of the Vitu~kin approximation theorem [10]
for mean approximation by harmonic functions. We assume that ~ is
fixed, 4 < e<~ . We let X be a compact subset of ~ of posi-
tive Lebesgue measure, and we assume n>5 . If ~£~ , let
]~, (~)~ I~ ~ ~ : I I~- ~I< ~ }. All funct ions w i l l be real -valued. If t°O, 4} ,Ulet ~ "denote the vector space Of all polynomi-
als On ~ which are homogeneous of degree ~ , with inner product
Zf k ~ {4,~} is fixed, define the (positive) function G k
L~ t~? ~) k' aS %he inverse Pourier transform of Ck(~)=
= (i i" i~I ~ )- I~ A C ~ define the Bessel capacity , and for each
~ ,f)1, (~) = Jwl~ [ II ~ II Lp,(~) measurable and ~ 0 On
Q on At i f kp' < M, , the r e e= s s a co -
ta~t C >0 such that C -~ .< ~Kp (B~t°))/~IP'-k "< C
f o r 0 < % ~< ~ . See [7 ] . [8] .
We say thmt X has the ~P h a r m o n i c a p p r o x i m a -
t i o n p r o p e r % y ( ~ h.a.p.) provided that for each
> 0 , stud each function ~ ~ LP(X) which is harmcnlo on the
interior ~ A , there exists a harmonic function ~ on an open
neighborhood of X such that l[ ~- { II J(X) < ~ "
THEOREM 1. I f any one of the f c l l ow in~ condi t ions holds, the~
X ~s ~he L P h..,,p.
a) ( [ 8 ] , [ z ] ) p'>~v,
b) ([8]) e'< ~ and there exists a constant ~'0 such that
~4,p,(B~tm)\X)>~!2~ ~tP'-4 if me~X and 0<~.<~ @
c) ( h i , [51 ) ~or each k ~ {i ,~} one of the fo lZc~±~ two con-
a~t~o~ is .et : (±) kp' > v~ o__~ (~i) kp' .< m and there ~is~8,...~
se~ E k with ~K,p' (Ek) : 0 such that
467
0 See also [5 , Theorem 6~; it follows from [8, Theorem 2.7] that
the condition in c) for k= ~ is necessary for the ~ h.a.p., but
Hedberg has pointed out that the condition in c) for ~=4 is not
necessary (see [I , Section 2 ] .) To characterize the sets having
the ~P h.a.p, we define other capacities. We use the notation
< T, ~ > tO denote the action of the distribution 7 of compact
support on the function ~ ~ ~--(~n) . T et E(gC) = 0~/[~ ~-
be a fundamental solution for a . Let ~ be a subset of the open
where the supr~am is taken over all (real) distributions ~ on ~
such that the support of ~ is a compact subset of A , E,T~
~(~,~)aad IIE*TII[~(~]. If He~ \ {0] ,we define ~p,H(A)~)--
= s~PI<T , H> 1 , where the supremum is taken over all distributions
T ~ on ~ satisfying the following four conditions: (i) the sup-
port of V is a compact subset of ~ ; (ii) <T,~> =0 ; (iii)<T,P)
---- 0 for each P c ~ satisfying [ H, D}=O ;
(iv) E .To LP( ,Bc) ane i lE,Tl lu{_q.) ..< 4 . ~ o r references
to related capacities of Harvey-Polking, Hedberg, and Maz'Ja, see
[1, Section 2, Remark 3].
The capacity ~ p~ is closely related %o the Bessel capacity
~,p' . Moreover, if ~ , {~,~] , one can prove that there
exists a constant C> 0 such that
result follows from the proof of [I, Theorem 2.1], with obvious
changes since here our functions are real-v~lued; the proof is conSt-
ructive, extendimg techniques of Lindberg [6] which are based on
those of Vitu~kin [10].
468
THEOREM
a) ~ has the h,a,p.
b) If H~o U~ \{0} , and if G
of ~ satis~.~ing G~ 2 ~ , then
c) There exist numbers ~ > 0 and
2. The followin~ conditions are equivalent.
an d I~ are o,p,en subsets
[p,H CG
~ 7 0 such that
.
From Theorem 2 it is possible to deduce part a) of Theorem 1
(see ~I, Section 2]) and part b) of Theorem I; however, we have not
been able to deduce part c) of Theorem 1, and this forms a motivation
for the first problem below. We also note that a motivation for prob-
lems I and 3 is provided by corresponding results in the theory of
mean approximation by analytic functions; see the references in the
first paragraph of ~I~. A number of other papers related to the pre-
sent note are also given in the references in [I].
PROBLEMS. I. Can one characterize the compact sets havin~ the L P h,a.p, by means of conditions of Wiener type? Specifically, let
us say that X has property C*) provided that for each ~{~,~}
one of the following two conditions is met: (i) kp'>~ or
(ii) kp1% • and there exists a set E k with ~K,p' CEk) =O
such that
0
and :aSX\E k.
,{01
If ~ has the ~ h.a.p., then ~ has property ~*) ; this follows
from Theorem 2, (I) and the Kellogg property ~5, Theorem 2]. Our
question is whether the converse holds: if ~ has Rro~erty (~) ,
does it have the L e h.a,p.? We remark that if this question were
answered affirmatively, then part c) of Theorem I would follow by
use of (I).
469
2. If p=~ and ~5 , a different criterion for the ~P
h.a~. follows from work of Saak [9]. What is the relation between
saak,s work and Theorem 2?
3. If H ~ 0 U~ ~{0} and ~ is an open set. then
o~j~(', ,~') is an increasing set function defined on the subsets
. Can one characterize the compact sets ~vin~ the ~ h.a.p.
b,~ means of inoreasim~ set functions which are countabl,~ subadditive
a_nd have the propert,y that all Bore! sets are oapacitable? (To say
that a set ~ is capacitable with respec~ to a set function
eans t h a t K compact, °
open, ~ DE ~. ; See [ 3 ] , [ ~ ] . Per the case p=~ , see [9 , Le.~ma 2].
REFERENCES
I. B a g b y T. Approximation in the mean by solutions of ellip-
tic equations. - Trans.Amer.~ath.Soc.
2. B y p e H ~ O B B.~. 0 np~6mm~H~ ~yH~m~ ~3 npocTpa~CTBa W~ (i~I ~z~ma~ ~ys=~ ~ npo~sBox~oro o ~ H ~ o ~ ~ O -
ZeoTBa ~ . -- Tpy~N MaTeM.~--Ta NM.B.A.CTeF~oBa AH CCCP,1974, ISI, 51-63.
3. c h o q u e t G. Porme abstraite du theoreme de capacitabili-
re. -Ann.Inst.Pourier (Grenoble), 1959, 9, 83-89.
4. H e d b e r g L.I. Spectral synthesis in Sobolev spaces, and
uniqueness of solutions of the Dirichlet problem. - Aota Math.,
1981, 147, 237-264. 5. H e d b e r g L.I., W o 1 f f T.H. Thin sets in nonlinear
potential theory.Stockholm, 1982. (Rep.Dept.of Math.Univ. of
Stockholm, Sweden, ISSN 0348v7652, N 2~.
6. L i n d b e r g P. A constructive method for ~? -approxima-
tion by analytic functions. - Ark.for Mat., 1982, 20, 61-68.
7. M e y • r s N.G. A theory of capacities for functions in Le-
besgue classes.- Math.Scand., 1970, 26, 255-292.
8. P o 1 k i n g J.C. Approximation in ~? by solutions of el-
liptic partial differential equations. - Amer. J.Math., 1972,
94, 1231-1244.
9. C a a E %.M. ~oc~ Ep~Tep~ ~ odxacT~ c yc~o~mBO~ sa~a-
Ee~ ~p~xxe ~ SJ~nT~qecENx ypaBHeH~ BNC~HX ~op~EoB. -- Ma-
470
~em.c6., 1876, 100(142), ~ 2 (6), 201--208o I0. B H T y m E ~ H A.Y. AHaJn~T~ecEa~ eMEOCT~ MHO~eOTB B s~a~ax
Teop~ ~p~0~eH~. - Ycnex~MaTe~.~ayE, I967, 22, B~n.6, 14I--
-199.
THOMAS BAGBY Indiana University
Department of Mmthmmatics
Bloomington, Indiana 47405, USA
EDITORS' NOTE. Many years before the appearance of C6S the cons-
tructive techniques of Vitushkin was applied to the L P -approxima-
tion by analytic functions by S.O.Sinanyan (CMHaH~H C.0. ANHpOECHMa-
~HS aHa~HTMqecEH~ ~tHE~H~MH H HO~MHOMaMH B cpe~HeM no n~o~H. - Ma~eM.c0., I966, 69, ~ 4, 546-578. )See also the survey MexbHHEOB
M.C., CwaH~ C.0. BonpocH TeopHH rrpH6~ixeH~i~ SyH~ O~HOI'O Komn~ex-
CHOre nepeMeHHoPo. - B KH: "I~TOrH HayKH ~ TeXHHF~". CoBpemeHHme npo_
6~eM~ MaTemaTMKH, T.4, MOcKBa, 1975, HS~-BO B~T~, 143-250.
471
8.11. old
RATIONAL APPRO~TION 0F ANALYTIC P~CTIONS
1. Local approximations. Let
l ¢ ) O
and let ~ be a complete analytic function corresponding to the ele-
ment ~ , For any ~ ~ define ~(~)= s~p{~(~-~): ~ ~
where ~ ($) is the multiplicity of the zero of ~ at ~ is
the set of all rational functions of degree at most ~.
Por any ~ there exists a unique function 9~, ~ ~
such that ~n(~)= ~ (I-~) " It is called the ~-th diagonal
Pad~ approximant to the series (1). Let ~ @ > ~ be an arbitrary fi-
xed number and let I.I=@ -%(') ; then ~ is the function of the
best approximation to ~ in ~with respect to the metric: ~(~)=
detailed discussion on the Pad~ approximants (the definition in [2] slightl~ differs from the one given above).
For any power series (I) we have
A being an infinite subset of ~ depending on ~. A functional analogue of the well-known Thue-Siegel-Roth theorem
(see [3], Theorem 2, (i)) can be formulated in our case as follows:
is ~ is an element of an algebraic nonrational function ~;hen for
any ~ 8e~ ~ , the inequality ~ (~) > ~8~ holds on2y for a
finite number of indices ~ , ~rom this it follows easily that in
01.L~ e a s e
-4
C3)
Apparently, this theorem is true for more general classes of ana-
lytic functions.
CONJECTURE I. If ~ is an element of a multi-valued analytic
f un~ction ~ with a finite set of singular points then (3) is valid.
472
In connection with CONJECTURE 1 we note that if
~-47~(I ) =+~ (4)
then ~ is a single-valued analytic function; but for any A,A>0 the inequality ~ - 4 9~(~) > A is compatible with the fact
that ~ is multi-valued (the first assertion is contained essential-
ly in [4],[5], the second follows from the results of Polya [6]).
Everything stated above can be reformulated in terms of sequen-
ces of normal indices of the diagonal Pad6 approximations (see[7],
[I] ). In essence the question is about possible lacunae in the se-
quence of the Hankel determinants
5~.
Thus (3) means that the sequence { ~ } has no "Hadamard lacunae" and (4) means that {F~} has "Ostrowski lacunae" (in the terminology
of [8]), Apparently many results on lactmary power series (see [8])
We their analogues for diagonal Pade approximations.
2. Uniformapproximation, We restrict ourselves by the corres-
ponding approximation problems on discs centered at infinity for the functions satisfying (1). Let ~ >~ l , ( is ho- lomorphic on E ) and ~R = {E: } " . Denote by 2 , ~ ) the best approximation of ~ on E by the elements of ~n : "
~(~)=H {I I~-%~E: % ~ ~ ~, U'~ is the sup-
n o : l ~ o n E e
Let ~ be the se t of a l l compact8, F , F C ~)R (with the con- nec t ed complement) such that ~ admits a holomorphic (single-valued)
continuation on C \ F , Denote by C D (F) the Green capacity of
F with respect to ~ (the capacity of the condenser (E, F) ) and define
•
473
For every
This inequality follows from the results of Walsoh ([9], oh.VIII).
CONJECTURE 2. For ar4y
c t7/ Inequalities (5), (6) are similar to inequalities (2). To clari-
fy (here and further) the analogy with the local case one should
pass in section I from 9~ to the best approximations k~ . In par-
ticular, equality (3) will be written as
i D CONJECTURE 3. I_~f $ is an element of an analytic f~qtio B
which has a finite set of sin~u, lar points then
(7)
| If tmder the hypothesis of this conjecture $ is a single-valued
analytic function, both parts of (7) are obviously equal to zero.
CONJECTURE 3 can be proved for the case when all singular points
of 4 lie on ~ ( fo r the case of two singular points see [10]).
In contradistinction to the local case the question of validity
of (7) remains open for the algebraic functions also.
REFERENCES
1. Perron
1957. 2. Baker
1975.
O. Die Lehre von den Kettenbx4/chen, II, Stuttgart,
G.A. Essentials of Pad~ Approximant, New-York, "AP",
474
3. U c h i y a m a S. Rational approximations to algebraic func-
tions. - Jornal of the Faculty of Sciences Hokkaido University,
Serol, 1961, vol.XV, N 3,4, 173-192.
4. r o H ~ a p A.A° ~OEa~BHOe yC~OB~e O~HO3Ha~HOCTE aHaJn~T~eCEEX
~y~. -MaTeM.C6., 1972, 89, 148-164.
5. r o H ~ a p A.A. 0 CXO~MOCTH 2n~ROECEMsrU~ ~e. - ~2TeM.C6.,
1973, 92, 152-164. 6. P e I y a G. Untersuchungen uber Lucken und Singularitaten
yon Petenzreihen.- Math.Z., 1929, 29, 549-640.
7. r o H ~ a p A.A. 0 C~O~MOCTH annpoEcHMa~ Ha~e ~ HeEoTopHx
ExaCCOB MepoMop~m~X $ ~ . -- MaTeM.Cd., I975, 97, 605 -
- 627. 8. B i e b e r b a c h L. Analytische Fortsetzung. Berlin - Hei-
delberg, Springer-¥erlag, 1955.
9. W a I s h J.L. Interpolation and approximation by rational
functions in the cemplex domain. AMS Coll.F~Bl., 20, Sec.e~.1960.
I0. r o H ~ a p A.A. 0 cEopocTH pan~oHax~HO~ annpoEc~Mau~ EeEo-
Top~x aHaJL~TEeCE~X ~y2E~. -MaTeM.C6., I978, I05, I47-I88.
A,A. GONC~AR CCCP, 117966, Mocxma
(A.A.r0RNAP) yx.Bamzxoma 42, CCCP.
475
8 . 1 2 . old
A CONVERGENCE PROBLEM ON RATIONAL APPROXIMATION
IN SEVERAL VARIABLES
1. The one-variable case, ~e 6 . Let me first give the back-
ground in the one-variable case. Let ~(~)~ ~ C~ ~ , Re 6 ,
be a formal power series and P/Q , Q ~ 0 , a rational function
in one variable ~ of type (~,~) , i.e. P is a polynomial of
degree ~ ~ and Q of degree ~ 9 . It is in general not possible
to determine P/Q so that it interpolates to ~ of order at least
~* 9 @ I at the origin (i.e. having the same Taylor polynomial of
degree ~v 9 as ~ ). However, given ~ and ~ , we can always
find a unique rational function P/~ of type (~,~) such that
? interpolates to ~Q of order at least ~+ ~+ ~ at the
origin, i.e. (~-P)(~)~0(~+$+1). This function P/~ , the[~- 1
P a d e a p p r o x i m a n t to ~ , was first studied systema-
tically by Pade in 1892; see [I]. In 1902 Montessus de Ballore [2]
proved the following theorem which generalizes the well-known result
on the circle of convergence for Taylor series.
THEOREM. Suppose ~ is holomorphlc at the origin and meromor-
Rhic in ~I<~ with ~ poles (counted with their multiplicities),
Then the [I,,#3 -Pade approximant to ~ , ~ / ~ , conver~es uni-
formly to ~ , with ~eometric de~ree of convergence I in those com-
pact subsets of I~I <~ which do not contain ar47 poles of ~ .
With the assumption in the theorem it can also be proved that
P /Q diverges outsiae I I= if is chosen as large as
possible [3, p.2693 and that the poles of Pw / ~w converge to the
poles of { in l~l <~ • Furthermore, when ~ is sufficientS,y
large, ~ /Q~ is the -n~que rational function of type (~, ~)
which interpolates to ~ at the origin of order at least I,+9 + I.
Montessus de Ballore's original proof used Hadamard's theory of po-
lar singularities (see [4]). Today, several other, easier proofs are
known; see for instance [51,[6] ,[7] and [8].
Pad6 approximants have been used in a variety of problems in nu-
merical analysis and theoretical physics, for instance in the numeri-
cal evaluation of functions and in order to locate singularities of
functions (see [I] ). One reason for this is, of course, the fact that
the Pad6 approximants of ~ are easy to calculate from the power
series expansion of ~ . In recent years there has been an increas-
ing interest in using analogous interpolation procedures to apprexi-
476
mate functions of several variables (see E9~). I propose the prob-
lem to investigate in which sense it is possible to generalize Mon-
tessus de Ballore's theorem to several variables.
2. The two-variable case, ~(~i,~); ~,~e~.
We first generalize the definition of Pad6 approximants to the two-
variable case. Let ~(~)~ ~ G~K~ ~ ~ be a formal power series
and let ~/~ , ~0 , be a rational function in two variables ~I
and ~ of type (~,$) , i.e. P is a polynomial in ~I and ~ of
de~ree ~< ~ and ~ of degree ~ ~ . By counting the number of coef-
ficients in P and ~ we see that it is always possible to deter-
mine P and 0 so that, if(~f-P)(~)~ K ~ , then
~K~-O for (~,K)6 ~ , where ~ , the interpolation set, is a
chosen subset of ~ x ~ with ~(We~)(~+$)+~(9+~)(9+~)-
elements. There is no natural unique way to choose ~ but it seems
reasonable to assume thatI(~,K):~+K~W} c~ and that(~,K)~
=>(~,W%) 6 ~ if ~ ~ } and ~ K . In this way we get a
r a t i o n a i a p p r o x i m a n t P/@ of type (w, 9)
to corresponding to . With a s table choice of , P/Q
is unique [7 , Theorem 1.I~. The definition, elementary properties,
and some convergence results have been considered for these and simi-
lar approximants in [9], ~0] and [7]. The possibility to generalize
Montessus de Ballore's theorem has been discussed in [6],[73 and E11]
but the results are far from being complete.
PROBLEM I. In what sense can Montessus de Ballore's theorem b@
~eneraliz~d to several variables?
I% is not clear what class of functions ~ one should use. We
consider the following concrete situstion. Let ~=~/6 , where
is holomorphlc in the polydisc~=(~,~): l~I<~, ~=~,~ ~ and &
is a polynomial of degree 9 , ~(0) ~ 0 • By the method described
above we obtain for every ~ a rational approxin~nt PW/Qw of
type (~,~) to # corresponding to some chosen interpolation set
~= ~w" In what region of ~ does Pw/~ converge to ~ ? Partial
answers %o this problem are given in [7] and [11] (in the latter with
a somewhat different definition of the approximants). If @=~ , ex-
plicit calculations are possible and sharp results are easy to obtain
[7 , Section 4~. These show that in general we do not have convergen-
ce in{~: I~l<~, $=~,~\{~:e(~)~---O} . This proves that the general
Analogue of the Montessus de Ballore's theorem is not true. It may be
477
added, that it is easy to prove - by just using Cauchy's estimates -
that there exist ration~l functions ~$ of type (~,~) interpola-
ting to ~ at the origin of order at least ~+ I and converging
ifo=ay, as , to in compact subsets
A disadvantage, however, of ~$ compared to the rational ap-
proximants defined above is that %~ is not possible to compute
from the Taylor series expansion of ~ (see ~ , Theorem 3.3~).
In the one-variable case the proof of Montessus de Ballore's
theorem is essentially finished when you have proved that the poles
of the Pad~ approximants converge to the poles of ~ . In the seve-
ral-variable case, on the other hand, there are examples ~ , Section
4, Counterexample 2];when the rational approximants P~/~ ~ do
not converge in the whole region ~ \| ~: ~(,)=0} in spite of fact that the singularities of P l ~ / ~ converge to the
singularities of ~/~ . This motivates:
PROB~ 2. Und.er what c ondi, t.!ons does Q~ conver~e to ~ ?
The choice @f the interpolation set ~t~ is important for the
convergence. For instance, if $-----J and ~i_~_~_~_oo , we get con-
vergence in ~\ ~ ~: ~(~)=0 } with a suitable choice of ~
ET, Section 4~. On the other hand if we change just one point in ~-
without violating the reasonable choices of ~ indicated in the de-
finition of the rational approximants - we get examples [7, Section
4, Cotuuterexample 1],where we do not have convergence in any poly- disc around ~0 .
PROBLE~ 3. How is the convergence P~/Q. ............... ~ ~ influenced b~
the choice of the interpolation s~t ~ .~
Since we do not get a complete generalization of Montessus de
Ballore's theorem it is also natural to ask:
PROBLEM 4. If, the sequence .o,f rat!ona,l,, ,approximants aces not
conver~e t is there a subs equence t~t c,onv~rges _t0 ~ ?
(Compare ~7, Theorem 3.4] )
l~inally, I want to propose the following conjecture.
and the interpolation set ~ is suitably chosen. (Compare Ell,
Corollary 2~ and the case 9= ~ referred to just after PROBLE~ 2 above. )
478
REFERENCES
I. B a k e r C.A. Essentials of Pad6 Approximants. New York,
Academic Press, 1975.
2. d e M o n t e s s u s d e B a I I o r e R. Sur les
fractions continues alg~briques. - Bull. Soc•Math. France ~ 1902~
30, 28-36.
3- P e r r o n O. Die Lehre yon den Kettenbr~chen. Band II.
Stuttgart, Teubner, 1957.
4. G r a g g W.B. On Hadamard's theory of polar singularities. -
In: Pad6 approximants and their applications (Graves-Morris,
P.R., e.d.), London, Academic Press, 1973, 117-123.
5. S a f f E.B. An extension of Montessus de Ballore's theorem
on the convergence of interpolation rational functions. - J.
Approx.T., 1972, 6, 63-68.
6. C h i s h o 1 m J.S.R., G r a v e s - M o r r i s P.R. Gene-
ralization of the theorem of de Montessus to two-variable appro-
ximants. -Proc.Roy~l Soc.Ser.A., 1975, 342, 341-372,
7- K a r 1 s s o n J., W a 1 1 i n H. Rational approximation by
an interpolation procedure in several variables.- In: Pad~ and
rational approximation (Saff, E.B. and Varga, R.S., eds.), New
York, Academic Press, 1977, 83-100.
8. r o H q a p A.A. 0 CXO~HMOCT~ O606~eHR~X annpo~c~Many~ ~a~e
Mepo~opSHax ~yHE~. -~4aTez.cS., 1975, 98, 4, 563-577. 9. C h i s h o 1 m J.S.R. N -variable rational approximants. -
In: Pad6 and rational approximation (Saff, E.B. and Varga, R.S.,
eds.), New York, Academic Press, 1977, 23-42.
IO. F o H ~ a p A.A. JIOEaKBHOe yC~OB~e O~O3HaqHOCTE aHa~I~T~lqeo-
K~X ~ y ~ Hec~o~X nepeMeHHax. - ~Te~.C6., 1974, 93, ~ 2,
296-313. 11. G r a v e s - M o r r i s P.R. Generalizations of the theorem
of de Montessus using Canterbury approximant. - In: Pad6 and ra-
tional approximation (Saff, E.B. and Varga, R.S., eds.), New York,
Academic Press, 1977, 73-82.
HANS WALLS Ume~ U n i v e r s i t y
S-90187 Ume~, Sweden
479
COMMENTARY BY THE AUTHOR
In a recent paper A.Cuyt (A Montessus de Ballore theorem for multivariate Pad~ approximants, Dept. of Math., Univ. of Antwerp,,_ Belgium, 1983) considers a multivariate rational approximant ~/ to ~ where ~ and Q are polynomials of degree ~÷~ and ~$~ , respectively, such that all the terms of P and Q of degree less than~$ vanish. It is then possible to determine P and Q so that ~Q- has a power series expansion where the terms of degree ~+~+~ are all zero. Por this approximant P/Q she proves the following theorem where P/Q- P~ / Q~ and ~ and ~ have no common non-constant factor: Let ]=F/6 where F i , holo- morphic in the polydisc {Z:IZ~I(R~} and ~ is a polynomial of de- gree ~ , ~(0) ~0 , and asstm~e that ~(0)~0 for infini- tely many ~ . Then there exists a polynomial Q(Z} of degree $
uch u oo uo oo of{P,/Q4 that converges ,~niformly to ~ on compact subsets of [;~.lzi, I <
480
8 .13 . old
BADLY-APPROXI~L~BLE FUNCTIONS ON CURVES AND REGIONS
Let X be a compact Hausforff space and ~ a uniform algebra
on X : that is, A is uniformly closed, separates points, and con-
tains the constants. Pot example, if Xc ~ then we might take
A~ P(X) , the uniform limits on X of polynomials. We say that a
function ~ , ~ C(X) , is b a d 1 y - a p p r o x i m a b 1 e
(with respect to ~ ) to mean
where U • ~ i s the supremum norm ove r X . The problems d i scussed
he re concern f i n d i n g c o n c r e t e d e s c r i p t i o n s o f t he b a d l y - a p p r o x i m a b ! e
functions for some classical function algebras. They are the func-
tions that it is useless to try to approximate.
In this section, we let ~ be a bounded domain in C , with
boundary X , and let A~(X) be the algebra of boundary values of
continuous functions on ~U X that are analytic in ~ . In case
is the open unit disc, then A*(X) is the "disc algebra" (regarded
as consisting of functions on X and not on ~ ).
POREDA'S THEOREm. EI~. I_~f X consists of a simple closed Jordan
curve, then ~ , ~*s G(X) , is badl,T-approximable ' with respect toA*(X)
if and only if ~ has nonzero constant modulus, and ¢~ ~P~ 0 .
Here, ¢14~ ~0 is the index of ~0 , defined as the winding num-
ber on X of ~ around 0.
THEORE~ A. [2] I_~f @ , ~06 ~(X) has nonzero constant mo~dulus
and if ~ ~ 0 , then ~ i s badl,7-approximable with respect to
THEORE~ B. [2] Each badly a~oroximable (with respect to ~*(X))
function in ~ (X) has constant modulus on the boundary of the comp-
lement of the closure of ~ .
THEOREM C. [2] Suo2ose that X consists of ~÷ I dis,~oint
closed Jordan curves. If ~ is badly-approximable with respect to
A*(~) , then ~ has constant modulus, and ~Q[< N •
An example was given in ~ to show that the range 0 ~ ~
< N is indeterminate, so that one cannot tell from the winding num-
ber alone, on such domains, whether or not ~0 is badly approximable.
481
PROBLEM I. ~ind necessary and sufficient conditions f o r a func-
tion ~ ~gbe badly-ap~roximable with respect to ~*(X) if X is a
finite union of disjoint Jordan curves.
Note: In the case of the annulus,~=I~:I~l=$ or I~=~ } ,
where 0 <%~ , supposing ~ is of modulus ~ on X , it is shown
in ~] that ~ is badly-approximable with respect to ~*(X) if and
only if either $~ ~< 0 or i ~ 0 and
.l
PROBLEM II. The analogue of Problem I for ~*(X) , which is the
limits on ~ of rational functions with p0!es off ~ , where one
permits G to have infinitely many hole s.
PROBLEM III. Characterize the b adl~-approximable funct~gns with
respect to P(X) , where X is any compact set in ~ .
PROBLEM IIIr. The same as problem III but in the specielgase
X= D m
Despite appearances, Problem III' is just about as general as
Problem III. An answer to Problem III could be called a "¢o-MergelyBn
theorem" since Mergelyan's theorem [3] characterizes the "well-appro-
ximable" functions on X .
THEOREM. E4]- If ~ is badly-approximable with respect t 9
P( O~e6 ~) then ~ ~--~ II@~ ~ ~ II ~ JJ e@ wher_~e If" ~e@ is the . supre-
mum norm over ~ . The converse is false~
PROBLEM IV. Obtain I fo r sets X , Xc~ ~ , ~ , any signi-
ficant result about badly-approximable functions with respect to a~¥
algebra like P(X), A(X) , ~ ~(X).
RE~ERENCES
1. P o r e d a S.J. A characterization of badly approximable func-
tions. - Trans.Amer.Math.Soc. 1972, 169, 249-256.
2. G a m e 1 i n T.W., G a r n e t t J.B., R u b e 1 L.A.,
S h i e 1 d s A.L. On badly approximable functions. - J.Approx.
482
theory, 1976, 17, 280-296.
3. R u d i n W. Real and Complex Analysis, New York, 1966.
4. K r o n s t a d t E r i c. Private communication, September
1977.
5. L u e c k i n g D.H. On badly approximable functions and uniform
algebras. - J.Approx.theory, 1978, 22, 161-176.
6. R u b e 1 L.A., S h i e 1 d s A.L. Badly approximable func-
tions and interpolation by Blaschke products. - Proc.Edinburgh
Math.Soc. 1976, 20, 159-161.
LEE A. RUBEL University of Illinois at Urbana-
-Champaign Department of Mathematics
Urbana, Illinois 61801 USA
483
8.14. EXOTIC JORDAN ARCS IN ~ old
Let ~ be a simple (non-closed) Jordan are in ~(~), ~(~)
be the closure in C(~) of polynomials in complex variables,
0(~) be the uniform closure on ~ of algebra of functions holo-
morphic in a neighbourhood of ~ . Denote by A(~) a uniform al-
that ~C~)~ gebra on such and let ~A(~) be
its spectrum (maximal ideal space). For an arbitrary compact set
in C ~ the spectrum ~A(~) depends essentially on the choice of
the subalgebra A(K) arc; Until recentl~however,~ it seemed plau-
sible that for Jordan the spectrum ~A(~ ) depends on
only.
Consider also the algebra ~(~) of uniform limits on ~ of
rational functions with poles off ~ , and the algebra m (~)
which is the closure in C(~) of the set of all functions holomor-
phic in a pseudoconvex neighbourhood of ~ . Then we obviously have
In 1968 A.Vitushkin (see D, 2]) discovered the first example of
a rationally convex but not polynomially convex arc ~ in ~2 . In
other words in this example
~C~)=~, but ~pC{)~ ~.
In 1974 the author (see [~, p.116; [~, p.174) found an example
of Jordan arc [ in ~ which~being holomorphically convex cannot,
nevertheless, coincide with an intersection of holomorphically con-
vex domains, i .e. __~oC~)=~ but ~HC~) ~ ~ . A curious problem remains, however, unsolved. Namely, whether
~(~)= ~M(~) for every Jordan arc.
CONJECTURE I. There exists a Jordan arc ~ i_~n ~ satisfyin~
Consider now the algebra A(K,S) of all functions continuous
on the Riemann sphere ~ and holomorphic outside a compact set K ,
K e-~ . To prove conjecture I it is sufficient,for example, to
prove the following statement which simultaneously strengthens
the classical results of J.Wermer (see[4],[6]) and R.Arens (see [5],
[6]).
484
CONJECTURE 2. There exists a Jordan arc ~ o_.nn ~ such that
A(~ ~ ~) contains a finite!~generated subalgebra with the spect-
rum S •
All known exotic Jordan arcs in C~ are of positive two-dimen-
sional Hausdorff measure. It would be very interesting therefore
to p~ove that there is no exotic arc of zero two-dimensional Haus-
dorff measure.
PROBLEM. Suppose that a simple(non-closed) Jordan arc ~ i_~n
C ~ has zero 2-dimensional Hausdprff measure~ Is ~ polynomiall2
convex (i.e. ~O(;)=~ )?
Recall that H.Alexander [7] has proved that every rectifiable
simple arc in C ~ is polynomially convex.
RE I,~EREN CE S
I. B E T y m E ~ H A.r. 0(~ o;~Ho~ ss~a~e PyrrHa. - ~o~.AH CCCP,
1973, 213, ~ I, 14-15.
2. X e H K ~ H r.M., q ~ p E a E.M. PpaH~we CBO~CTBa rO~O-
Mop~HRX ~ HecEoJIBEFLX EO~JIeKCH~X nepeMeHE~X. - B ~H. : COB-
peMeHH~e npo6xemH MaTeMaT~, 4, M., BHHH~4, 1975, 13-142.
3. w e I I s R.0. ~unction theory on differentiable submanifolds.
- In: Contributions to analysis. A collection papers dedicated
to Lipman Bers, 1974, Academic Press, INC, 407-441.
4. W e r m e r J. Polynomial approximation on an arc in ~ . -
Ann.Math.~ 1955, 62, N 2, 269-270.
5. A r e n s R. The maximal ideals of certain function algebras.
- Pacific J.Iv~th.~ 1958, 8, 641-648.
6. G a m e 1 i n Th.W. Uniform algebras. Prentice-Hall, INC,
N.J., 1969.
7. A 1 e x a n d e r H. Polynomial approximation and hulls in
sets of finite linear measure in C~ . - Amer.J.Math., 1971,
93, N I, 65-74.
G. M. HENKIN
(r.M.X~(~H) CCCP, 117418, MOCEBa
yX, EpacMKoBa 32,
~eHTpax~m~ SEOHO~nEO--
MaTeMaT~ecE~HCT~TyT AH CCCP
485
8.15. REMOVABLE ~TS ~OR BOUNDED ANALYTIC FUNCTIONS old
Suppose E is a compact subset of an open set V , VEC •
Then E is said to be removable, or a Painleve null set [I], if
every bounded analytic function on V \ E extends to be analytic on
V . This is easily seen to be a property of the set E and not V • • t
Palnleve [2] asked for a necessary and sufficient condition for a
compact set E to be removable. The corresponding problem for harmo-
nic functions has been answered in terms of logarithmic capacity and
transfinite diameter. Ahlfors [3] has restated the question in terms
of the following extremal problem. Let
be the analytic capacity of E . Then E is removable if and only
if ~ <E)=0 . A geometric solution to this problem would have appli-
cations in rational approximation and cluster-value theory. See, for
example, [4] and [5]. Also [6] contains an interesting historical
account.
It is known that Hausdorff measure is not "fine" enough to cha-
racterize removable sets. Painlev~ (and later Bes!covitch ET]) proved 4
that if the 4-dimensional Hausdorff measure. H (E) , is zero then ~+~ > >
[(E)=0 . It is also classical that if M CE) 0 , for some 8 0,
then ~(E) >0 . However examples, [~,E9], show that it is possible
for M4CE)>0 and ~CE)=0 ~).
if ~ is the ray from the origin with argument @ , let
lPoCE)I denote the Lebesgue measure of the orthogonal projection
of E on ~@ . Let
o
This quantity first arose in connection with the solution of the
Buffon needle problem as given by Crofton ~ in 1868. If the diame-
ter of E is less than ~ , it is the probability of E falling on
a system of parallel lines one unit apart. See ~ for an interesting
geometric interpretation. Vitushkin [4] asked if C~<E) = 0 is
~) see also pp.346-348 of the book ~ - Ed.
486
equivalent to ~ (E)--0 . It is not hard to see that if
~ ] ' ( E ) - - 0 , then CR(E' ) - - -0 . ~Iarstrand [13] has proved that if MI~6(~) > 0 then C~(~) ~ 0 - In order to answer Vitushkin's questi-
on, one thus needs to consider only sets of Hausdorff dimension ~ .
A special case is the following theorem asserted by Denjoy [14]
in 1909.
THEOREM. If E is a compact subset .o.f a rectifiable curve
then ~(E)=0 if and only if H4(E)=0 Although his proof has a gap, Ahlfors and Beumling [I ] noted
that it is correct if ~ is a straight line. They extended this re-
suit to analytic curves ~ . Ivanov [163 proved it for curves slight-
ly smoother than CI • Davie [17] proved that it sufficed to assume
r is a C ~ curve. Recently, A.P.Calder~n [18] proved that the Cauchy
integral operator, for C 4 curves, is bounded on L P , ~< p < oQ .
Denjoy's conjecture is a corollary of this theorem. Here is an
OUTLINE OF THE PROOF.
Let ~ be a finitely connected planar domain bounded by C , a
union of rectifiable arcs C4,...,C~. Let F~ map the unit disk con- c
formally onto C~ and let C i = F~(I~I=~). We say that ~ , analytic
in ~ is in E~(~) if and only if ~ I~(~)~ I~l < co
and def ine II IIE, = f ~ , where C is t raced twice i f C
it is an arc.
LE&~ I. ( [19] ). I_.ff C consists of finitely many analytic cur-
yes, then
=
In this classical paper, Garabedian introduces the dual extremal
problem: ~(IIglIE~: ~ E E4(~), g(oo)= 4) to obtain the above rela- tion. It was noticed by Havinson [15] that the result remains true
for rectifiable arcs. If ~L~(C) let G(~)---~ ~(~) ~ " ' c ~-Z ~
L E ~ 2. ( [203). I f C i s the un,i,on o,f,,,,,,,,finitely many C 4 -
.curves and i f the Cauchy i n teg ra l ha.s b.ounda.ry va.l..ues G ~, J ~ J(C), th~n G ~ E~(~) .
487
This follows by writing ~ = )'~ where each ~ is analytic
off one of the contours in ~ . Then use the well-known fact
that ~ I~(~)I 2 16~,~1 increases with ~ if ~EH~(~).
C 4 of_. LE~L~ 3. ([21]). Let C be a curve. If for all ~ ,
we have ~E Eg(~) , then the length and capacity of a subset E o_~f
C are simultaneousl,y positive or zero.
This follows by approximating the set E by a subset E of C
consisting of finitely many subarcs, then applying Lemma 1 to the cha-
racteristic function of E •
Thus by Calder~n's theorem, Denjoy's conjecture is true for C ! curves. Davie's result finishes the proof. Incidentally an older theo-
rem of [25], p.267, immediately implies Davie's result. @
About the same time that Besicovitch rediscovered Paimlev~'s the-
orem (see above), he proved one of the fundamental theorems of geome-
tric measure theory. A set E is said to be r e g u 1 a r if it is
contained in a countable umion of rectifiable curves. A set E is
said to Be i r r e g u 1 a r if
H4(EnB( 't')) H4(EfqBC$'t')) f o r H - , , . , , . E,
4 where B ( ~ , t ) = { ~ : t ~ - ~ l . < t } . Besicovitch [22] proved that i f HCE)< oo then E =E UE where E~ is regular and E~ is irregular. Later
[23], he showed that if E is irregular, then the orthogonal projec-
tion of E in almost all directions has zero length. Thus if a4(E)<oo and C~(E)>0 there is a rectifiable curve F so that the length
of EO F is positive. Since Denjoy's conjecture is true, ~CE) is
positive whenever H4CE)<oo and C~<E) >0 •
All examples where the analytic capacity is known concur with
Vitushkin's conjecture. For instance, let E be the cross product
of the Cantor set, obtained by removing middle halves, with itself.
It is shown in [9] that ~(E)=0 . For each ~, DO ~E , one can
find annuli centered on ~ which are disjoint from E and propor-
tional in size to their distance from ~0 . Thus E is irregular and
C~E)=O . We remark that the projection of E on a line with
slope 4~ is a full segment. Another relevant example is the cross
product of the usual Cantor tertiary set with itself, call it F .
The Hausdorff dimension of F is greater than one so that ~F)>0
488
and ~(F)>o • However every subset F~ of F with H 4(F) <
is irregular and hence satisfies C~ (F)=0 . This shows we cannot
easily reduce the problem to compact sets E with H~(E) < ~.
I f ~(E)> 0 , one possible approach to prove CR(E)>o is
tO consider the set E -1-T-z E,0(e(2 A point ~ is not
in ~ if and only if the line passing through ~ and whose distan-
ce to the origin is I~I , misses the set E . It is not hard to see C R(E)> o i f i f E has posit ive area. Uy [ 24] *) has
recently shown that a set F has positive area if Bad only if there
is a Lipschitz continuous function which is analytic on ~ \ F . so one
might try to construct such a function for the set ~ , A related que-
stion was asbed by A.Beurling. He asked, if ~ (E)>o and i f E has
no removable points, then must the part of the boundary of the normal
fundamental domain (for the universal covering map) on the unit circ-
le have positive length? This was shown to fail in [26].
Finally, I would like to mention that I see no reason why C~(E)
is not comparable to analytic capacity. In other words, does there
exist a constant K With 4/K "C~(E)~ ~(E)~ ~ C~(E) ? If this were
true, it would have application to other problems. ~or example-, it
would prove that analytic capacity is semi-subadditive.
REFERENCES
I. A h 1 f o r s L.V., B e u r 1 i n g A. Comformal invariants
and f~uction-theoretic null sets. - Acta Math., 1950, 83, 101-129
2. P a i n 1 e v ~ P. Sur les lignes singuli~res des fonctions ana-
lytiques. -Ann,Fac.Sci. Toulouse, 1888, 2.
3, A h 1 f o r s L.V. Bounded analytic functions. - Duke Math.J.,
1947, 14, 1-11.
4. B m T y m ~ ~ H A.F . A ~ J ~ Z T m ~ e c ~ e eM~CTI, MHoxec~'B B s s ~ a x
Teolm~ ~pm6~ze~. - Ycuex~ ~mTeu.HsyK,I967,22,~, I4I-I99.
5. z a 1 c m a n L. Analytic capacity and Rational Approximation
- Lect.Notes Math., N 50, Berlin, Springer, 1968.
6. C o 1 1 i n g w o o d E.P., L o h w a t e r A.J. The Theory
of Cluster Sets. Cambridge, Cambridge U.P., 1966~
7. B e s i c o v i t c h A. On sufficient conditions for a functi-
on to be analytic and on behavior of analytic functions in the
neighborhood of non-isolated singular points. - Proc.London Math. Soc°, 1931, 32, N 2, I-9.
See [27] for a short proof. - Ed.
489
8. B ]~ T y m E B H A . r . lIpilMep MEtoxecT~t ZZO~ZZZTe~Z~Ot ~ m ~ , ~ro
Hy~eBoM SN~Jt~TB~eoEo~ eNEOOTg.-~0~.AH CCCP, 1959, 127, 246-249. 9. G a r n e t t J, Positive length but zero analytic capacity -
Proc.Amer.Math. Soc., 1970, 24, 696-699.
l O . H B a ]K o B ~ . ~ . BS1SI~aI~MB MHozecTB lit ~ y H l ~ ] ~ . M . , "Hayza",I975. 11. O r o f t o n M.W, On the theory of Local Probability. -Philos.
Trans.Roy.Soc., 1968, 177, 181-199.
12. S y 1 v e s t e r J,J. On a funicular solution of Buffon's
"Problem of the needle" in its most general form~ - Acts Math ,
1891, 14, 185-205.
13. M a r s t r a n d J°M. Fundamental geometrical properties of
plane sets of fractional dimensions. - Proc.London Math.Soc.,
1954, 4, 257-302.
14. D e n j o y A° Sur lea fonctions analytiques uniformes ~ sin-
gularit~s discontinues. -CoR,Acad.Sci,Pmris, 1909, 149, 258-260,
15. X a B M H C O H C.H. 06 ~aa~TM~ecEo~ e~EOCT~ MHoxecTB, CoBue-
O~Ol) HeTpMBaa~OOT~ ImSJta~KUX ~laccoB aHaJt~T~qecz~x ~ym~c~ J
~e~e ~Bap~a B npoMsBo~m~x 06~aOTSX. - MaTes.c6., 1961, 54,
~I, 3-50. I~ B a H o B ~.~. 06 aHa~Tsqecxo~ eKI~OOT~ ~J~He~h~x IE~O~eOTB.
--Ycnexa MaTea.Hay~, I962, I7, I43-I44. D a v i e A.M. Analytic capacity and approximation problems. -
Trans.Amer.Math.Soc., 1972, 171, 409-444.
C a 1 d e r ~ n A.P. Cauchy integrals on Lipschitz curves and
related operators. - Proc.Nat,Acad.Sci, USA, 1977, 74, 1324-1327
G a r a b e d i a n P.R, Schwarz's lemma and the Szeg'o kernel
function. -Trans.Amer.~lath,Soc=, 1949, 67, 1-35.
X a B ~ H B.H. l~aHa~Hue CB0~OTB8 asTerpa~oB Tana Koma a rapao- ~ecEa Conp~eHHuX ~mU~a~ B 06~a0TSX C0 c~pe~seao~ rpaHa~e~.
-~aTe~.c6., 1965, 68, 499-517.
Xa Bz H B.H., Xa B ~ H c o H C.~. HeEOTOI~e o~eH~a aHa-
~a~ec~o~ eaEoc~z. -~o~.AH CCCP, 1961, 138, 789-792. B e s i c o v i t c h A. On the fundamental geometrical proper-
ties of linearly measurable plane sets of points I. - Math.Ann.,
1927, 98, 422-464. II: Math.Ann., 1938, 115, 296-329.
B e s i c o v i t c h A. On the fundamental geometrical proper-
ties of linearly measurable plane sets of points III.- Math.Ann.,
1939, 116, 349-357.
U y N. Removable sets of analytic functions satisfying a Lip-
schitz condition. -Ark.Mat., 1979, 17, 19-27.
e d e r e r H. Geometric measure theory. Springer-Verlag, Bar-
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
490
lin, 1969.
26° N a r s h a 1 1 D°E. Painlev~ null sets, Colloq, d'Analyse Har-
monique et Complexe. Ed,: G.Detraz, L,Gruman, J.-P.Rosay. Univ.
Aix-Marseill I, Marseill, 1977.
27. X p ~ ~ e B C.B. HpOcTOe ~oz, asa~em~cTBo ~eopeuu o0 yCTIm~M~X ooo6e~ooT~ a E a ~ e c z ~ x ~ m o ~ , ~o~.~e~Bop.mo~x yO~OB~ ~Lmn- =az~a. - 8a~ .Ha~.oea .~0MN, I 9 8 I , I I 3 , I 9 9 - 2 0 3 .
DONALD E. MARSHALL Department of MathematicspUniversity
of WashingtonjSeattle, Washington 98195
USA
Research supported in part by National Foundation Gramt
No MCS 77-01873
491
8.16. ON PAINLEV~ NULL SETS old
Suppose that E is a compact plane set and that ~ is an open
neighbourhood of ~ . i set is called a P a i n 1 e v e n u 1 1
s e t (or 2.N. set) if every function regular and bounded in ~\ E
can be analytically continued onto E . In this case we also say
that E has z e r o a n a 1 y t i c c a p a c i t y.
The problem of the structure of P.N. sets has a long history.
Painlev~ proved that if E has linear (i.e. ~ - dimensional Haus-
dorff) measure zero, then E is a P.N. set, though it seems that
this result was first published by Zoretti [I]. " s Painleve' theorem
has been rediscovered by various people including Besicovitch [2]
who proved that if ~ is continuous on E , as well as regular
outside E , and if E has finite linear measure, then ~ can be
analytically continued onto E . Denjoy [3] conjectured that if E
lies on a rectifiable curve, then E is a P.N. set if and only if
E has linear measure zero. He proved this result for linear sets.
Ahlfors and Beurling [~ proved Denjoy's conjecture for sets on ana-
lytic curves and Ivanov [4 for Sets on sufficiently smooth curves.
Davie [6] has shown that it is sufficient to prove Denjoy's conjectu-
re for C 4 curves. On the other hand Havin and Havinson [~ and Ha-
vin [8] showed that Denjoy'sconjecture follows if the Cauchy integ-
ral operator is bounded on ~ for ~ curves. This latter result
has now been proved by Calder6n [gB so that Denjoy's conjecture is
true. I am grateful to D.E.Narshall [10] for informing me about the
above results.
Besicovitch [11] proved that every compact set ~ of finite
linear measure is the union of two subsets ~I ' ~ . The subset E~
lies on the union of a finite or countable number of rectifiable
Jordan arcs. It follows from the above result that ~4 is not a P.N.
set unless ~4 has linear measure zero. The set ~ on the other
hand meets every rectifiable curve in a set of measure zero, has
projection zero in almost all directions and has a linear density at
almost none of its points. The sets E4 and ~ were called respec-
tively r e g u 1 a r and i r r e g u 1 a r by Besicovitch
[11]. Since irregular sets behave in some respects like sets of mea-
sure zero I have tentatively conjectured [12, p.231] that they might
be P.N. sets. Vitushkin [13] and Garnett [14] have given examples
of irregular sets which are indeed P.N. sets, but the complete conjec-
ture is still open.
A more comprehensive conjecture is due to Vitushkin [15 , p.14~
492
He conjectures that E is a P.N. set if and only if ~ has zero
projection in almost all directions.
It is not difficult to see that a compact set ~ is a P.N. set
if and only if for every bounded complex measure distributed on E ,
the function
E is unbounded outside E ~). Thus E if there exists a positive unit measure
f t~-gl E
is bounded outside E i.e. if E
p.73]. This is certainly the case if
respect to some Hausdorff function
9,-----i-~ < oo
0
(I)
is certainly not a P.N. set
on E such that
has positive linear capacity [16,
E has positive measure with
, such that
([17]). Thus in particular E is not a P.N. set if E has Haus-
dcrff dimension greater than one. While a full geometrical charac-
terization of P.N. sets is likely to be difficult there still seems
plenty of scope for further work on this intriguing class of sets.
RE FEREN CE S
I. Z o r e t t i L. Sur les fonctions analytiques uniformes qui
possedent un ensemble parfait discontinu de points singuliers. -
J.~th.Pures Appl., 1905, 6, N I, 1-51.
2. B e s i c o v i t c h A. On sufficient conditions for a func-
tion to be analytic and on behavior of analytic functions in the
neighborhood of non-isolated singular points. - Proc.London ~ath.
Scc°, 1931, 32, N 2, I-9.
3. D e n j o y A. Sur les fcnctions analytiques uniformes a sin- ° ~
gularltes discontinues. - C.R. Acad. Sci.Paris, 1909, 149, 258-
- 260 .
~) See Ed. n o t e a t the end o f the s e c t i o n . - Ed.
493
4. X a B H H C o H C.~. 0d aHS~HT~eoEo~ eMEOCTM MHOXeOTB, COB-
MeoTHO~ HeTpHBH~HOOTI pasm~x IuIaOOOB a~TH~ecE~X ~yHEI~
aeMMe IUBap~a B ~poHsBO~X 06zaoT~x. - NmTeM.c6., 1961, 54,
I, 3-50.
5. H B a H 0 B ~.~. 0 lauoTese ~aHxya. - Ycnex~ MaTeM.HayK, 1964,
18, 147--149. 6. D a v i e A.M. Analytic capacity and approximation problems.
-Trans.Amer.Math.Soc., 1972, 171, 409-444.
7. X aB ~ H B.H., X a B ~ H C O H C.H. He~oTop~e o~e~Fa a~a-
~T~ecEo~ e~OCTH. -~oF~.AH CCCP, 1961, 138, 789-792.
8. X a B H H B.H. F p ~ e CBO~CTBa HHTeI~S2IOB T~na Ko,~ ~ rapMo-
HEec~ conp~eB~x ~ B o6~aCT2X CO cnp~eMo~ rp~e~. -
M~TeM.C6., I965, 68, 499-517. 9. C a I d e r ~ n A.P. Cauchy integrals on Lipschitz curves and
related operators. - Proc.Natl.Acad.Sci. USA, 1977, 74, 1324-1327.
10. M a r s h a 1 1 D.E. The Denjoy Conjecture. Preprint, 1977.
11. B e s i c o v i t c h A. On the fundamental geometrical pro-
perties of linearly measurable plane sets of points I. - Math.
Ann., 1927, 98, 422-464. II: Math .Ann., 1938, 115, 296-329. 12. H a y m a n W.K., K e n n e d y P.B. Subharmonic Functions
Vol. 1. London - N.Y., Academic Press, 1976.
13. B H T y m E ~ H A.~. ~ep ~Ho~ecTBa Ho~O~Te~Ho~ ~ , Ho
HyxeBO~ a~a~T~ecEo~ eMEOCT~. --~oEx.AH CCCP, 1969, 127, 246-
-249. 14. G a r n e t t J. Positive length but zero analytic capacity. -
Proc.Amer.~'2th.Soc., 1970, 24, 696-699.
15. B ~ T y m E Z H A.r. AHSJH~TEeCEB~ eMNOCTB MHo~eCTB B s8~aqax
TeOp~ np~6~eH~. - Ycnex~ MaTeM.HayE, I967, 22, ~ 6, 14I-I99. 16. C a r 1 e s o n L. Selected problems on exceptional sets. -
Van Nostrand Math.stud., N 13, Toronto, Van Nostrand, 1967. • . • ° •
17. E r o s t m a n 0. Potentiel d'equ~l~bre et capac~te des en-
sembles avec quelques applications a la theor~e des fonctions. -
Medded.Lunds~Univ.~t.S~., 1935, 3, 1-118.
18. B ~ Tym E ~ H A.F. 06 O~HO~ Bs~a~e ~a~ya. - HsB.AH CCCP,
cep.MaTeM., 1964, 28, ~ 4, 745--756. 19. B a ~ ~ c E ~ ~ P.8. HecEox~EO saMe~a~ o6 o~paH~xe~x aHa-
~ecE~x ~y~n~x, npe~cTaB~M~x ~HTe~pa~oM T~na Kom~-CT~T~eca.
-C~6.MaTeM.~., I966, 7, ~ 2, 252--260.
W.K.HAY~,~N Imperial College, Department of
Mathematics, South Kensington,
London SW7 England
494
EDITORS' NOTE. As far as we know the representability of a 1 1
functions bounded and analytic off E and vanishing at infinity by
"Cauchy potentials" (I) is guaranteed when E has finite Painlev~'s
length whereas examples show that this is no longer true for an ar-
bitrary E (~18],[19]). We think THE QUESTION £f existence of po-
tentials (I) bounded in ~ \ ~ (provided E is no t a P.N. set) i_~s
one more interestin ~ problem (see also § 5 of [ 4]).
495
8.17. old
ANALYTIC CAPACITY AND RATIONAL APPROXIMATION
Let E be a bounded subset of C ^and B(E,~) be the set of a l l funct ions ~ in ~ ana ly t ic on ~\ m and wi th ~(co)= 0 ,
I~1 < 4 . Put A ( E , O = { ~ ~ BQE,~ ) " ~ is continuous on~O 3 . The number
is called the analytic capacity of E . The number
~A(E,O ~--~
is called the analytic C-capacity of E •
The analytic capacity has been introduced by Ahlfors [I] in
connection with the Painlev~ problem to describe sets of removable
sin~alarities of bounded analytic functions. Ahlfors [I] has proved
that these sets are characterized by ~(E) = 0 . However, it would
be desirable to describe removable sets in metric terms.
CONJECTURE I. A compact set E , E cC , has zero anal~tic
capacity iff the projection of E onto almost evel V direction has
zero length ("almost every" means "a.e. with respect to the linear
measure on the unit circle). Such an E is called irregular pro-
vided its linear Hausdcrff measure is positive.
If the linear Hausdor~f measure of E is finite and ~(~) = 0
then the average of the measures of the pr@jections of E is zero.
This follows from the Calderon's result [2] and the well-known theo-
rems about irregular sets (see [ 3], p.341-348). The connection bet-
ween the capacity and measures is described in detail in [4].
The capacitary characteristics are most efficient in the ap-
proximation theory [~, [6] ,[7], [8S. A number of approximation prob-
lems leads %o an unsolved question of the semiadditivity of the ana-
lytic capacity:
%(EUF) ~ c [ ~ ( E ) + ~ ( F ) ] ,
496
where C is an absolute constant and E, F are arbitrary dis-
joint compact sets.
Let A (K) denote the algebra of all functions continuous on a
compact set K , K:-C , and analytic in its interior. Let ~(K)
denote the uniform closure of rational functions with poles off
and, finally, let ~o~ be the inner boundary of ~ , i.e. the
set of boundary points of ~ not belonging to the boundary of a
component of C \ K . Sets K satisfying A(~) = ~(~) were
characterized in terms of the analytic capacity [6]. To obtain geo-
metrical conditions of the approximability a further study of capa-
cities is needed.
CONJECTURE 2. If ~(~°~)=0 then A(~) =~(K) .
The affirmative answer to the question of semiaddivity would
yield a proof of this conjecture. Since ~([)= 0 provided ~ is
of finite linear Hausdorff measure this would also lead to the proof
of the following statement.
CONJECTURE 3. If the linear Hausdorff measure of ~o~ is zero
( K b e i n g a compact ..sub.s.et o f ~., ) t h e n A(K] --C(K). f
The last equality is not proved even for K s with ~°K of
zero linear Hausdorff measure.
It is possible however that the semiadditivity problem can be
avoided in the proof of CONJECTURE 3.
The semiadditivity of the capacity has been proved only in some
special cases ([9], [10-13] ), e.g. for sets ~ and ~ separated by
a straight line. For a detailed discussion of this and some other
relevaal% problems see [14] o
REFERENCES
I. A h 1 f o r s L.V. Bounded analytic functions. - Duke Math.J.,
1947, 14, 1-11.
2. C a 1 d e r ~ n A.P. Cauchy integrals on Lipschitz curves and
related operators.- Proc.Nat.Acad.Sci., USA, 1977, 74, 1324-1327.
3. H B a H 0 B ~.~. Bap~am~ MHo~eCTB ~ ~yHEm~, M., "HayEa",
1975.
4. G a r n e t t J. Analytic capacity and measure. - Lect.Notes
Math., 297, Berlin, Springer, 1972.
5. B ~ T y m E ~ ~ A.r. AHaX~T~ecEa~ eMEOOT~ ~o~ecTB B sa~avax
497
Teop~npEdx~meH~. - Ycn~xHMaTem.HayE, 1967, 22, ~ 6, 141-199.
6. M e ~ ~ H ~E 0 B M.C., C ~ H aH ~ ~ C.0. Bonpoc~ Teop~
np~6~m~eH~a ~y~z~ O~HOrO KOM~eECHOPO HepeMeHHOrO. - B EH.:
CoBpeMem~ge npo6xeMHMaTeMaT~ T.4, MOCKBa, BHHETH, I975, I43-
-250,
7. z a I c m a n L. Analytic capacity and Ration~l Approximation.
- Lect.Notes Math., 50, Berlin, Springer, 1968.
8. G a m e I i n T.W.. Uniform algebras, N.J., Prentice-Hall, Inc.
1969.
9. D a v i e A.M. Analytic capacity and approximation problems.
- Trans.Amer.Math.Soc., 1972, 171, 409-444.
IO. M e x ~ H ~ E 0 B M.C. 0~eHEa ~HTerpaxa Ko~ no a~a~TH~ecEo~
EIDEBO~. -- MaTeM.Cd., 1966, 71, ~ 4, 503--514.
II. B ~ T y m E ~ H A.F. 0~e~a ~Te~a~aEom~. - MaTeM.c6., 1966,
71, ~ 4, 515--534.
12. ~ ~ p 0 E 0 B H.A. 06 O~HOM CB0~CTBe aHa~TEecEo~ eMEOCTE. --
BeCTH~E ~IY,cep.MaTeM., ~ex., aCTpOH., 1971, 19, 75-82.
13. ~ ~ p 0 E O B H.A. HeEoTop~e 0BO~CTBa aHa~IETEeCEo~ eMEOCT~. --
BeCTH~E~I~J, cep.MaTeM., MeX., aCTpOH., 1972, I, 77--86.
14@ B e s i c o v i t c h A. On sufficient conditions for a func-
tion to be analytic and on behaviour of analytic functions in
the neighbourhood of non-isolated singular points. - Proc.London
Math.Sec., 1931, 32, N 2, I-9.
A. G. VITUSHKIN
(A.r.BHTY~Gm)
CCCP, 117966, MOCKBa,
yx.BaB~oBa, 42,
MEAH CCCP
M. S.MEL 'NIKOV
(M.C.MF/6m~0B) CCCP, 117234, MOCEBa,
Mexa~aEo-MaTeMaT~ecE~
(~m~ym_~TeT MOCEOBCEOrO
yH~BepC~TeTa
498
8.18. ON SETS OF ANALYTIC CAPACITY ZERO old
Let K be a compact plane set and Aoo (K) the space of all
functions analytic and bounded outside K endowed with the sup-norm.
Define a linear functional ~ on ~oo(K) by the formula
I~-$
w i t h ~ > ttM3/J6 {1~ I : ~ ~ K ] The norm of L is c a l l e d t h e
a n a 1 y t i c c a p a c i t y o f k . We denote it by ~(K).
The function ~ is invariant under isometries of C . Therefore
it would be desirable to have a method to compute it in terms of
Euclidean distance. E.P.Dolzenko has found a simple solution of a
similar question related to the so-called %-capacity, C1]. But for ~ the answer is far from being clear. I would like to draw
attention to three conjectures.
CONJECTURE 1. There exists a positive number C such that for
a%7 compact set K
where ~(K,~) the line through
CONJECTURE 2. There
an,y compact set K
T denotes the l e n g t h of the p ro~ec t i o n of K onto
0 and ~ ~ T • exists a positive number C such that for
Y(K)~c I ~ (K,~)#~(~) . T
These CONJECTURES are in agreement with known facts about ana-
lytic capacity. For example, it follows immediately from CONJECTURE
I that ~ (K)>0 if ~ lies on a continuum of finite length and has
positive Hausdorff length. In t~rn, CONJECTURE 2 implies that ~(K)=0
provided the ~avard length of K equals to zero. At last, let
be a set of positive Hausdorff M-measure (a surveE of literature
on the Hausdorff measures can be found in ~3] ). If ~ ~(~2~" ~t < co O
then the Favard length of K is positive. This ensures the existence
499
of a compact K4 , K~cK , such that ~(K0>O
function ~, ~ (~ )= I. - ~ , is continuous on the Hausdorff ~-mea~ure. Hence ~(K) >~ ~ (KO > 0 easily follows from CONJECTURE 2.
with
and
and the
, ~ being
which also
CONJECTURE 3. Pot an E increasing function ~ :(0, +°°)--~(0,+°°)
I ~(~)/~ ~----oo there Qxists a set K satisfyin~ ~(K)>0
0 .
To corroborate this CONJECTURE I shall construct a function
~nd a set E such that ~(~=0, ~ ( E ) > O but ~ ( E ) - O •
Assign to any sequence 6~ {8 . } , ~ev~O (~), a compact set E(8). • Namely, let ~a(8) [0,4J . I f ~ . is the union of disjoint segments , of length^j ~.(8) then ~.+~C~) is the union of ~n sets ~L\ ^j - ~n , A n being the interval of length
I
~(6)~-8~) concentric with the segment ~ . Put
and let ~ be a constant ~e~uenoe, ( ~ ) , =C . ~inally let E =
= E(~°). Tt is known (see [~]) that ~(E) = 0 . This ~mplies the
existence of a function ~ such that
~ (~(13)=0 and ~'(E(6))<Sv(t) t-,-o
for any sequence 8 satisfying ~4 < ~ . Then ~(8) has the de-
sired properties for properly chosen 8 , as will be shown later.
To choose ~ pick numbers ~4 ~0 and ~4~ such that ~Co~i) < V~
~(~(6&~)) < ~ and (~+o~)~ ~-~ . Set ~j =d.,~ for j ~ 4, ~,,,.9~" Proceeding by induction, pick ~K+I to provide
the inequality ~(~K+~) < ~ ~ ~ (~+%j)~J
and next pick f~K+~ such that
(~+o~k+4) >~ and
500
Set now ~j -----~I<+~ for j =N ,+~ , . . . ,N~ (N~ ~-- ~ + . . . + ~s).
The sequence ~ defines a function ~ equal to ~ at
(8) , M~ and linear on each segment [~i(~)~ ~]-t (~)] It is easy to verify that ~ ~I/~Ct)~ ~ 0 and E(~) has positive
~ -measure. It remains to cheek only that ~[m(6)] = 0.
For this purpose let ~ ~ Aoo(E(8)) and let
Jl Jl where contour ~ embraces~ E(~)f] (A~ × A~) and separates
it from ~ . The set ~(~) being the union of ~n squares
~ with the side ~ ( )_~_~. and lying at the distance at oJ least ~(8) 4+8~ one from another, it is clear that ~ are
uniformly bounded and
I~,l
This implies k
i
tL(I)I..<7'<. ,~pIILII J-'!4. < . . . . ~[a~kj<")] %
4 J
m,J
and finany L C~)= 0 •
REFERENCES
I. ~ o ~ x e H E o E.II. 0 "cm~p~" oco(~em~ocTe~ a~ax~TEec~x
(~y~n~m~. - Ycnex~ MaTeM.~ay~, 1963, 18, ~ 4, 135-142.
501
2. C a I d e r ~ n A.P. Cauchy integrals on Lipschitz curves and
related operators. - Proc.Nat.Acad.Sci.USA, 1977, 74, 1324-1327.
3. R o g e r s C.A. Hausdorff measures. Cambridge, Cambridge Uni-
versity Press, 1970.
4. G a r n e t t J. Positive length but zero analytic capaeity. -
Proc.Amer.~th.Soc., 1970, 24, 696-699.
L. D. IV~OV
(~JI.~Bm{OB)
CCCP, 170013,
~CF~X~ rocy~apc~BeHmm~
ymmBepc~TeT
502
8.19. ESTIYi~TES OF ANALYTIC CAPACITY ola
Let ~ denote the Lebesgue measure on ~i=---~ . In what follows
we let E range over ~ and % over the interval C0~) C~ . We
always suppose that ~ ~ ~ and A >0 . ~ will be used as a ge-
neric notation for compact subsets of ~ . For any locally integrab-
le (complex-v~lued) function ~ on ~ we denote by
its mean value over the disc ~(~,,)=~:~ ~ I~-~I<~}. ~P'~
will stand for the class of all functions ~ on that are locally
integrable to the power ~ and satisfy
])(,~,~} +co
(cf. [ t ] for references on related function spaces). Investigation of
removable singularities for holomorphic functions in these classes
gives rise naturally to the corresponding capacities ~p,A defined
as follow~ (compare [2]). If ~,k(~,~) denotes the class of all
~g that are holomorphic in the complement of (including E co ) and satisfy the conditions
then
where -
It is an important feature of these capacities that they admit
simple metrical estimates which reduce to those of ~elnikov (cf.
0hap.V in [3]) for a special choice of the parameters when they
yield Dolzenko's result on removable singularities in Holder classes.
Writing ~ M for the diameter of M , M C C , and defining
for • , ~ >~0 , and ~ , ~ >0 ,
503
where the infimum is taken over all sequences of sets M~, M~c
with ~ M~ <~ such that E c U M~ , we may state the following
inequalities (cf. [4] ).
THEOREIvi I. Let ~ ~p÷A<Xp +Z , ~ = ~ (~IA) and define
F . Then there are constants 0 and ~ such that
(
for all E •
FROBLE~ I. V~at are the best values of the constants c~k
occurrin ~ in (I)?
Theorem 1 is of special interest in the case ~=~ , because
it characterizes removable singularities of holomorphic functions of
bounded mean oscillation (cf.[5]) as those sets E whose linear
measure
g-*O
vanishes. This is in agreement with the example of Vituskin (cf.[6]).
The capacity of E corresponding to the broader class of functions
of bounded mean oscillation may be positive even though E has zero
analytic capacity corresponding to bounded functions which is de-
fined by
where now ~(E~) is the class of all functions holomorphic off E
and vanishing at co whose absolute value never exceeds 4 . Never-
theless, by the so-called D e n j o y c o n j e c t u r e
(which follows from combination of results in [7] ,[8],[9], [10] ) the
e quivalenc e
=0 IcE)=O
504
is true for E situated on a rectifiable curve. The upper estimate
of ~(.) by means of I(.) is generally valid (cfo [11]) while the
lower estimate of ~(E) by means of a multiple of ~(E) is possible
only for E situated on sets ~ of a special shape.
PROBLEM 2. L,et, ~ C ~ ~e,,a compact,,set,~, F,ind ~eometri 9 condi-
tions on Q ~uaranteein ~ the existence of a constant C such,,that
~(E)~c I (E) , E c Q (2)
The following theorems 2,
THEOREM 2 (cf. [12]). Let
simple and continuousl~ differ entiable , I~I= is
i I~'(b-~'(~l~ ~t -= ~<+~ 0
3 may serve as sample results.
and
Then (2) holds and G can be computed by means of $ (see also
~7 in [11] ). T~OREM 3 (Cfo [13] ). I_~f Q has only a fi~i~t,e number of c,~mp,o-
nents and
i Q +oo,
T
where ~ (~) is the number of points in
then (2)holds with $=(~)-4 (~V(Q) +()-!
If Q is a straight-line segment then
renke's equality ~(E)-¼~(E) holds (cf.
following
PROBLEM 3. Is it possible to improve
iC~v(G)~ +0 -~ ?
~/( Q ) = 0 and Pomme-
[13]). This leads to the
in Theorem 3 to
505
REMARK. It was asserted in [4] that Theorem 3 holds with this
value of the constant. Dr. J.Matyska kindly pointed out that there
was a numerical error in the original draft of the corresponding
proof in [I~.
I. Peet re J.
1969, 4, 71-87.
2. H a r v e y R.,
REfeRENCES
On the theory of
Polking
~.p,A -spaces. - J.~unct.Anal.,
J. A notion of capacity which
characterizes removable singularities. - Trans.Amer.Math.Soc.,
1972, 169, 183-195.
3. M e x ~ H~ Z 0 B M.C., C H HaH ~ H C.0. Bonpocw weop~
spH6~N~eHN~ ~ y ~ O~HOrO EOME~eECHOPO nepeMesHoro. - B EH.:
CoBpeMe~e npo6~eM~ Ma~eMaTH~ T.4, MOCEBa, BHHHTH, 1975, 143-
-250.
4. K r ~ I J. Analytic capacity. - In: Proc.Conf."Elliptische
Differentialgleichungen" Rostock 1977.
5. J o h n F., N i r e n b e r g L. On functions of bounded
mean oscillations. - Comm.Pure Appl.Math., I£61, 14, 415-426.
6. B H T y m E H H A.r. HpmMep MHozeCTBa HO~O~Te~HO~ ~I~HH, HO
HyxeBO~ a~a~ec~o~ ~ocT~. - AoEx.AH CCCP, I959, I27, 246-
- 2 4 9 .
7. c a 1 d e r ~ n A . P . Cauchy i n t e g r a l s on L i p s c h i t z c u r v e s and
r e l a t e d o p e r a t o r s . - P r o c . N a t l . A c a d . S c i . USA, 1977, 74, 1324-1327 .
8. D a v i e A.M. Analytic capacity and apprQximation problems. -
Trans.Amer.l~th. Soc., 1972, 171, 409-444.
9. X a B H H B.Ho, X aB ~H C 0 H C.H. HeEoTopMe oneHE~ aKa~H-
T~ecEo~ eMKOCTH. --~OF~.AH CCCP, 1961, 138, 789-792.
I0. X a B ~ ~ B.H. l ~ ~ e OBO~OTBa ~HTe~pa~OB TEa EO~ ~ ~sp-
~om~ecF~ conp~e~ ~ys~z~ B 06XaCT~X CO c~p~eMo~ ~pa~e~.
- MaTeM.c6., 1965, 68, 499--517. 11. G a r n e t t J. Analytic capacity and measure. - Lect.Notes
Math., 297, Berlin, Springer, 1972.
~2. FLB a H O B ~.~. 0 rgnoTese ~as~ya. - Ycnex~ MaTeM.~ayE, 1964,
18, 147--149.
13. P o m m e r e n k e Ch. ~ber die analytische Kapazit~t. -
Arch.Math., 1960, 11, 270-277.
506
14. F u k a J., K r ~ 1 J. Analytic capacity and linear mea-
sure. - Czechoslovak ~th.J., 1978, 28 (103), N 3, 445-46].
Jos~F ~_~T, Matematick~ ~stav ~SAV,
Zitn~ 25, 11567, Praha 1,
gssa
507
8.20. old
~BER DIE REGULARIT~T EINES RANDPUNKTES
F~R ELLIPTISCHE DIFFERENTIALGLEICHUNGEN
In den letzten Jahren wurde dem Kreis yon Fragen, die um alas
klassische Kriterium yon Wiener ~ber die Regularit~t eines Randpunk-
tes in Bezug auf harmonische Funktionen gruppiert sind, viel Aufmerk-
samkeit geschenkt [I ,2]. Nach dem Satz von Wiener i~t die Stetig-
keit im Punkt 0~ 0 ~ ~ , der LBsung des Dirichletproblems f~ur die
Laplace-Gleichung im ~-dimensionalen Gebiet ~ ( ~ >2) unter der
Bedingung, da~ auf ~ eine in 0 stetige Fuzzktion gegeben ist,
~quivalent zur Divergenz der Reihe
ca, K~4
C
Hierbei ist C~----{,~: , ~ /2~ ~ ; ~ j " mud ~K die
harmonische Kapazitat der kompakten Menge K • Diese Behauptung wurde (manchmal nur der Tell der Hinlamglich-
keit) auf verschiedene Klassen yon linearen und quasilinearen Glei-
chungen zweiter Orduung ausgedehnt (eine Charakterisierumg dieser
Untersuchungen und Bibliegraphie kann man im Buch [3] finden).Was die
Gleichungen hoherer als zweiter Ordnung betrifft, so gab es fur sie
bis zur letzten Zeit keine Resultate, die analog zum Satz yon Wiener
sin& In der Arbeit [4] des Autors wird das Verhalten der Losung des
Dirichletproblems fur die Gleichung ~¢¢ = ~ mit homogenen Rand-
bedingungen, wobei ~ e C ~ (~) ist, in der Umgebung einer Rand-
punktes untersucht In [4] wird gezeigt, &a~ fur ~= 5,6,7 die
Bedingung
2 K(,,-4)
wobei C~p~ die sogenannte biharmenische Kapazitat ist, die Stetig-
keit tier Losung im Punkt 0 garantiert Fur ~= 2,5 folgt die Ste-
tigkeit der LSsuug aus dem Einbettungssatz von S.L. Sobolev, aber im
Fall pt= 4 , der ebenfalls in [4] analysiert wird, hat die Bedingung
f'6r die Stetigkeit eine andere Gestalt.
HYPO~HESE 1~ Die Bedin~un~ ~<8 ist night we sentlic h.
Dem Autor ist nur ein Argument fur diese Annahme bekannt. F~t~
811e F~ ist die LBsung der betrachteten Aufgabe f~tr einen beliebigen
508
Kugelsektor im Eckpunkt stetig. Die Einschr~qkung ~<o~ tritt nut bei
einem ~er Zemmsts suf, auf denen die Beweise in [4] beruhen Sis ist
sber notwendig fur dieses Lemms~ Es geht hierbei um die Eigensehaft
des Operators ~ , positiv mit dem Gewicht I~I~-~ zu sein Die-
se Eigenschsft erlaubt es, f'~r ~= 5,~,7 folgende Abschatzung der
Greenschen ~nktion ~es biharmonischen Operators in einem beliebigen
Gebiet anzugeben:
Ig(=,9)l ~< c(~)l~-~l "-~ (2)
wobei $C~ ~ ~ und C(~) eine nut von ~ sbhangige Konstante ist.
HYPOTKESE 2. Die Absch~tzun~ (2) ~ilt such f't~r ~ ~ S .
Es versteht sich, de9 man analoge Pragen such f~ allgemeinere
Gleichungen stellen kanno Ich mochte die Aufmerksamkeit des Lesers
aber auf eine Aufgabe lenken, die auch fur den I aplace-0perator nicht
gelost ist. Nach [5], [6] genugt sine harmonische ~unktion, deren vex~-
allgemeimerte Randwerte einer HElder-Bedin~gung im Punkt 0 genugen,
derselben Bedingung in diesem Punkt, falls
2 c p(C _ k n) > 0. (3) N.-, , . o o N~K>~4
Es ware i~teressant, folgende Annahme zu rechtfertigen @der zu wider-
legen~
HYPOTHESE 3. Die Bedingun~ (3) ist notwendig
Wir wenden uns zum Schlu~ nichtlinearen elliptischen Gleichungen
zweiter Ordnung zu. Wie in [ 7] gezeigt wurde, ist der Punkt 0 regular
fEr die Gleichung ~IZ ( I ~ t&l P'~ ?~ ~) = O, 1<p<~,falls
Z [ 2 '/(P-'>
ist, wobei p-~p (K) = ~ { II P " = is%. Dieses Resttltat wurde unl~ngst in ~ r Arbeit [ 8] a~ die se~ allgemeine Klssse VOlt Gleichtt~gen C~IY A (~,~, ~ ~)== B(=,~,~ ~) ubertragen. Ds die Bedingung (4) fur p=~ mit dem Kriterium yon Wie- ner zusalmnenfallt, ist es naturlioh, folgende Hypothese aufzustellen:
HYPOTHESE 4. Die Bedin~un~ (4) ist notwendig
509
In [9] wurden Beispiele behandelt, die zeigen, da~ die Bedin-
gung (4) in einem gewissen Sinne genau ist~ Fur die HYPOTHESE 4 spre-
chen auch neuere Ergebnisse ~ber die Stetigkeit nichtlinearez Poten-
tiale [10], [11].
Z0~N~A~ ~.S WP~ASS~,RS.
(~T~ JAEEB SP~.R)
"Anscheinend ist noch keine der formulierten Aufgaben gelBst" -
mit diesen Worten hatte der Autor vor, die ErBrterung des obenange- f~Lhrten Textes zu beginnen. Aber als der Kommentar fast fertig war, horte dieser Satz auf, wahr zu sein. Es tauchte folgendes GEGENBEI- SPIEL ZUR HTPOTEESE 3 auf,
und ~ die Vereinigung der Kugelschichten ~ \ ~ + ~ , i " 0,|,..., die dutch Offnungen OJ~ in den Spharen ~ ~9, ~ ) 4, verbunden sind Die Offnung OJ9 stellt eine geodatische Kugel mit beliebigem Mittel-
punkt und dem Radius ~-I/~ dar, Es ist klar, da9 die Kapazitat der Menge CZ-~ k ~ nut fur
K = ~, ~-I yon Null verschieden ist und da~ fur diese K die Un~eioh~ c~p (Cz~ \ ~ ) >/C ~-K(~-~) gi l t . Deshalb ist f~r N>~
N~K~4
Polglida divergiert die Reihe von Wiener f{Lr das betrachtete Gebiet,
aber die Bedingung (3) ist nicht erfullt. Wit zeigen, da~ trotzdem eine beliebige in ~ harmonische Punk-
tion, deren Randwerte der Holder-Bedingung im Punkt 0 genugen, eben- falls der H~lder-Bedingung im l>~ukt 0 genugt~
Es sei IA Losung des Dirichletproblems /ki& ~ 0 in ~ , i&-~=0 auf 8~ , wobei ~ eine stetige Funktion ist, die tier Bedingung
~(~C) ~0(I~l ~) , ~>0 , gen~gt. Man kann annehmen, da~ 4>~(~)> 0 ist. Wit bezeichnen mit ~ einen beliebigen Punkt des Ge-
biets ~ und mit $ eine Zahl, f{~r die f~-4 ~ i~l >i ~ ist. Es sei ~----O au~erhalb von C~OS(~_4) und ~q-----~ ~-4 Fer- her sei i&~ eine harmonische ~ku%ktion in ~ , die auf ~ mit ~ ubereinstimmt. Wegen 0~ ~ C~_! auf ~ ist
510
~j ~ Wir f~hren eine in der Kugel ~_~ c harmonisohe Funktien ~ ein, die auf 8~ 2 gleich ~ @ ist.
ofze b r O k,-<4 ~-~\ GJ~_~ . Stellt man ~ in der Gestalt eines Poisson-
Integrals dar so. erhalt man hieraus die Ungleichung 0 ~ ~
e (~.~ + 2 -(~'~)/~'z ~'~in ~_~ • Da nach dem Maximumprinzip"
0~ < ~J-'U~ ~ ist, er-halt man" 0 ~< U($C)-t~(SC)~ ClSCI ~/4 . Polg-
lich genugt die Punktion t~ im Punkt 0 der H~lder-Bedingung mit dem
Exponenten ~I~.
Modifiziert man das konstruierte Beispiel, so kann man leicht
zeigen, da$ man auf Grund der Geschwindigkeit des Wachstums der Par-
tialsummen der Reihe yon Wiener keine unteren Abschatzungen einer
harmonischen Punktion mit Null-Randbedingungen in der N~he des l~-
tes 0 machen kann. Indem man namlich die Art und Weise tier Konver-
genz der Radien [~ und der Durchmesser der Offnungen OJ~ gegen
Null vorschreibt, kann man eine beliebig schnelle Konvergenz der Funk-
tion ~ gegen Null im Punkt 0 bei beliebig langsamer Divergenz der Rei-
he von Wiener erreichen. Somit ist die HYPOTHESE 3 widerlegt, abet
desto interessanter bleibt die Frage nach den den Rand charakterisie-
renden notwendigen und hinreichenden Bedingungen fKtr die Holder-Ste-
tigkeit einer beliebigen harmonischen ~unktion mit Holder-stetigen Randwerten.
Was die anderen Fragen betrifft, die vor f~mf Jahren gestellt
wurden, so gibt es auf sie bisher noch keine Antwort *)~ In der letz-
ten Zeit wurden mit ihnen verbundene neue Informationen gewonnen, die,
wenn man es richtig betrachtet, nicht so sehr in die Tiefe wie in die
Breite gehen.
Im Zusammenhang mit den HYPOTHESEN ~ und 2 erw~hmen wir die Ar-
beit [~2], in der die grundlegenden Ergebnisse des Artikels [4] (die
ausftuhrlich in [13]dargestellt sind) auf die erste Randwertaufgabe
fur die polyharmonische Gleichung (-A)"~t$=f ubertragen wurden. Lei-
der verlangte auch bier die Methode, die auf der Eigenschaft des
Operators (-A)m~ ~>~ beruht, positiv mit dem Gewicht ~($C-~) zu
sein, wobei ~ die Pundamentallosung ist, alle Dimensionen mit Aus-
nahme der folgenden drei zu opfern: ~= ~, ~Y~+~,2~+2 • Als frag-
w~trdigen Ausgleich gestattet uns dies, die folgenden beiden Hypothe-
sen zu fo~ieren, die sich an die HYPOTHESEN I und 2 anschlie~en~
~) ~merhung bei der Korrektur: I.W.Skrypnik teilte soeben auf der Tagung "Nichtlineare Probleme der Mathematischen Physik" ( 13 April, L0~I, Leningrad) mit, da~ er die Notwendigkeit der Bedingung (4) fur ~>~ bewiesen hat. Damit ist die HYPOTHESE 3 teilweise gestutzt.
511
HYPOTHESE I'. ~r ~ >2, ~2,$+5 ist die Gleich~
E 2K(n,- 2m,) K)4
wobei cllp~ die m-harmonisohe Kapazitat ist, hinreichend fur die
Steti~keit der LBstulg des Diri0hletproblems mit NullraudbedinKunKen
der Glelohun~ Q-A) "~ 4~= ~ ~eCi(n)im Ptmkt 0.
HYPOTHESE 2'. FS_r ~>2, ~2~+S F~ilt frar die Greemache F~nk-
tion G~ des Operators (-/~)"~die Abschatzun ~
i C-,,,(=,o)l ~ cl :- l wobei di9" Konstante C yon ~ und ~aber nicht vom Geblet ' abh~n~t.
In der letzten Zeit wurden neue Erkenntnisse ~ber das Verhalten
der LSsung der ersten Randwertaufgabe f~r stark elliptische Gleichun-
gen der 0rdnung 2m in tier Nahe konischer Pumkte erhalten. Im allge-
meinen (s. [14] ) haben die Hauptglieder tier Asymptotik solcher LSsun-
gen in der Umgebung des Eckpunktes des Konus die Gestalt
N
cl l Z ( f lt, l l) (x/i O K=O
Dabei ist ~ Eigenwert des Dirichletproblems fSr einen gewissen po-
lynomial vom Spektralparameter abh~ngigen elliptischen Operator in dem Gebiet, das durch den Konus auf der Einheitssph~re ausgesohnitten
wird. Die Flmktion (5) hat genau dann ein endliches Dirichlet-Inte-
gral, wenn ~e ~ > ~- ~/2 . sie ist des weiteren stetig und gen'ugt
sogar einer H~ider-Bedingung, falls Re ~ > 0 ist. Wenn im Band
0 > Be ~ > ~- ~/2 Eigenwerte des genannten Operators existieren,
dann besitzt die Ausgangsrandwertaufgabe verallgemeinerte L~sungen,
die in einer beliebigen Umgebung des Eckpunktes des Konus unbe- schr~akt sind, und yon einer Regularitat nach Wiener kann man selbst .
bei einem konisohen Pumkt mioht reden. Es zeigt sioh ([15], [16]),~
wir auf solche unerwarteten Erscheinungen schon bei stark ellipti- schen Gleichungen zweiter Ordnung mlt konstanten Koeffizienten
=--- a ~ -
512
stolen, falls nicht alle Koeffizienten reell sind; In [16] (eine aus- fS/L~liche Darstellun~ erscheint evtl- in ,,MaTe~aTN~eoE~ O6Op~NE")
wird das homogene Dirichletproblem au~erhalb eines d'6nnen Konus
l'~'~=(t~ Z-'~E~ ~: ~,,.)' O~ U . ,~ I~ q~untersuoht, wobei ~ ein kleiner _ F,, (. )~' iiJ. r TM ~ - i ~ - I "J ~ ~ r v - , farame~er, CLT~---- ~. ~ e IK : ~ ~ e u I i ~nd w e in Gebiet im IK ist.
~s wir~ ~ie ~,~su~ ~(~)---I~IX(~)~(£,~/I~I) aes sta~k elliptischen Systems Qz~ (~) ~ (6~ ~C) =o betrachtet, wobei ~(~)
eine ~atrix mit homogenen Polynomen der Ordnung ~ als Elementen
und ~ (6) ----- 0 (~) f'/r ~-~+ 0 ist. Hauptergebnis ist eine asymptoti-
sche ~ormel fur den Eigenwert k (£) , welche f't~r den eimfachsten ~all
der Gleichung (6) die Gestalt
X (~) = £n-s{ __ . , - z I ~ ca, pp.(D~t,O)
• .n .(~-a)/~ ]
(,~) (d,et; I~iKII~,K-,) ÷O(1) ] ( (~ei: II ~,<" "-' ~('-~)/~ ',i,K=l/
hat:. Hierbei ist [Sgl die Oberflache der (K+I)-dimensionalen Ein-
heitskugel und C .r~, n~ eine komplexwertige Funktion des Gebiets o,p p,, - ~ ' "
Of , welohe eine Verallgemeinerung der harmonischen Kapazitat dar- atellt:
,/
En-'\ u: i, K'4 e'
wobei O# eine im Unendlichen verschwindende LSsung der Gleichuag
~z(0~,0) ~(~)----0 in ~-l\OJ ist, die auf 80# gleich 0 ist. Naeh
[IG] k~n~ man ~e =oeffizienten ~ so ~men, ~a9 ~e U~leio~u~ 0 > ~eX > (2-~)/2 erfullt ist. Im Fall ~=3 gilt
h(2)= t21 ~o~ 21)-'(~ + o0)) f~ ~-- +0.
Folglich erf~It j ede verallgemeinerte LBsung die H~Ider-Bedimgung,
falls der 5ffnungswinkel des Konus K~ genugend klein ist. Es ist
nicht ausgeschlossen, da~ die Forderum~ nach einem kleinen ~ffnungs-
winkel unwesentlich ist. Dies ist gleichbedeutend mit folgendem Satz
HYPOTKESE 5. F~ ~=3 ist ein konischer Punkt fur einen belie-
b_i~en elliptisehen Operator P~(D,) nit, komplexen Koeffiziente ~ re- gular math Wiener.,
513
F~tr den biharmonischen Operator im ~ und fur die Systeme yon
und Stokes im --]R~ wurden derartige Ergebnisse in " -[17J, " -[18] er- Lam~
halten,
REFERENCES
I. W i e n e r N. The Dirichlet problem - J,Math. and Phys~ 1924,
3, 127-146~
2. W i e n e r N~ Certain notions in potential theory - J~Math
and Phys. 1924, 3, 24-51
3. ~I a H ~ ~ c E.M. YpaBHe~S BTO10OrO IIO10S~ ~JU~nT~ec~O~O ~ na- pa6o~m~eci¢oro T~a, M., Hayz, a, 197I.
4. M a 3 ~ ~ B.r. 0 nOBe~eH~ B6~SZ rpSH~H pemeH~R saXaqR ~psx-
Jle ~ 6RPspMo~,eoEoro oIIepSTOpS. - ~oEJI.AH CCCP, 1977, I8, ~ 4,
15-I9.
5. M a 2 ~ s B.r. 0 !DSI~JI~pHOOTm H8 I~H~I~8 IDSmSHR~ @JI~IITBRS01~X ylmBHe~mi~ ~ KoH~olxmoro OTO6paxe~ms. - ~or~I.AH CCCP, I963, 152,
6, I29V-I300.
6. M a s ~ s B . r . 0 ~oBe~eH~ ~ 6 ~ s rlmHs~u peme~s sa~a~s ~ p s x - ae ; ~ s ~ z ~ n ~ a ~ e o ~ o ~ o ypa~se~as B~OpO~O nops;m~a B ~BepreHTSO~ ~O-
pMe. -~a~ea .saae~Fm, 1967, ~ 2, 209-220. 7 . M a s z s B . r . 0 ~ e n p e p u z ~ o c ~ z B r l m S a ~ o ~ ~ o ~ e pemeHz~ r m a s a -
JI~He~X @~aiiT~ecF~x ypaBHeHZ~. - BeOTH.~[rY, 1970, 25, 42-55
(nonlmBF~: BeO~H.~rY 1972, I , 158). B. G a r i e p y R., Z i e m e r W,P. A regularity condition at
the boundary for solutions of quasilinear elliptic equations. -
Arch,Rat.Mech.Anal., 1977, 67, N I, 25-39.
9. E p o ~ ~ ~.H., M a ~ ~ s B.r. 06 OTOyTOT~ ~enpepB~ooT~
Henl~elmBHOCTZ uo re~epy pemeHz~ KBSS~He~x ~m~Tz~ec~mx ypB~-
HeH~ B6JL~3~ HepSryJlSpHOH TOV~ER. - TI~ MOCE.MaTBM.O--BS, 1972,26, 73-93.
I0. H e d b e r g L. Non-linear potentials and approximation in the
mean by analytic functions. - Math,Z,, 1972, 129, 299-319,
II.A d a m s D.R., M e y e r s N. Thinness and Wiener criteria
for non-linear potentials. - indiana Univ,Math.J., 1972, 22, 169-
197. I2. M a s ~ s B.T., ~ o H~ e B T. 0 perw~qpHoo~ no B~epy rlm-
KZ~HO~ TOPAZ ~S uo~ralX~OH~eoxoro onepaTopa. - ~o~.Bo;az,.AH,
514
1983, 36, 2 2. I3. m a z ' y a V~G. Behaviour of solutions to the Dirichlet prob-
lem for the biharmonic operator at the boundary point, Equadiff
IV, Lect.Notes Math , 1979, 703, p 250-262
14. E o H ~ p a T ~ e B B.A. ElmeB~e sa~s~ ~ smmn~ecF~x YlmB--
HOHI~ S 06JIaCTSX C EOH~0CF~M~ ~Jl~ yraOBHMH TO~I~SMH. T I ~ MotE. aa~eM.o-Ba, 1967, I6 , 209-292.
15. Mas~s B.T., Hs 38 p0B C.A., HaaMeHeBC~m~
B.A. 0TCyTCTB~e Teope~ TmnS de ~op~s ~as C~a~HO saa~T~ecz~x
ypSBHeH~ O KOMII~IeEoHHM~ l~os~uaeHTa~. - 3aTr.Hayq.OeM~H.ZOMM, I982, IIS, 156-I68.
16. Ma s ~ s B.F., Ha s a po B C.A., IIaa ~ e He BC ]~ ~
B.A. 06 o/o~opo/~ux pemeH~x sa~a~z Jl~p~x~e Bo B~e=HOOT~ TOH~O~O
~¢o]¢~0a. -/~oI~a.A~ CCCP, 1982, 266, • 2, 281-284.
17. Ms s ~ B.F., Haa ue H e Be ~ ~ ~.A. 0nl~a~ne~mE-
csw~Ma ~ 6aral~OHa~eCl~OrO ypaBHeHas B 06Y~aGTB C EOHJ~NOCI~M~ TO-- ~ . - 14SB.BY3oB, 1981, ~ 2, 52-59.
18. M a s ~ s B.L, H a a m e H e B C ~ a ~ B.A. 0 CBO2Cr~ax l~-
meH~ TpeXMepH~X sa~sq TeOp~ ynlmjrocT~ ~ r~pOJ!~H~K~ B O6aSC-
TSX C ~SOaSpOBaHHHM~ OC06eHHOOTm~S. -- B 06. : ~HSS~E8 O]UtOmHO~ cpe~, HoBoc~6spc~, 1981, BUn.50, 99-121.
V. G.MAZ'YA
(B.r.MAS~) CCCP, I98904, JIelmHrlm~,
He TpO~mOl~ea, ~eHgHrl~o~m~
~ooy~81oc TBSHHR~ ~H~BepC~TeT,
Ma TeMa TaXo-aexa~eo~caJl ~ a ~ v e v
515
8.21. THE EXCEPTIONAL SETS ASSOCIATED WITH THE BESOV SPACES
For ~ real and 0<p,~<oo , we will use Stein's notation
P~ for the familiar Besov spaces of distributions on ; see IF] AA and [S~ for details. The purpose of this note is to generally survey
and point out open questions concerning the general problem of deter-
mining all the inclusion relations between the classes ~ ~, $
~>0~ of exceptional sets naturally associated with the spaces
A~ for various choices of the parameters &,p )~ ; c.f.[A~S],
These exceptional sets can be described as sets of Besov capacity ze-
on ~ ) ~ some fixed smooth dense class in the spaces A P~ ~ ,
l~l~p¢ the no= (q~si-no=) of ~ in A~, K compact, A~p~ is extended to all subsets of as an outer capacity. Then
E~B~p~ iff A~p~<E) 0 Thus o~ ~oB~ is: ~ive~ an ~ir-
bitrar~ compact se t K such that A~ p @(K)=O , fo r which ~ ,
~rite A~s << A~p~ . The s~bol ~ will mean that both di-
rections, << and >> , hold.
Now when 4 ~ p,~ < OO , there is quite a bit that can be said
about this problem. First of all, one can restrict attention to
~ ~ 4 ~/& . Functions in A~ for p>~/A are all equivalent
to continuous functions and hence AA, p, $ (E) >0 iff E~ . Continuity also OCCURS, f o r example, when ~ - I'l,/& and ~= 4 . Se- condly, i n the range 4 < p .< ~/~ . , ~ < ~ c o , t h e r e appear to be presently four methods for obtaining inclusion relations. They are:
I. If ~ C ~j~ ( continuous embedding), then clearly
. Such e~Bbeddings but not ve~ypoften. However, since & . ~ >0 , with ~ deno-
ting the usual class of Bessel potentials of ~P functions on ~
(see [ $] ), and since the inclusion relations for the exceptional
sets associated with the Bessel potentials are all known [AM], it is
easy to see that A>,%, s << AA, F,~ when ~ < %p (no addi- tional restrictions on S and ~ ) and that the reverse implication is false,
2, Using the mln-max theorem, it is possible %o give a dual for-
516
. . . . r~<~, . , ,~ , . . .A_~/ , mulation of the Besov capacities: A~ ,,,,,(K~ ~/P --,'~uD~ LI,.4~.u,~( £°r'~ ÷
¢ K and .< 4 },"'r"r" - _.. i t fices i
to prove inclusion relations between the positive cones in the d u~l I n . IA ~P~+ spaces. This method is facilitated by the characterizatio :~u'-~/
iff
0
f o r p,~, > 't , and
<oo
for p>4,
3. The reason that one cannot expect all inclusion relations to
follow from the first two methods is the simple fact that the capa-
citary extremals (in the primal and dual problems) generally have
additional regularity. One can take advantage of this by comparing
the Besov capacities AA, p, P to the Bessel capacities ~,p
i.e. the capacities associated with ~ . Recently, P.Nilsson
observed that the positive cones in ~,~l and coincide.
Hence ~&~p, p ~ ~&,~ . And again since all the inclusion rela-
tions for the Bessel~exceptional sets are known (a result that re-
lies on the regularity of the Bessel extremals), it follows that the
corresponding relations carry over to ~A,p~ • A
(The equivalence
of ~ ,p ,p and BA, ~ for all p>~ has been known since T .
Wolff's recent proof of the Kellogg property in non-linear potential
theory; see ~HW~, also ~ ~4] ).
4. It is possible to apply the method of smooth truncation to
the class of Bessel potentials of non-negative functions that belong
to the mixed norm space ~P$ (~ ~ x ~) to obtain still further
inclusion relations. This is due to the fact that the Besov capacities
can be viewed as restrictions of such mixed norm Bessel capacities
to subsets of ~ . We refer the reader to [ AD.
In addition to the above relations, it is also possible to show
that ~A,4)~ ~-& = Hausdorff ~-@ dimensional measure (04~<~) .
517
we summarize the results of I. through 4. in the following dia-
the cross indicates ~,p,~ (~) : 0 and the shaded re- grams;
gion, the pairs ( { , ~) for which A~,%,5 (K) =0, }%=o~p, as a consequence.
Qm~STZO~S: That H~'~'P(K~:O implies A~.,p,~(.K)=O for
p.~ ~ is quite easy; is this still t~e for ~< p ? Do these
diagrams represent all inclusion relations? I f so, how does one
account fo r the difference in the cases Ap < ~ and ~e= • ?
When 0<p<~ , 0< $ < 0o , very little seems to be known.
One obvious thing to try is to compare - ~&,p,e with ~A,p - the
latter is now defined using Bessel potentials of the real Hardy
spaces ~? on , 0<p<~ . This seems to be a good idea in
view of all the recent developments on the structure of Hardy spaces,
especially the atomic decomposition. Indeed, it is just such an ap-
proach that leads to H~'~'P<<~,p , ~,p<~,, 0< p .<~, ~ d then via trace theorems [ ~4] to H ~-~e << A~,~,p . However,
it is not presently known if AA,p, P ~BA, P holds for 0 < W <
though it is probably true, One of the main difficulties now is that
the obvious dual capacity is no longer equivalent to the primal one
(and the min-max theorem does not apply since the spaces in question
are no longer locally convex). This all does, however, suggest compar-
ing the Besov capacities to yet another class of capacities,namely EP a, those naturally associated with the Lizorkin-Triebel~ spaces & ;
see [ P] and [ ~D - And since it is known that ~ coincides with
Bessel potentials of ~P (0< p < co) the ~ -capacities~ are a natu-
ral extension of the Bessel capacities. Thus we might expect some
rather interesting results here in view of the things descussed
above. However, it should be noted that for fixed & and p , the F
518
capacities agree with ~&,~ whenever $ satisfies ~ $ p~$ <
or ~ ~ ~w ~ . Hence the ~ -diagram summarizing the inclusion rela-
tions for the ~ -exceptional sets will be considerably different
than that for the ~ -exceptional sets ~&,~,$ . QUESTION:
What does it look like?
REFERENCES
[A I]
[A2]
[A~]
[A~S]
[~w]
[J1 ~
is]
A d a m s D.R. On the exceptional sets for spaces of poten-
tials. -Pac.J.Math.~1974, 52, I-5.
A d a m s D.R. Lectures on ~? -potential theory. Ume~ Univ.
Reports, 1981.
A d a m s D.R. , M e y e r s N.G. Bessel potentials.
Inclusion relations among classes of exceptional sets. - Ind.
U.Math.J.~ 1973, 221, 873-905.
A r o n s z a j n N., M u I I a F., S z e p t y c k i P.
On spaces of potentials connected with ~? classes. - Ann.
Inst.Pourier, 1962, 13, 211-306.
H e d b e r g L.I., W o 1 f f T. Thin sets in nonlinear
potential theory. - Ann. Inst. Fourier, 1983, 33.
J a w e r t h B. The trace of Sobolev and Besov spaces,
0 < p < ~ . - Studia Math., 1978, 62, 65-71.
J a w e r t h B. Some observations on Besov and Lizorkin-
Triebel spaces. -Math.Scand., 1977, 40, 94-104.
P e • t r e J. New thoughts on Besov spaces. Duke Univ.
Press, 1976.
S t e i n E. Singular integrals and differentiability pro-
perties of functions. Princeton U. Press, 1970.
DAVID R. ADAMS Department of Mathematics
University of Kentucky
Lexington, KY 40506
USA
519
8.22. COMPLEX INTERPOLATION BETWEEN SOBOLEV SPACES
~ t w ~ ' P ( ~ ) = t ~ : ~ E L P ~ ) , 0 ~ ~ ~ , the usual Sobolev space. The space W K'°° seems to be poorly understood. Prob-
lem 1.8 gives one example of this. Another example is furnished by
considering the complex method of interpolation, (., ~)%~ Let~<~.
z...~8 (W~,Po~), WK,'~C~))e=W~,P(~) , ~ - Po ? This is easy and true when ~=~ . Using Wolff's theorem [I] it is
easy to show that a positive answer for one value of Po is equiva-
lent to a positive answer for all values of ~o . The question is also
easy to answer if one replaces the ~ endpoint by a BUO endpoint.
The corresponding problem for the real method of interpolation is
solved in ~2~
REPERENCES
1. W o i f f T. A note on interpolation spaces~ - Lect.Notes Math.,
1982, N 908, 199-204. Springer Verlag.
2. D e V o r e R., S c h e r e r K. Interpolation of linear
operators on Sobolev spaces~ - Ann.Nmth.~979, 109, 583-599.
PETER N,JONES Institut Uit tag-Leffler
Aurav~gen 17
S-182 62 Djursholm
Sweden
Usual Address :
Dept. of Mathematics
University of Chicago
Chicago, Illinois 60637
USA
CHAPTER 9
UNIQUENESS, MOMENTS, NORMALITY
Problems collected in this chapter are variations on the follow-
ing theme: a "sufficiently analytic" function vanishing "intensively
enough" is identically zero. The words in quotation marks get an
exact meaning in accordance with every concrete situation. ~or in-
stance, dealing with the uniqueness of the solution of a moment prob-
lem we often exploit traces of the analyticity of the function
The theme is wide. It encompasses such phenomena as the quasi-
analyticity and the uniqueness of the moment problem, and borders on
normal families (see e.g. Problem 9.5), various refinements of the
maximum principle and approximation. Its importance hardly needs any
explanation. The Uniqueness marks (more or less explicitly) all con-
tents of this book. After all, every linear approximation problem
(and the book abounds in such problems) is a dual reformulation of a
uniqueness problem.
Every problem of this Chapter (except for 9.3 and 9.7) deals
not only with "the pure uniqueness" but with other topics as well.
Problem 9.1 is connected not only with zeros of some function classes
but with a moment problem (as is Problem 9.2) and with Pourier -
Laplace transforms of measures; in Problem 9.6 the uniqueness in ana-
lytic Gevrey classes is considered in connection with peak sets for
521
Holder analytic functions. "Old" Problems 9.8 and 9.9 deal (from
different points of views) with differential and differential-like
operators (both have evoked a great interest, see respective commen-
tary). Problem 9.4 has certain relation to spectral operators and to
the "anti-locality" of some convolution operators (in contrast with
"the locality" of convolutions discussed in 9.9). Problem 9.5 is a
quantitative variation on the title theme and 9.13 gravitates towards
spectral analysis-synthesis of Chapter 7. Problem 9.10 is aimed at
approximation properties of exponentials and concerns also some as-
pects of quasianalyticity, as does Problem 9.12. Problem 9.11 deals
with an interesting "perturbation" of the~ ~I>-~ -theorem.
The theme of this chapter emerges in some Problems of other
Chapters (37, 43, 44, 49, 512, 77, 7.17, 718, 8.4, 10,1, 105,
10.6, S.4, S.6).
522
~.I I .
eld SOME OPEN PROBLEMS IN TEE THEORY O~REPRESENTATIONS
OEANALYTIC FUNCTIONS
I. Denote by~ the set of functions (O , satisfying the follow-
ing conditions:
In the factorization theory of meromorphic functions in the unit
disc ~ , developed in [I], the following theorem on solvability of
the Hausdorff moment problem, proved in [2], played an important
role: the Hausdorff moment problem
0
where
.~o-4, ~ = (,r~I~(~)~'l~)-~ (",=~,~, -'') O
and ~ , ~ , has a solution in the class of nondecreasing and
bounded functions & on [0,~] .
Assuming ~ ~ ~ (i=~,~) , consider the Hausdorff moment prob- o lem of the form
Q
where
4
0 0
CONJECTURE. The moment problem (1)-(27 has a solution in the
class of nondecreasing and bounded functions on [0,~] , or at least
523
w
in the class of functions ~ with bounded variation 0.V (~) < + OO [ ,~]
provided the functions ~j are monotone on [0,1] and 034/0) ~ i_~s
non-increasin6 on [0,I].
The proof of this conjecture, which is true in the special case
~(Z) -~-~ , would lead to important results on embeddings of class-
es N {60i} (j=1,2) of meromorphic functions in ~ , considered in
[..1]. II. Denote by ~L~
following conditions:
I) the function
~(~) >0 ; 2) the integrals
Ak=kl g
are finite.
Putting A o =
the set of functions ~ , satisfying the
is continuous and non-increasing on ~0>co)
, consider entire functions of Z :
(.-k I - 2
0
00
~k
and wC ®)
e
~et, finny, {~k}4 (o < l~kl .< Izk,41<~) s e q u e n c e o f comp lex numbers such t h a t
°I Z ~c~) d,z <o~.
be an arbitrary
(3)
524
CONJECTURE. Under condition (~) the infinite product
co
conver~es on any compa,,ct set, not co,ntaining points,,,,,,,,of
provid~e,d ~ satisfie,s, the additional condition
¢zf+~.
co
The validity of this conjecture for some special cases and in
particular for
There ~CO<~<~), ~(O<~<o~) and ~(O<~<~o) arear- bitraryparameters, was proved in [3].
III. Let ~ be a complex function on [0, oo) such that
0
Then it is obvious that the function O0
0
is regular on the Riemann surface
and that
~, ~ [ O, oo).
525
In view of this the following conjecture n~turally arises.
Let ' 4 be an analEtic function on ~ CONJECTURE. satisfyina
I~1<=o
Then,,,there exist,,s, a,,functio n
that
~ o~ [0, ~) satisfying (4) such
0
(5)
Note that in the special case, when
the function admits the expansion
k~o
~ [O,m)
Thus in this case
points 0,1,2,... of the axis [0, oo) only.
zv. Denote by HP(~) (O<p<~), - I<~< ~) analytic functions ~ on ~ such that
I zeros of
by ~(~) and let
(5) holds with a measure concentrated at the
the class of all
D
a~P(&) and {~i} ~ (O<l~iI-<I~i$41 < I) be the sequence of enumera t ed i n a c c o r d a n c e w i t h t h e i r m u l t i p l i c i t y . Denote
the n~mber of ~j S i~ the ai~c I Zi -< ~ ( 0 < ~ < ~)
0
526
It is well-known (of.[4]) that if ~6HP(A) and I#O then
I(~_~ ~ J ~ ' ~ ' ~ <~. (~
o
I n t h i s c o n n e c t i o n i t i s n a t u r a l t o s t a t e t h e f o l l o w i n ~
be an arbitrary sequence in ~ satisfying CONJECTUI~. Let {~j}4 . . . . . . . . . . . . . . . ao
(6). Then there exist a s eguence of numbers {Oj} 4 (0~< 0 i < ~f#)
,,~ ,, ~c~o~ i , , l . ' ~° . I , ~H~ ( .~ . o u ~ ~h~t
e j°i (j°44,-..) ~*(*i )--0.
Note that a statement equivalent to (6) and some other results
about zeros of functions of the class ~P(A) (if ~ ~0 ) were ob-
tained in [5] much later than [4]°
REFEREN CE S
I. ~ z p 5 a m ~ H M.M. TeopE~ ~8S{TOpHSaL~ ~yi~l~, MepoMop~I4S~X B Kpy2e. -- r~aTeM.cS., 1969, 79, }~ 4, 517-615. (Math.USSR,
Sbornik, 1969, 8, N 4, 493-591).
2. ~ a p 5 a m ~ H M.M. 0605meEH~ onepaTop P~Ma~a--~EyB~ ~ He--
Eo~op~e ero npEMeHeH~. -- MsB.AH CCCP, I968, 82, I075-IIII. (~ath.USSR, Izvestia, 1968, 2, I027-I06~.
3. ~ a p 5 a m ~ H M.M, 05 O~HOM 5eCEOHe~HoM nl00~SBe~eHYd~. -- HSB.
AH ApMH~ICEO~ CCP, MaTeMaTI{Ea, 1978, 13, ~ 3, 177- 208. (see also: Soviet Math.Dokl,, 1978, 19, N 3, 621-625)
4. ~ a p 6 a m a H M.M. K npoSxeMe npe~cTaB~OCTE aHa~HTE~ecEEx
~ . -- CO06~.HHCT.MaT. E Mex.AH Ap~HCEO~ CCP,I948,2,8-89~ 5. H o r o w i t z C. Zeros of functions in the Bergman spaces. -
Duke ~th.J., 1974, 41, 693-710.
M. M. DJRBASHYAN CCCP, 375019, EpeBsH yJl. ~ap eEalW4rT~Lf~ 246,
~#ICTETyT MaTeMaTHEE
AH ApM. CCP
527
CO~NTARY
The question posed in section III has a negative answer. We be-
gin with the following observation. The logarithm ~0~ being a biholo-
morphic map of the Riemann surface G~ onto C , it is clear
that every function ~ on ~ defines a (unique) entire function
satisfying
= z e G®.
N
C l e a r t y , ~ s a t i s f i e s the assumption M~ (~) < + O0
i f
if and only
On the other hand
0
I e. o
if and only if
(8)
Here, as in section iIl, ~ stands for a complex Borel measure on
[0,+~o) such that I®e~l~(~)l<+~ for every ~£~ . It follows 9
that ~ has a finite full mass (put ~=0 ).
It is shown in [6] (ch III, §4 ) that there exists a true pseudo-
measure ~ (i.e. a distribution, with the ttuiforml~y bounded Fourier
transform and not a measure) supported by ~=[0~ U 0 ~
The support of ~ being compact, the Fourier transform of 5 coin- cides with the restriction onto ~ of an entire function ~ of expo-
nential type, I ~(~)I ~- A e I~i . Moreover, it follows from the
L. Schwartz' version of the Paley-Wiener theorem (see [7], Ch.6, §4
for example) that ~I ~C~)I = II ~ II L=(~) and that
A
528
if ~ ~ 0 . Therefore F(Z) ~-~ ~ (~) satisfies (7) but F
cannot coincide with the Laplace transform (8) of a finite Borel mea-
sure because this contradicts the fact that $ is a TRUE pseudo-
measure and to the uniqueness theorem for ~ourier transforms of dis-
tributions. (The solution ~s found by S.V.Hruscev).
RE~ERENCE S
6. K a h a n e J.P. S~ries de Fourier absolument convergentes.
Berlin, Springer-Verlag, 1970.
7. Y e s i d a K. Functional analysis. Berlin, Springer-Verlag,
1965.
529
92. MOMENT PROBLEM QUESTIONS
Let ~0 be the non-negative integers and ~: the set of all
multi-indices&=(&~, ~&~) with each & ~ E ~ o . Por any 0~= R, &, d.4 ~I~ " o
=(¢~,...,0C1~ ) E ~ write ~ = O~ 4 . . . ~ l r , where ~ =~ .
Denote by ~o the complex vector space of all polynomials,
p(~)=~@~@, considered as functions from
is called a m o m • n t
exists a bounded non-negative Borel measure
The moment sequence is said to be d e t e r m i n e d if there
exists a unique representing measure. We refer the reader to the re-
cent expository article by B.Fuglede [2] for a discussion of this
problem together with an up to date set of references.
If ~= ~ , it is well known that if the moment sequence is de-
termined, then ~ is dense in ~(~) . Indeed, more is known.
If j(.~.o=~ , and ~ is an extreme point of the convex set of repre-
sent:i~g measures f o r # lb&, t hen ~ i s dense in ~. ( i ~ ) • In 1978, the t h e o r e t i c a l p h y s i c i s t , P r o f e s s o r John O h a l l i f o u r
of Indiana University proposed (in priwate conversation with the
author) the following question:
C . A multi-sequence
s e q u e n c e if there
~(~) so that
QUESTION I. ~or ~ > ~ , is it still true that if a multi-para-
meter moment sequence ~4 has a unique representin~ measure ~
then ~0 is dense in ~&(~) ?
To turn to a second question suppose ~% and $& are moment se-
quences from *~)~is the moment sequence formed from the convo-
lution measure ~ (~, ~), then it was shown in ~I ] that if (~ ~ ~ )~
is a determined moment sequence r then so are the individual moment
sequences ~ and 9% . Very recently, the statistician, Persi Dia-
conis of Stanford University proposed (again in private conversation
with the author) the following question:
QUESTION II. If ~ and 9~ are determined moment sequences, is
it true that (~,9)~ is a determined moment sequence?
530
REFERENCES
I. D e v i n a t z A. On a theorem of Levy-Raikov. - Ann. of Math.
Statistics, 1959, 30, 538-586.
2. P u g 1 e d e B. The multidimensional moment problem. - Expo.
Math.,1983, 1, 47-65.
ALLEN DEVINATZ Northwestern University
Department of Mathematics
Evanston, 11 60201
USA
EDITORS' NOTE, Christian Berg(Kcbenhavnsumiversi~ets matematiske
Institut, 2100 K~bemhavn, Danmark) informed us that he has answered
QUESTION 2 in the negative. Moreover, he has constructed a measure~
such that the sequence ~ is determined, but the sequence (~,~
is not.
531
9.3. old
SETS OF UNIQUENESS FOR ANALYTIC FUNCTIONS WITH
THE FINITE DIRICHLET INTEGRAL
be a class of functions a~lytic ~ ~ . A closed sub- uni-
Let
set ~ of the closed disc ~0¢ ~ is said to be a
q u e n e s s s e t for ~ (briefly ~(~) ) if
>
(it is assumed that ~C~) ~ ~lw ~ (~) at ~ 67~ E ). %~4-0
The structure of ~(~) -sets is well understood for many important
classes ~ (see[l] ,[2]; [3] contains a short survey). The same can-
not be said about the family ~(~A) , ~ being the space of all
functions analytic in ~ with finite Dirichlet integral
< The description of
D seems to have to do not only with the Beurling - Carleson condition
(see (I) below) but with capacity characteristics of sets.
We propose two conjectures oonce=ing subsets of T in ~(~A)" Associate with every closed set ~ cT a (unique) closed set
~ , ~* CF so that o~(F\~*)--0 (cap stands for the ca-
pacity corresponding to the logarithmic kernel) and every non-empty
relatively open (in ~ ) part of F~ has positive logarithmic capacity.
CONJECTURE I. A closed subset .... ~ o_~f T does not belon~ to
~(~A r~ CA) ( C A stands for the disc-al~ebra) i,f,f
a)
T
The difficulty of this problem is caused by the fact that func-
tions in 0 A posess no local smoothness on T . The cenjecture agrees with all boundary uniqueness theorems for
~A we are aware of. These are two.
THEOREM (Carle~on [4]). Suppose that FcT , ~04 F= F ,
~F=0 and for some ~>0 the ine~ualit ~ CA(F~I(~))>~
holds for an arbitrar~ ~ and eve~ ~>0 , I(~,S) being
532
the arc ,of length ~ centered at ~ . ,T, hen F 6 ~ (~) i,,f,,f
T
Here C% denotes the capacity corresponding to the kernel 1~1# A set F satisfying the conditions of the Carleson theorem coinci- des with F* because ~&(E) >0 implies ~(E) >0 .
~oREM (~,~ - ~avin [6]). Suppose that F cT ,~F=F
and that there exists a ram%l[ ~ of mutuall[ disjoint open arcs I
satisfying Fc U I and I~I ........
I¢~i ~Cl) _+oo. (2)
Then F ~ $(~A) "
Evidently (2) ~plies Z ~I- ~ 4 ~(I) - +co . Any
f~ily { ~} of open mutiny disjoint arcs , which a~os~ co or° n
%~.~# ~@~. This remark is an easy consequence of the sub- add~tivity of ~ . Therefore (2) implies the divergence of
the series ~ ~ ~ , [~#} being the family of complementary i n t e r v a l s o f v F , ~ rov ided ~ F = O .
To state the second conjecture consider a class ~ of non- negative functions defined on T . A closed subset E is said to belong to ~(~) if
WIE ---,-IN T
Let ~+ , _be the set of all traces on T from W~(~) (i.e. the functions in mable generalized gradient).
of non-negative functions ~(~) with square sum-
533
CONJECTURE 2. 8( 4) = (3)
Equality (3) (if true) permits to separate the difficulties
connected with the analyticity of functions of ~A from,, those of purely real character (such as the investigation of \ ~ 0 ~
for non-negative in W~ ).
The inclusion ~C~+) c ~(~,_, is obvious because
~II~ ~ ~+ for $ ~ OA . The proofs of the theorems cited above are based precisely upon this inclusion (and upon Jensen's inequality). Here is another remark suggesting that (3) is a "right"
analogue for the Beurling - Carleson theorem. Thi~ptheorem asserts
that ~p~(&))= ~(~+&). Here ~(&) ~-~- C A N~C&)
stands for the space of functions in the disc-algebra satisfying the Lipschitz condition of order ~ .
The well-known Carleson formula for the Dirichlet integral of an analytic function [7] permits to reformulate conjecture 2.
Suppose that for a 6iven ~ C~ there exists a non-zero
i_.nn ~ [ ~ ) sat isfyin~ A
T
,IE. - -0 ,
then there exists a function ~ ~+(T)
(4)
(5)
satisfyinK (4), (5) and
Tq' 0o. (6)
Some estimates of the Carleson integral in (6) are given in [8].
The sets of ~ A~(~A ) located in ~ have been considered in [4]
534
REFERENCES
I~ Beurling
72.
2. C a r 1 e s o n
the unit circle. V V *~
3. Hrusc ev
Arkiv for Mat.,
4. Ca rl e s on
A. Ensembles exceptio~mels. Acta Math., 1940,
L. Sets of uniqueness for functions regular in
Acta Math., 1952, 87, p.325-345.
S.V. Sets of uniqueness for the Gevrey classes.
1977, 6, p. 253-304.
L. On the zeros of functions with bounded Di-
richlet integrals. Math.Zeitschrift. 1952, 56, N 3, p.289-295.
5. S h a p i r o H.S. and S h i e 1 d s A.L. On the zeros of
functions with finite Dirichlet integral and some related func-
tion spaces. Math.Zeit. 1962, v.80, 217-229.
6. M a s ~ ~ B.r., X a B E H B.H. "Hp~o~eH~ (p,~) -~r~OCT~
E HecEoJIRF~ sa~aqaM Teop~ ~CF~H~TeJ~HNX MHO~eCTB". MaTeM.c6op--
H2E 1973, 90 (182), B~n.4, 558-591. 7. C a r i e s o n L. A representation formula for the Dirichlet
integral. - Math.Zeit. 1960, 73, N 2, 190-196.
8. A~ e E c aH~p 0 B A.B., ~ p 6 am~H A.3., Xa-
B ~ H B.H. "0 ~opMy~e Eap~ecoHa ~ ~Terpaxa ~mx~e".
BeCTH~E ~Y, cep.~ar., Mex., aCTp., 1979, I9, 8-I4.
V. P. HAVIN (B.H.XABMH)
CCCP, 198904 HeTpo.~Bopen, MaTeMaTEo-Mexa~ecK~ Ey TeT2W
s.V.HRu SNv CCCP, I9IOII ~eH~m~a~-II
*OHTa~Ea 27,
~0M~AHCCCP
C0~ENTARY
A description of ~(~A) can be found i n [ 9] which, unfortu-
nately?is difficult to apply.
Conjecture I has been disproved by an ingenious counter-example
of L.Carleson [10]. Conjecture 2 remains open.
535
It is interesting to note that the closely related problem of
description of the interpolating sets for ~A has been solved in
[11]. Namely, a closed set EcT is said to be an interpolation set
if [~A ~ C~l ~ = C [ ~ ~ . Then ~ is a~ interpolation set
Ifz E =-- 0 .
RE~ERENCES
9. M a 1 1 i a v i n P. Sur l'analyse harmonique des certaines cla-
sses de s@ries de Taylor.- Symp.Math.ist.Naz.Alto Mat. London -
N.Y., 1977, v.22, p.71-91.
I0.C a r 1 e s o n L. An example concerning analytic functions
with finite Dirichlet integrals. - 8a~.Hay~H.ceM.JION~4, 1979, 92,
283-287.
ii.H e a a e p B.B., X p y ~ ~ B C.B. 0nepaTop~ raHEeaa, Ha-
Eay~mEe np~6x~meH~ H CTaZ~oHapH~e PayccoBcEHe nponecc~. - Ycnex~
~TeM.Hayz, I982, 37, B~n.I, ctp.53-I24.
536
9.4. ~ALYTIC EUNCTIONS STATIONARY ON A SET, THE UNCERTAINTY
PRINCIPLE FOR CONVOLUTIONS, AND ALGEBRAS OF JORDAN OPERATORS
I. The . statement of the problem, We say a Lebesgue measurable fun-
ction ~ defined on the circle T is s t a t i o n a r y on the
set E~ E cT ? if there exists a function ~ absolutely
continuous on T and such that
a e~ on E
A measurable set E, E CT t y S (in which case we
non-constant function in M r (T) stationary on E.
PROBLEM. Give a description of sets of the class (S)
We mean a description yielding an answer to the following
QUESTION I. Does every E with rues E > 0 belon~ to (S) ?
There are natural modifications of the PROBL~. E.g. we may ask
the following
QUES~IO~ 2 .
,in the disc-algebra
~ ~$ ,stationary on E
sehitz conditio n of order less than one?
Note that every ~ E A satisfying the first order Lipschitz
condition and stationary on a set of positive length is co~_stant.
Using a theorem of S.V.Hru~ev [I], it is not hard to prove[2,3]
that a closed set E, E c W 7 has the property S if ~¢esE>0
and if moreover
, is said to have t h e p r o p e r-
w r i t e E ~ (S) ) if there is no
E C T, ~e8 E > O. Is there a non-constant
A (i.e. ~ ~ C (W) , ~(~)-0 for all
? What about ~ ~ A satisfyin~ a Lip-
iF<{ T r< + co , (c)
the sum being taken over the set of all complementary arcs ~ of E
QUESTION 3. Suppose E E (S) . Oo.es E contain a closed
537
subset E~ of positive length satisfyin~ (C)?
This question may be, of course, modified in the spirit of QUES-
TION 2. A deep theorem by S.V.Hruscev (deserving to be known better
than it is~ [I] , Th.4.1 on p 133) suggests the positive answer.
We like our PROBLEM in its own right ~nd feel it is worth solving
because it is nice in itself. But there are two "exterior" reasons
to look for its solution.
2. The uncertaint ~ principle for convolutions. Let K be a dis-
tribution in ~ X a class of distributions (in ~ ) Suppose the
convolution ~( ~ ~ has a sense for every ~ ~ X The set
E~¢is called a (~X) - s e t if
(K* E =o, F ! E = o .F--o.
(The exact meaning of the convolution K * ~ and of the restricti-
ons ~I E, ( K* ~)I E becomes clear in concrete situations, see
[2], [d). If the class of (K,X) -sets is sufficiently large (e.g con-
tai~s all non-void open sets) then we may say that the operator
-*- K~ ~ obeys "the uncertainty principle", namely, the knowledge
of both restrictions ~I E and ( ~ W ~) I E determines ~ uniqu-
ely. For example every set E c ~ of positive length is an
(~l,Z)-set, ~ being the Hilbert transform (i.e
-~ ~_~ ). Other examples see in
[2], [4], [5]. There are interesting situations (e.g. ~ (~C)=~
= Izl-~(~e ~, O<<<~,X a suitable class of distributions) when
we only know that there are many (K, X) -sets but have no satis-
factory characterization of such sets, The most interesting is, may
be, the case of K(~)'I~I -~+4 (~6~ ~ ) closely connected with
the Cauchy problem for the Laplace equation.
In an attempt to obtai~ a large class of relatively simple K~s
obeying the above "uncertainty principle" and to understand this prin-
ciple better the kernels with the so-called semirational symbols have
been introduced in [2],
Consider a Lebes~ue measurable function k : ~-*" C and put
rier transform of ~ ,^and define ~ * $ kfor ~ ~ e k by the
identity (~ ~ ~)A_ ~k - We call the symbol of the ope-
rator ~ ~: K ~ ~" • The function k is called semirational
538
if there exists a rational function ~ such that k I (- oo,0)=
= ~ I ( - o o , O ) , k ( ~ ) + - ~ ( ~ ) a.e. on a n e i ~ b o ~ h o o d of +~o. In [2] it was proved that every closed set E c ~ of positive
length satisfying (C) (where ~ runs through the set of bounded com-
plementary intervals of E ) is a (K,~k) -set provided k is
semirational (a simpler proof see in [3]). It is not known whether
condition (C) can be removed An interesting (and typical) example
of a convolution with a semirational symbol is the operator K,
a perturbation of the Hilbert transform. We do not know whether every
set E with rr~eS E> 0 is a ~ K,L =) -set (though we know it is
when C= 0 or when E satisfies (c)).
All this is closely connected with our PROBLE~ (or better to say
with its slight modification).
DEFINITION. I) A Lebesgue measurable function q on the line
is said to be ~-s t a t i o n a r y on the set E~ E c~, if there are functions ~4 ~..-,~ ~ W; (~) (i.e ~(~) , ab-
solutely continuous and with the L¢(~) -derivative) such that
~IE =~41E, ~, ~IE, .... ~-~1 ~ I E IE O. 2) A set E, is said have t r o p e r -
t y S~ (or E E ( S ~ ) ) i f
~ H~(IR), ~ ~-stationary on E ~ ~-=-0.
It is not hard to see that
E ~ C~s E, E e t C ) , l~es E > O ~ E e ( S ~ ) , ~=~,2,... c o
and that if E ~ ~-i St then E is a (k, ~k) -set for eve-
ry semirational k [3].
moreover if there exists a ~ H2(~)~ ~@ 0, stationary on the
set E , then E is not a (K,~k) -set for a semirational k
(which may be even chosen so that k agrees with a linear fmnction on (-co,O) ) .
Another circle of problems where ~ -stationary analytic functi-
ons emerge in a compulsory way is connected with
539
3. Jordan operators. We are 6oing to discuss Jordan operators (J.o.)
~of the form
~+~ where ~ is unitary, ~ = O , ~ G - Q ~ ( in this case we sayT is of order ~ ). It is well known that the spectrum of any such~
lies on ~ so that ~ is invertible Denote by ~(~) the weakly
closed operator algebra spanned by ~ and the identity I We are
interested in conditions ensuring the inclusion
T-' ~ P,, (T). (**)
EXAMPLE. Let E be a Lebesgue measurable subset of T and H be the direct sum of (~+I) copies of L ~ (T\E) The operator
I = Y( E,~)defined by the (~4 ) × (~* {) -matrix
I ~. I .
( ~ being the operator of multiplication by the complex variable
is a J.o. of order ~ . It is proved in [3] that
J - ' e g ( l ) < :- E ~(S;).
Here (~) denotes the class of subsets of T defined exactly as
C8~) in section 2 but with ~ replaced by the class of all functi-
ons absolutely continuous on~.
The special operator I- I c E~) is of importance for the
investigation of J.o. in general, Namely [3],if ~ is our J.o. (~)
of order ~ and ~ stands for the spectral measure of ~ then
•herefore i~ E e ~=,A CS'~) ( ~ particular ~ ~eS E >0 ~d E ~ CC) ) then (**) holds whenever ~U(E) 0
Recall that for a unitary operator T (i.e. whenT = ~Q= 0
540
in (*)) the inclusion (~*) is equivalent to the van lshing of ~ on
a set of positive length A deep approximation theorem by Sarason [61
yields spectral criteria of (**) for a normal T . Our questions con-
cer~ing sets with the property ~ and analogous questions on clas-
ses (~), (~) are related to the following difficult PROBLEM:
which spectral condltions ensure (**~ for T ~ ~ ~ ~ where
is normal and ..... ~ is a nilpotent com~ntin~ with ~ ?
REPERENCES
1. X p y m e B C.B. Hpo62eMa O~HoBpeMeHHO~ annpoEc~Ma~EE z cT~paH~e
Oco6eHHOCTe~ ~Hwerpa~oB T~na Kom~. - Tpy~ MaTeM.EH--Ta AH CCCP,
I978, IS0, I24-195.
2. E p ~ K E e B., X a B ~ ~ B.H. Hp~R~ Heonpe~e~H~ocT~
onepaTopoB, nepecTaHOBO~HRX CO C~BErOM I. -- 8anEcE~ Hay~H.Ce~H.
~0M~, I979, 92, I34-170; H. - ibid., I98I, II3, 97-I34.
S. M a E a p o B H.F. 0 CTa~zoHapH~x ~yR~IE~X. - BeCTH~K ZIY (to
be published).
4. H a v i n V.P., J o r i c k e B. On a class of uniqueness
theorems for convolutions. Lect,Notes in Math , 1981, 864, 143-
170.
5. X a B E H B.H. HpzH~H Heonpe~e2~HHoc~z ~x~ O~HoMepHax HoTe~a-
2OB M.P~cca. -~oE~.AH CCCP, 1982, 264, ~ 8, 559-568. 6. S a r a s o n D. Weak-star density of polynomials. - J.reine umd
a~ew.Math., 1972, 252, ~-15.
V. P. HAVIN
(B.H.XABHII)
B. JSRICKE
N. G. MAKAROV
(H.LMAKAPOB)
CCCP, 198904, ZeH~Hrlm~, IleTpo~mope~,
JleR~HrpajIcE~ rocy~apCTBeH~ yH~Bep--
CE TeT MaTeMaT~Ko-MexaH~ecE~
~BEyJIB Te T
Akademie der Wissenschaften der DDR
Zentralinstitut f~ur Mathematik und
Mec~
DDR, 108, Berlin
Mohrenstra~e 39
CCCP, 198904, JleHEHrpa~, HeTpo~Bopen,
JIeHE HrpaJIc~zi~ rocy~apc TBeHHNI~ yH~BepOZTeT MaTeMa TI~Eo-~exaH~ eCKEI~ ~Ey2BTe T
9.5.
541
PROBLEM IN THE T~ORY OF FUNCTIONS
In 1966 I published the following theorem:
There exists a constant @ • 0 such that a%7 collection of poly-
nomials Q of the form
with
k )
is a norma ! famil,y in the complex plane.
See Acta Math., 116 (1966), pp.224-277; the theorem is on page
273.
This result can easily be made to apply to collections of poly-
nomials of more general form provided that the sum from I to oo in
its statement is replaced by one over all the non-zero integers. One
peculiarity is that the constant @> 0 r e a I I y m u s t b e
t a k e n quite small for the asserted normality to hold. If ~ is
I a r g e e n o u g h, the theorem is f a I s e.
The results's proof is close to 40 pages long, and I th~k very
few people have been through i%. Canoone find a shorte r and clearer
proof? This is my question.
Let me explain what I am thinking of. Take amy fixed ~ , 0 <2<
< ~ and let ~ be the sllt domain
CbO
C\ U
I f ~ i s any po lynomia l , w ~ t e
By d i r e c t harmonic e s t ~ t i o n i n ~3 one can f i n d w i thou t too much t r o u b l e t t ~ t
542
_ ~ '7+~ ~'
where K~ (~) depends only on ~ and ] . (This is proved in the
first part of the paper cited above. ) A natural idea is to try to ob-
tain the theorem by making ~ ~ 0 in the above formula. This,
however, cannot work because Kf (~) tends to oo as ~-~0
whenever $ is not an integer. The latter must happen since the set
of integers has logarithmic capacity zero.
For polynomials, the estimate provided by the formula is too
crude, The formula is valid if, in it, we replace ~I~(~)[ by
an~ function subharmonic in ~ having sufficiently slow growth
at ~ and some mild regularity near the slits K ~-~, ~÷~] •
P o 1 y n o m i a 1 s, however, are s i n g 1 e - v a 1 u • d
in ~ . This single-valuedness imposes c o n s t r a i n t s
on the subharmonio function ~[ ~(~)I which somehow work %o diml-
niah ~8(~) to something bounded° (for each fixed • ) as ~--~ 0,
provided that the sum figuring in the formula is sufficiently small.
The PROBLE~ here is to See quantitativel,7 how the constraimts cause
this d~m~n~shin~ t o take place.
The phenomenon just described can be easily observed in one
simple situation. Suppose that U(~) is subharmonic in ~ ,%hst
~(~)~< ~[~l there, and that E~(~)] + is (say) continuous up to the
slits ~-~, ~+ ~ . If U(x)~< ~ on each of the intervals
El,!,- ~ , 'H, * jo] , t h e n
This estimate is best possible, and the quantity on the right blows
up as ~ ~ 0 . However, if U(~)=~i ~(~)l where ~(~) is
a s i n g i e v a I u e d entire function o f exponential type
A < ~ , we have the better estimate
M c A • A
w i t h a c o n s t a n t C A i n d e p e n d e n t o f ~ . The i m p r o v e d
543
result follows from the theorem of Duffin and Schaeffer. It is no
longer true when A ~ ~ -- consider the functions ~ ( ~ ) ~ ~ with L ~ oo •
The whole idea here is to see how harmonic estimation for func-
tions analytic in multiply connected domains can be improved by
taking Into account those functions' simgle-valueduess.
PAUL KOOSIS Institu% Mittag-Leffler, Sweden
McGill University, Montreal, Canada
UCLA, Los Angeles, USA
544
9.6. PEAK SETS FOR LIPSCHITZ CLASSES old
The Lipschitz class AA , 0< % ~ ~ , consists of all functions
analytic In D , continuous on ~D and such that
I{(~,) { ( ~ ) I . < ~ I ~ q l ~ q ~ T (~) A closed set E , E cT , is called p e a k s e % f o r A~ (~ s~bo~ E ~ % ). i~ the~ e~s~s a ~unction ~, I ~ A~ (the so called p e a k f u n c % i o n) such that
I{1<~ on T \ E ~ l I E = - 1 .
THE PROBLEM is to describe the structure of ~& -sets. B.S.Pav-
icy [I] discovered a necessary condition;this condition was redis-
covered by H.Hutt [2] in a more complicated way. Write E ~ ~-
={z; ~T. ¢i,,~ (~;, E) ~ a}. THEOREM 1 (Pavlov [ 1 ] , Hut% [ 2 ] ) ° I f Ee~ then
~ ( E ~ ) = O ( ~ ) , ~ ~+0. (2)
COROLLARY. ~ -sets ' are finite.
SKETCH OP THE PROOF. Let ~ be a peak function for E and
@=~-~ . Then @6A& , R6~>/O in D and ~}E=O. Hence
~ I/~ ~ 0 in D ° ~nd Herglot°z theorem [3] ~ys t°hat ~/~ is C~chy integral of a finite measure, Now condition (2) follows from the
weak type estimate [43
. &
and from the evident inequality 19(~) l .<c~.~&(w,E). • Until quite recently only some simple examples of ~A -sets
el..
were known [2]; condition (2) holds for these examples wzth a v v'
reserve". But recent S.V.Hruscev's results [5] on zero sets for
Gevrey classes permit us to obtain a very exact sufficient conditi-
on. Define the G e v r e y c 1 a s s GA as a class of all ana-
lytic functions in D such that ~ I{(~)(~)1-< (co~)~+t(~!) ~*~, ~=0,4, .... A set E , E CT is called z • r O S e t
f O r Q& (or E ~ Z (G&~(~)), i f there i s a non-zero func t ion ~ , ~eG~ , ~ th ~ IE=-O , ~ = 0 , ~ , . . . . ~ v [ ~ ]
has completely investigated 7. (G~)-sets and gave a lot of examples.
545
T~O~ 2. z (%) c ~ SKETCH OF THE PROOF. I t has been shown in [5] t h a t eve ry n o n -
empfy E in Z (G~) defines a positive f u n c t i o n U on T with the
following list of properties:
~) ~{~,Ef~U{~), ~;~T;
c) 0<cI<U(~D/U{~p <c~<+~ provided ~i,~\~ and I~#-~I<~(~4)~) . Here
~----~ {-U-~U} stands for the outer function with the modulus
~zpt-U) ' on T . i t i s e a . y to . e e that { ~ G . , ~ ) I E - O , ~:0,~,... (see [5] for deta i ls ) .
Set @ --~-----(60~#) -4 i n ~ 9 i Then
Let us prove now that @6 ~& . Obviously (I) holds if either
~4 and ~Z lie in different complementary intervals of E or
-~/~ I ~ , - % 1 > ~ { U ( ~ # ~ / % U{~) }.
~f ig4 g~l < ~ U{g4) -~A then by b) I~4 g~l<~2(g~ E) and bya) and c)
1 {}(gD-~{ ~)1 -( e~2 igc ~1U(gJ +~ U (g£~ 4 ~ o ~ I g~ -~ 1 ~ .
It is clear that ~=(~+~)(~-~)-~EA& and for ~ .
Theorem 2 gives a number of examples of
lowing conjecture seems now plausible.
c o ~ c T ~ . ~ = Z { G~. There exist some (not very clear) connections between the free
interpolation sets for A m and ~& [6]; these connections corroborate
our conjecture. The Am-function from theorem 2 is a "logarithm" of
some ~m-function. Probably, it is a general rule and it is possible
is a peak function
~& - sets and the fol-
546
to link two interpolation problems in a direct way. A possible way to prove our conjecture is a conversion of the
proof of Theorem 2 The "strongly vanishing function" ~=6~<- ~{-~)
is of course not in G& for an arbitrary peak function ~ 6 ~ .
But in theorem 2 such function I is not arbitrary; it is extremal
in some sense [5]. Perhaps, it is possible to obtain ~6~ for
some extremal peak function ~ . The extremal functions are often
analytic on T\ ~ , and such an ~ may be a smooth function.
The following necessary condition may be a first step to the
conjecture: if ~ 6 ~ then dist ('~ ~)-&6 14. (T).
The description of ~ -sets is interesting for the investi-
gation of the singular spectrum in the 2riedrichs model [1,7].
REFERENCES
I. H a B ~ o B B.C. TeopeMN e~ERCTBeHROCT~ ~ ~yR~ C noao~-
Te~BRO~ MHHMO~ ~aCTBD. --Hpo6~eMH MaTeM.~HSEEE, HB~--BO J~V, 1970,
4, 118-125. 2. H u t t H. Some results on peak and interpolation sets of ana-
lytic functions with higher regularity. - Uppsala Univ.Dep.Math.,
Thesis, 1976.
3. H p ~ B a x o B H.H. l~paR~R~e CBO~CTBa aRax~T~ec~x #yH~.
M.-~., I~TT~, 1950.
4. Z y g m u n d A., Trigonometric series, Cambridge Univ.Press,
London, New York, 1969.
5. H r u ~ ~ B° v S.V. Sets of uniqueness for the Gevrey classes.
- Ark.f$'r ~t., 1977, 15, N 2, 256-304.
6. ~ ~ H ~ ~ z ~ E.M. CBOSO~Ra~ ~RTepnoa~nEs B ~accax P~x~epa. -
MaTeM. c60pHHE, 1979, 109, N I (Math.USSR Sbornik, 1980, 37,
N I, 97-117).
7. P a v 1 o v B.S., P a d d e e v L.D. This Collection, 4.4.
E.M. DYN ' KIN CCCP, 197022, SeR~Rrpa~
y~. npo~.HonoBa, 5
9xe~TpoTex~ec~ I~HCT~TyT
m~.B.H.Yx~HoBa (~eB~a)
547
9.7. A PROBLEM BY R.KAUI~AAN old
Let ~ be a bounded Lipschitz domain and C~(@) (*~ ~) the
class of functions analytic in @ , with ~ -th derivative uniformly
continuous on @ . Do the classes C~(@) have the same zero-sets
in c~0~ @ , or on SG ?
R. KAUPNAN University of Illinois
at Urbana-Champaign
Department of~athematics,
Urbana, Illinois, 61801
USA
548
9-8- old
QUASI-ANA~YTICITY O~ FUNCTIONS WITH RESPECT TO A
DIFFERENTIAL OPERATOR
Suppose ~ is a domain in ~ , E is a closed subset of ~ ,
{~I is a sequence o~ positive n=bers such that ~ ~q~l~=~
( ~m is the best monotone majorant for M~ ), h is a differential
operator of order ~ with C~°(~) coefficients. A function
~ C ®~) is said to belong to the class ~(~) if the follewimg inequalities are satisfied ~ .
IIC~IIL~/~ ~ .<C~,, ~--0,~,~,... (O=c~). Denote by ~o the maximal set among the subsets of ~ enjoying the following property:
if a function ~4 , 14 ~ C°°(~), has a zero of infinite mul-
tiplicity on E and satisfies the equation ~ ~4=0 then 141~o =0 .
I CONJECTURE that under an appropriate definition of the order
of the operator h the following is true.
CONJECTURE. I_~f ~ belom~s to the class %(M@~) and has a
zero of infinite multiplicit.7 on E , then ~Ino =0 .
In other words functions quasi~nalytic with respect to the ope-
rator ~ behave with respect to the uniqueness theorem as solutions
of the homogeneous equation ~ ~ = 0 .
We will SKETCH THE PROOF of the Conjecture IN TWO CASES.
I. Suppose ~ is an elliptic operator such that the operator
D~-L W is elliptic, F= {Z0} is a point Of ~ , and ~=~[.
Consider the solution . ~(~,X) of the problem (D~-t~)~=O,
for small ~ : ® ~k~
The function ~(~) has zero of infinite order in ~ - ~£ =0 ,
hence ~(4, ~t) ~ 0 and ~(~) ----- 0 •
2. L=D ~ +D ~ -D ~ il -IR ~ E ie a two-d~ensio-
549
~ I smooth su.rface, ~ e C ~ (I~ 5 ) and satisfies }I L.~( ~ II L~(_O_~
<C~K , K=O,~ , ~,...(so that here ~ =~ ). Denote by ~0 the clo-
sure of the operator L defined on the set of functions vanishing
of infinite order on ~ . It is clear that ~ is a symmetric
o~erator. Suppose the vector ~ , ~0 is s~oh t~t U o ~ = ~ .
Consider for ~ A > 0 the vector-valued function ~ --~ ~
-----~+(A-$)~ where ~ is the resolvent of a self-adjoint
extension of the operator ~0 " Then ~ ~A= ~ and
IXK(q'X,~t)t:I(.L;?~X, $)I:I(~x,L~ol)I ,<C~,, k=o,4,~,...
Hence, (~A,~)-(~,~)=O . Similarly (~,~)=0 where ~o~ =-¢~. Hence ~ belongs to the invariant subspace of the operator ~o on
which ~9 is self-adjoint. By the theorem of Gelfand and Kostju~en-
ko [I] ~ belongs to the linear span of the generalized eigen-
functions of the operator ~0 , i.e. of solutions of the equation
~,0~ =~ vanishing on "Qo . Hence ~1~ o =0 . @
REFERENCES
I. r e x ~ ~ a H ~ H.M., K o c T 10 q e H E 0 A.r. PasxozeHHe
no COdCTBeHH~M ~y~4q~M ~H~epeH~Ea~BH~X E ~pyr~x onepaTOp0B. -
~OF~.AH CCCP, 1955,108, 349-852.
2. B e p e s a H C E H ~ D.M. Pas~ozeH~e ~o O06CTBeHHRM ~JHEH~--
CaMOOOnp~eHH~X onepaTopoB. KHeB, "HayEoBa ~7MEa", 1965.
V. I. MATSAEV
(B.I,I.~) CCCP, 142482 qepHorOaOBEa MOCKOBCEa~ OOJI.,
MHCTHTyT X~ecEo~ ~SHEH AH CCCP
C O~ENTARY
The CONJECTURE is confirmed now in various particular cases. Here
are some quite recent (unpublished) results.
Let I~ be a domain in ~, A be a differential operator of C @° (II) -coefficients order with , pe(~,+-], M=~MkIk ~ a
550
sequence of positive numbers. Put ~ = Mkz/~] ,
.,4p(M,.O.)a~{YreC ()-IIA~.~IIL,(~_) ~. M~ ,j-o,,,...}. Suppose
Consider a family ~. re, p-< C~K+p (k,p=~,2,. . . ) •
= C C["I,-,. '~ ~) of positive numbers,
and a compact set K ~ KC~L • Put
CCM,~,,k') 'L~t{ ~ e C-(..0.). II o¢'f-llco<) ~<
C~,K ~ ~,, ~L~,~,+, ,~,~4 r (~ + E.o~i ~ j) ,
[½] being the in teger part of ~ . V~hen ~-----4 we wr i te CCM,K) instead of C ( H , ~ , K ) .
THEOREM 1 (A.G.Chernyavski i )Suppose that 1)A=~- O ~ P o ( t , ~ ) ~ , - - " ~ ' ' ~ " Icq +
~eC (M,G), where G _is the cEl inder { C ½ , ~ ) ' ~ [ - 4 , ~ ] ,
=e~"-' k:cl~;~} and ~oeCt,~)÷oinG" 2) the c~ass ~ (M,G) is quasi~a~,tic, i.e.
the,#
Fix now a multiindex For every multiindex o6 put
that
~ - - ( ~ ) ~ sup (,) ..... ~ K M K
U ~ O in a nei~hbourhood of the crispin.
with decreasing natural coordinates. I OC:~I = ~ O ~ K / ~ • Suppose
Ioc:Xl~t
where
and
C~c~ ~ C (M,K, ~) on every compact K?KC~,
C~ ~----J ~4/A i ( i = 4,...,~) o Assume moreover that
551
THE0~ 2. (M.M.~alamud). Under the above conditions
~ ) ~%~ (M)c c ( M, dr, b~ (..0.)) ('I<p<~) (the class C(M,A, bp(~t)) is defined like GCM,~,K)
above with ~" ~b'(~) instead of ll.~C(~) ).
2) Su==ose A~ . . . . . A~> A~** ,~ . . .> t , and E = ~ o ~ E ~ A . I f P ~ ( E ) = P ~ ( ~ ) ( ~ being the Pro~ection (~,..., x~) ,
--,-(0 ..... O, 9c¥.~..., X~,)) then
In I) M is not supposed to satisfy (* ) . The condition concer-
ning ~(~) cannot be dropped from 2). If ~ is elliptic (in that
case $~ ..... $~ ) then every singleton [Xo} can serve as the set
2),and K~A ~d %p(M,2) are both quasianalytic ifC(M,Lp(~)) is (this result is due to M.M.~alamud and A.E.Shishkov, see
also [3] ). ¥.P.P&lamodov has pointed out (a private communication to the
editors) that one can co~firm the CONJECTURE in the case of an ultra-
hyperbolic operator using methods of §17, Ch.VI of [4] .
We conclude by a llst of works connected with our theme.
REFERENCES
3- L i o n s J.-L., M a g e n e s P. Probl~mes aux limites non-
homog~nes e% applications. ¥ol.3, Paris, Dunod, 1970-
4. C o u r a n % R. Partial differential equations, N.Y., London,
1962.
5. ~ ~ 6 z q D.M., T K a q e H K o B.A. A6CTRaKTHS~q Hpo6~eMa
z~aszaH~T~OCT~. -- TeopEH ~/HK~Z~, ~.aHa~. z ~x npza., I972,
16, 18-29.
6. q e p H 2 B C X z H A.L KBaszaHa~ZTz~ecKze ~accu, nopo~eH~e
rznep6oJm~ecKzMz oneparo~ c HOCTOR~HE~ EOS~Zn~eHTaM~ B ~ .
- ibid., 1982, 27, 122-127.
7. q e p H 2 B C E Z ~ A.F. 06 O~OM O606ZeH~Z Teope~ e~HCTBe~o--
CTZ Xo~rpeHa. -CZ6.~aTeM.zypHa~, I98I, 22, ~ 5, 212-215.
8. K o t a k 4 T., N a r a s i m h a n M.S. Regularity theorems
for fractional powers of a linear elliptic operator - Bull.Soc~
Math.France, 1962, 90, 449-471.
552
9-9. LOCAL OPERATORS ON FOURIER TRANSFORMS old
If ~ is a square imtegrable function of a real variable, let
denote its Fourier transform. If ~ is a measurable function of
a real variable, the notation ~ is also used to denote a partially
defined operator taking ~ into ~-~* ~ whenever ~ and ~= ~I
are square integrable. The operator ~ is said to be 1 o c a 1 A
if, whenever a function ~ in its domain vanishes almost everywhere A
a set of positive measure, the function ~-~- ~. ~ also in vanishes
almost everywhere in the same set.
THE OPERATOR ~ IS CONJECTURED to be local i f ~ is the rest-
riction to the real axis of an entire function of minimal exponential
type.
If the operator ~ is local and if it has in its domain a non-
zero function which vanishes almost everywhere in a set of positive
measure, then IT IS CONJECTURED that ~ agrees almost everywhere
with the restriction to the real axis of an entire function of mini-
real 9~onent!al type which satisfies the convergence condition
If k is a function which satisfies the convergence condition,
if k>/ J , and if ~ ~ is uniformly continuous, then IT IS CONJEC-
TURED that for ever# positive number 6 a nonzero function ~ in
the domain of k ̂ exists which vanishes ' almost everywhere outside of
the interval (- ~, 6) .
If k is a function which does not satisfy the convergence con-
dition, if k>/J , and if $2 ~ is uniformly continuous, then IT IS
CONJECTURED that no nonzero function exists in the domain of ~ which
~ishes in a set of positive measure.
K~FEI~NCE
I. d e B r a n g e s L° Espaces Hilbertiens de Fonotions En-
tieres., Paris, Masson, 1972o
L.DE •RANGES Purdue University
Department of Nath.
Lafayette, Indiana 47907 USA
553
FROM THE AUTHOR'S SUPPLEMENT, 1983
The problem originates in a theorem on quasi-analyticity due to
Levinson [2~ . This theorem states that ~ cannot vanish in an inter-
val without vanishing identically if it is in the domain of ~ where
K is sufficiently large and smooth. L a r g e means that the inte-
gral in (*) is infinite, s m a I 1 that it is finite. The smooth-
condition assumed by Levinson was that ~ K I is non-decrea- ness
sing, but it is more natural E3~ to assume that ~IK I is uniform-
ly continuous (or satisfies the Lipschitz condition).
A stronger conclusion was obtained by Beurling E4~ under the A
Levinson hypothesis. A function ~ in the domain of ~ cannot vanish
in a set of positive measure unless it vanishes identically. The
Beurling argument pursues a construction of Levinson and Carleman
which is distinct from the methods based on the operational calculus
concerned with the concept of a local operator~ The Beurling theorem
can be read as the assertion that certain operators are local with
a trivial domain. It would be interesting to obtain the Beurling
theorem as a corollary of properties of local operators with nont~i-
vial domain ~ . . . ~ . The author thAn~S Professor Sergei Khrushchev for informing him
that a counter-example to our locality conjecture has been obtained
by Kargaev.
REPERENCES
2. L • v i n s o n N. Gap and Density theorems - Amer.Math.Soc.,
Providence, 1940.
3. d e B r a n g e s L. Local operators on Pourier transforms -
Duke Math.J., 1958, 25, 143-153.
4. B e u r 1 i n g A. Quasianalyticity and generalized distribu-
tions, unpublished manuscript, 1961.
CO~AENTARY
P.P.Kargaev has DISPROVED the PIRST and the LAST CONJECTURES.
As to the first, he has constructed an entire function k not
only of minimal exponential type, but of z e r o o r d e r
such that ~ is not local. Moreover, the following is true.
554
THEOREM (Kargaev). Let
to zero on [0 ,+@@) . Then there exist a function
("a divisor"), a set e c
followin~ properties:
be a positive function decreasi~
~th the , and a function I~(~)
i~e~e >O,
~ * ~ is bounded away from zero on e , where
e o o o ~ n ~ ( , ) = t h ( ~ + ~ ) ] -4 .o see k~ ~o o~ ~oro order, but K ~ "s ot l o c a l .
The LAST CONJECTURE i s disproved by the f a c t (a lso found by
Kargaev ) tha t there e x i s t rea l f i n i t e Borel meam~es ~ o nn ~ w i t h
ve~ large lacunae in ~p~ (i.e. there is a sequence
{(~ ~ ))~=4 of intervals free of I~I , ~ < ~ ,~=~,~,"-
~-~ tending to infinity as rapidly as we please) and with
vanishing on a set of positive length. Take b ~- ~ q ,
where ~ is a suitable mollifier and k:~C~(~,~p~))
Then hk is a Lipschitz f~ction and z~ ~=+~,
if • grows rapidly enough. Then the inverse Fourier
transform vanishes on a set of positive length and belongs to the
domain of ~ .
Kargaev's results All soon be published. The THIRD CONJECTURE is true and follows from the Beurling-~alli-
avin multiplier theorem (this fact was overlooked both by the author
and by the editors). Here is THE PROOF: there exists an entire func-
tion ~ of exponenti~ t~e ~ ~ ~0 satis~yin~ t lk I ~ on ~ . Then ~ ' ' ~ ~[X) is in the domain o f K •
555
9.10. NON-SPANNING SEQUENCES 0F EXPONENTIALS
ON RECTIFIABLE PLANE ARCS
Let A : (~) be an increasing sequence of positive numbers
with a finite upper density and let ~ be a rectifiable arc in ~.
Let C (~) denote the Banach space of continuous functions on
with the usual sup-morm. If the relation of order on ~sdenoted by<
and if Eo and Z 4 are two points on ~ such that ~0< Z! we set
The following theorem due to P.Malliavin
a necessary condition in order that the sequence (eXZ)~A.~
~on-spanning in C(y;
THEOREM. If the class
c°° (Mr.,,
and J.A,~Siddiqi [ 7]gives
be
i,,s non-empty .for some Zo,ZIC ? , where
M~= ~p_ ~ ~cA
~hen (eX~) ~ eA is ~on-spa~in~ in C (y) . It had been proved earlier by P.~alliavin and J.A.Sidaiqi [6]
that if ~ is a piecewise analytic arc then the hypothesis of the
above theorem is equivalent to the :5/Itz condition ~ ~-~ < co
In connection Wlth the above theorem the following problem remains
open.
PROBL~ I~ Given any non-quasi-analytic class o.f functions.....on
in the sense .of Den,~o~-Carleman , to f..ind a non-zero function. ~e.l.on-
~im~ to that class and hay in ~ zeros of infinite order at two peint.s
ix } With certain restrictions on the growth of the sequence
partial solutions of the above problem were obtained by T.Erkamma [3]
and subsequently by R.Couture [2],J.Korevaar and M.Dixon [4] &nd
556
M. Ltundin ~ 5 ].
Under the hypothesis of the above theorem, A.Baillette and J,A.
Siddiqi [1] proved that ( A~ e )AeA is not only non-spanning but
also topologically linearly independent by effectively constructim~
the associated biorthogonal sequence. In this connection the following
problem similar to one solved by L.Schwartz [8] in the case of linear
segments remains open.
To characterize the closed linear span of (eA~)AeA PROBLEM 2,
i_.nn C (V) when it is non-spanning. @
REFERENCES
I. B a i 1 1 e t t e A., S i d d i q i J,A. Approximation de fon-
ctions par des sommes d'exponentielles sur un arc rectifiable -
J.d'Analyse Math., 1981, 40, 263-26B.
2~ C o u t u r e R. Un th~or~me de Denjoy-Carleman sur une courbe
du plan complexe. - Proc,Amer,Math,Soc., 1982, 85, 401-406.
3. E r k a m m a T. Classes non-quasi-analytiques et le th~or~me
d'approximation de Muntz. - C,R,Acad.Sc. Paris, 1976, 283, 595-597.
4, K o r e v a a r J,, D i x o n M. Non-spanning sets of exponenti-
als on curves. Acta Math.Acad.Sci.Hungar, 1979, 33, 89-100.
5~ L u n d i n M. A new proof of a ~t[utz-type Theorem of Korevaar
and Dixon. Preprint NO 1979-7, Chalmers University of Technology
and The University of Goteborg,
6. M a 1 1 i a v i n P°, S i d d i q i J.A. Approximation polynS-
miale sur un arc analytique dans le plan complexe. C.R.Acad. Sc.
Paris, 1971, 273, 105-108.
7. M a 1 1 i a v i n P., S i d d i q i J.A. Classes de fonctions
monog&nes et approximation par des sommes d'exponentielles sur un
arc rectifiable de ~ , ibid., 1976, 282, 1091-1094.
L. Etudes des sommes d'exponentielles. Hermann, 8~ S c hwa r t z
Paris, 1958.
J.A°SIDDIQI I
Department de Mathematlques
Universit~ Laval
Quebec, Canada, GIK 7P4
557
T
It is a well-known fact of Nevanlinna theory that the inequality
in the title holds for boundary values of non-zero holomorphlc func-
tions which belong to the Nevanlinna class in the unit disc. But what
can be said about s!,m~ble functions S with non-zero Riesz projec-
tion~_~=~= 0 ? Here~_~ %e~ E ~(-~)~, l~J~1 • ~>Oc
Given a positive sequence ~ M ~ } ~>~ 0 dsfine
It is assumed that a) M~<~M~_,M~÷~,~:~,~ ; ~) '~,~ ~ : 0 , ~0
~ ' ' ' " '~, - - ~ 0 , 0 "
This does not restrict the generality because every Carleman class
coincides with one defined by a sequence satisfying a) and b). Let
T(~) ~-- ~t~ -~- , x~ 0 . ThenC~ M~} is a quasianalytic class
iff ~ ~ ~. ~ . In case C{Mil,} is non-quasiana-
lyric there are , of c@urse, functions ~ in C { M ~ } with 4 "~(~I~]~i'li,~-oo , in fact there exists an I in CIM } equal to zero on an open subset of
QUESTION. Suppose CI M~} is a quasi~l~tic class and let
~'(~) with ~ _ ~ M~} . Is it true that
Under some additional assumptions on regularity of~ M~} the ans-
wer is yes L lj
R~ERENCE
I. B o x ~ d e p r Ajl. Jlorapz~M HOqTE aHaJ~TZ~ecKo~ ~yHE~Z cy~M~py-
e~. - ~o~.AH CCCP, 1983, 265, 1317-1323.
A.L.YOL'BERG
(A.~.BO~SEPr) CCCP, 197022, JleH~HrpaA
yx.Hpo~eccopa HO~OBa 5,
JIeH~HrpaAcEE~ B2eETpOTeXH~eoEI~
~HCT~TyT
558
9.12. AN ALTERNATIVE POR ANALYTIC CARI,~N CLASSES
Given a sequence of positive numbers ~M~}~ o let C~ Mw} be the
Carleman class of infinitely differentiable functions on the unit
circle T satisfying
Yv ~p. J ~ ) ] ~ C~Q s H~ s~T
for ~=0,I,~ .... and some positive constants C~, Q~. A class of functions defined on T is called quasi-~alytic
it does not contain any function with ~(~)(~)~ 0 for some ~ in
and every ~0,I,... , besides~ 0 . Otherwise, the class is called
nonquasi-analytic. Clearly, each nonquasi-analytic Carleman class
contains a nonzero function vanishing on any given proper sub-arc of
. The well-known test of Carleman [I] provides a convenient cri-
terion in terms of ~ M~} to determine whether C{ M~} is quasi-ana-
lytic or not.
The analytic C leman classes cAI cl # =o, K=-I,-Z,... ~ can also be split into quasi-analytic and nonquasi-
analytic ones. There exists an analogue of Carleman's test for such
classes [2] , but in contradistinction to the classical Carleman clas-
ses a nonzero function in CA{~} , being the boundary values of a
bounded holomorphic function in the unit disc, cannot vanish on any
subset of ~ having positive Lebesgue measure. Nevertheless, for some
nonquasi-anslytic classes CA{M~J zero-sets of~. ,~functi°ns can be
rather thick. This is the case, for example, if M~-(~[) i÷~'~, ~=o,I,...,
where 0 ~ ~< ~ (see [3] ). Therefore it looks reasonable to formu-
late as a com~ecture the following alternative~
CONJECTURE I. For every, positive sequence { ~ ~ ~ ~ o either the
analytic Carleman class C A ~ Mw} is quasi-anal~tic or there exist a
non-empt,7 perfect subset ~ of ~ and a nonzero function & in
CA{ M~} such tha t el E---=- o
The alternative, if true, would have a nice application to dis-
sipative Sehr~dinger operators. Consider the class E of all bounded
measurable real functions ~ on [0,÷oo ) satisfying
Given ~6 let ~ be the Schrodinger operator in [,~(O,~=o) de-
fined by
559
, - o.
The operator ~ is selfadJoint for real V an~ real ~ and it can
have only a finite number of bound states, i.e, eigenvalues, if
~ B . Por complex ~ the situation changes considerably. Now the
number of bound states is finite if
and on the other hand for each ~ in (0,1) there exist a real-valued
potential ~ satisfying
and ~eC, ~ 0 such that ~ has i~finitely many bound states
(see [4] ). It can be even shown that the family of all closed sub-
sets of ~ ,which may serve as derived sets of the point spectrum of
$~ with the potential V satisfying (I), coincides with the fa-
mily of compact non-uniqueness sets in (0~+~o) for the Gevrey class
~ CA{M~} , M@=(~[) ~*~I~ (see [5] ). The above considera-
tions make plausible the following conjecture,
CONJECTURE 2. L,et ~ be, a positiv e f~otion on [0, ~ee) such
~ha$ $ ~ ~ ~( 6 5) is convex, Then either ever~ Shrodin~er operator
~ with the ~otential V sa,tisfyin~
Qo'~J~; • ~ 0 (2)
has only finite number of bound states or there exist V satisfyin~
(2) and~C~ ~ ~ 0 , such that the derived set of the point spect-
rum of ~ is non-emvty an d perfect~
REFERENCES
I. ~ a n d e i b r o j t S. S@ries adh~rentes~ R@gularisation des
suites. Applications. Paris, 1952. 2. R • - S a I i n a s B. ~unctions with null moments. - Rev. Acad.
Ci.Madrid, 1955, 49, 331-368.
3- H r u ~ ~ e v S.V. Sets of uniqueness for the Gevrey classes. -
Arkiv for Mat., 1977, 15, 253-304. 4. H a B ~ O B B.C. 0 HecaMoconpa~H~o~ onepaTope mp~Hrepa I, H,
~. - B EH.: "Hpod~.MaTeM.$Z~.", I986; I967; I968; B~n. I, 2, 3,
560
~e~Hrpa~, ~Y, 102-132; 133-157; 59-80. (English translation:
Pavlov B.S. The non-selfadJoint SchrBdimger operator I, II, III. -
in: Topics in ~ath, Physics, 1967; 1968; 1969; Consultants Bureau,
N.Y., 87-114; 111-134; 53-71.) 5. H r u ~ ~ e v S.¥. Spectral singularities of dissipative Schr~-
dimger operators with rapidly decreasing potential. - Indiana
Univ.Math.J. (to appear).
s.v.mu~v (c .B.XPY~2B)
CCCP, 191011 ~e~rpa~, ~-II
• OHTaHEa 27 ~0MM AH CCCP
561
9.13. ON A UNIQUENESS THEOREM old
The symbol H(~), ~ being an open set in ~ ,denotes the
set of all functions analytic in ~ . Let ~ , ~ c ~ (~>4)
and l e t C = ( e l , C~,... ,C,)~ ~ . D e f i n e t h e f o l l o w i n g s e t s
~ C ~ , c ) : { ~ : ~ + ~ c " : m ~ o , % q > q , j = ~ , . . . , , } ,
~ C O , c ) : ~ ~ CO ,c) ,
~D ( ~ , c ) =e~Uz (~' ~)~ CO,c) , ~D (o,c) - uz(.) ~ C o,c) .
~ppose t ~ for ~ ~ct ion j~ , ~ ~HC~(_Q,O)) the restriction (~,0)-- ~ is continuous on the set ~ (n, O) . Then
the function ~
~cm)= ,T. ~ .J~Cm), m~}, ~ ~=~~'. . . 'd~
is well-defined in ~ . The following uniqueness theorem has been
proved in [I].
If there exist C =(C 4,C 2~,,.,C~)~ ~+ and functions
C n : { ~ n : a~t(=, an)> nclt=~c~.c~+ ...+c, ~ }.
Note that the theorem is important for studying homogeneous con-
volution equations in domains of real (R ~) or complex (C") spaces
(see [I] , [2] , [3] ). 0ne might think that,=--0 on ~ , as it occurs in
the one-dimensional case. However there exists an example (see [I] )~
vhere all conditions of the uniqueness theorem are satisfied, but
562
~0 in ~ (for sufficiently large IICII ). Hence the appearance of the set ~C is therefore inevitable although ~C does not
seem to be the largest set where ~=0 •
PROBLEM. Pind the maximal open subset of the domain ~ where
REFERENCES
I. H a ~ a ~ E o B B.B. 06 o~o~ TeopeMe e~HCTBeHHOCT~ B Teop~ ~m~ ~mi~x EO~,~eEcHHX uepeMeHHHX X o~opo~mHe ypsmHeH~ TZ- na c~epT~Z ~ Tpy6~aT~X 06AaCT~X ~ -- HSB.AH CCCP, cep.~aTeM. 1976, 40, ~ !, 115-132.
2. HauaaEoB B.B. EH Ka B~ny~HX O6maCT~X 804-80V.
3. Ha~ax~oB B.B.
0~opo~e CHCTeMH ypasRem~ T~a cBepT-- ~. -~o~x.AH CCCP, 1974, 219, ~ 4,
0 pemem~x ypa~Rem~ 6ec~oHe~oro ~op~- • a B ~e~CTB~TeX~O~ C4AaCTH. -- MaTeM.c6., 1977, 102, ~ 4, 499--510.
V. V. NAPALKOV (B.B.HAIIA~OB)
CCCP, 45O057, Y#a yx. Ty~aeBa, 50 Fmm.lpc~ @~ax AH CCCP
Ce~op MaTeMaTNEM
CHAPTER 10
INTERPOLATION, BASES, MULTIPLIERS
We discuss in this introduction only one of various aspects of
interpolation, namely the f r • • (or Carleson) interpolation by
analytic functions.
Let X be a class of functions analytic in the open u~t disc
. We say that the interpolation by elements of X on a set Ec
is free if the set XIE (of all restrictions ~I E ,Se X ) can be desc-
ribed in terms not involving the complex structure inherited from ~ .
So, for example, if ~ satisfies the well-known Carleson condition
(see formula (C) in Problem 10.3 below), the interpolation by ele-
ments of H ~ on E is free in the following sense: a n y lunc-
H ~ rich, bounded on ~ , belongs to I ~ . The freedom of interpo-
lation for many other classes X means (as in the above example)
that the space ~I~ is ideal (i.e.~XI~,]~l~I~l on ~>
=>~I ~ )" Sometimes the freedom means something else, as is the ca-
se with classes ~ of analytic functions enjoying certain smooth-
ness at the boundary (see Problem 10.4), or wlth the Her~ite inter-
polation with unbounded multiplicities of knots (this theme is trea-
ted in the book H.E.H~KO~BCE~, ~eE~N~ 06 oHepaTope C~BEIB, MOCKBa,
HayEa , 1980, English translation, Springer-Verlag, 1984; see also
the article B~HoPpa~oB C.A., PyE~H C.E., 8an~cEE Hay~H~X CeMzHapoB
~0~, I982, I07, 36--45).
564
Problems I0.1-I0.5 below deal wit~ free interpolation which is
also the theme (main or peripheral) of Problems 4.10, 6.9, 6.19, 9°2,
11.6. But the imfoxlmation, contained in the volume, does not exhaust
the subject, and we recommend the survey BNHOPpa~OB C.A., XS3~H B.~.,
8an~cF~ Hay~H.ce~HapoB ~0MM, 1974, 47, 15-54;1976, 56, 12-58, the
book Garnett J., "Bounded analytic functions" and the recent docto-
ral thesis of S.A.Vinogradov "Free interpolation in spaces of ana-
lytic functions", Leningrad, 1982.
There exists a simple but important connection of interpolation
(or, in other words, of the moment problem) with the study and clas-
sification of biorthogonal expansions (bases). This fact was (at
last) widely realized during the past I~-20 years, though it was exp-
licitly used already by S.Banaoh and T. Carleman~ Namely, every pair
o f b i o r t h o g o ~ l families ~-----{ ~I}AG ~ , ~'/" { ' ~A}l~ ( &A are vec-
tors in the space V , ~ belong to the dual space, <~'~>=~A~ )
generates the following interpolation problem: %o describe the coef-
ficient space ~V (~S ~ ~,~>}Ae~ ) of formal Fourier
expansions ~ ~ ~, ~> ~ . There are also continual analo- A
gues of this connection which are of importance for the spectral
theory. "~reedom" of this kind of interpolation (or, to be more pre-
cise, the ideal character of the space ~ V ) means that ~ is an
unconditional basis in its closed linear hull. This observation plays
now a significant role in the interaction of interpolation methods
with the spectral theory, the latter being the principal supplier of
concrete biorthogonal families. These families usually consist of
eigen- or root-vectors of an operator ~ (in Function Theory
is oftem differentiation or the backward shift, the two being iso-
morphlc) : T~l= ~I A , ~E g . Thus the properties of the equation
~-~ ( ~ is the given function defined on 6" ) depend on the amo-
unt of m u 1% i p 1 i e r s of ~ , i~e. of operators ~-~
sending ~A to ~(~) ~A , where ~ denotes a function C--~ ~ or the
565
multiplier itself). These multipliers ~ may turn out to be functions
of ~ ~-~CT)) and then we come to another interpolation problem
(given ~ , find ~ ). The solution of this "multiplier" interpola-
tion problem often leads to the solution of the initial problem
~ ~ . Interpolation and multipliers are related approximately in this
way in Problem 10.3,whereas Problem 10.8 deals with Fourier multip-
liers in their own right. These occur, as is well-known, in numerous
problems of Analysis, but in the present context the amount of mul-
tipliers determines the convergence (summability) properties of stan-
dard Pourier expansions in the given function space. (By the way,
the word "interpolation" in the title of Problem 10.8 has almost
nothing to do with the same term in the Chapter title, and means the
interpolation o f o p e r a t o r s . We say "almost" because
the latter is often and successfully used in free interpolation). We
cannot enter here into more details or enlist the literature and re-
fer the reader to the mentioned book by Nikol'skii and to the artic-
le H1~a~6$v S.V. ,Nikol'skii N.K., Parlor B.S. in Lecture Notes in Math.
864, 1981.
Problem 10.6 concerns biorthogonal expansions of analytic func-
tions. The theme of bases is discussed also in 10.2 and in 1.7, 1 10,
1.12.
Problem 10.7 represents an i n t e r e s t i n g and vast aspect
of interpolation, namely, its "real" aspect. We mean here extensio~
theorems b. la Whitney tending t o the constructive description of tra-
ces of function classes determined by global conditions.
Free interpolation by analytic functions in ~ (and by harmonic
functions in ~ ) is a fascinating area (see, e.g., Preface to Gar-
nett's book). It is almost unexplored, not counting classical results
on extensions from complex submanifolds and their refinements. Free
interpolation in $~ is discussed in Problem 10.5.
566
10.1. old
NECESSARY CONDITIONS POR INTERPOLATION
BY ENTIRE ~UNCTIONS
Let ~ be a subharmonic function on C such that ~(¢+Izl)= ~(~(~)) and let A# denote the algebra of entire functions
such that I ~(~)I~ < A for some A2 > 0 Let V
denote a discrete sequence of points { a%l of C together with a
sequence of positive integers ~p~] (the multiplicities of {~n] )"
If~A#, ~#~ 0 , then V(~) denotes the sequence ~a~} of zeros of
and ~% is the order of zero of ~ at @~ .
In this situation, there are THREE NATURAL PROBLemS to study.
i. Zero set problem. Given ~ , describe the sets V(~) , ~A 3 .
Ii. Interpolation oroblem. If {~,~-Vc V(~) for some
~, ~ A# , describe all sequences { ~I$,K } which are of the
form
~cK)(~) O~ < J<< Kr ' f o r (T)
III. Universal Interpolation problem. If Vc V(~) for some
5, ~ Af , under what contitions on V is it true that for
e v e r y sequence ~KJ such that I~K I ~ A ~2(B~(@~))
there exists ~ , ~ Af , satisfying (I).
In case #(~)~(I~ ) (and satisfies some mild, technical condi-
tions), quite good solutions to problems I-III are known. This work
has been carried out by A.F.Leont'ev and others (see e.g. ~] for a
survey). However, when ~ is not a function of ~I , the general so-
lutions are not known.
T h e p u r p o s e o f t h i s n o t e is to call
attention to an interesting special case of III. Consider the case
~(~) =I 111~ I+ ~(~l~l~).Then A~ =~f , the space of all entire func-
tions of exponential type with polynomial growth on the real axis.
The space ~f is of special interest because, by the Paley-Wiener-
Schwartz Theorem, it is the space of Fourier transforms of distribu-
tions on ~ with compact support. The problems I-III are then dual
to some problems about convolution operators on the space ~c~(~)
(see eg or D]) Specifically, suppose
for some 8>0 , G~0 ,~m~ z , we have
567
V = t oI¢,~,i,i,}¢V(.;[-.) , where
=I Im [ + •
(2)
Then it is not hard to show that (2) is a sufficient condition that
has the universal interpolation property III. We wish to pose
the converse problem.
PROBLEM. Suppose" that Vc~(~) for some ~ , ~ ~r , and
t~t V is a universal interpolatin~ sequence: i~e~ III holds, Is ^#
it true that (2) must hold for some ~ , ~8(~)?
In all the cases known to the author where the PROBLEM has an-
swer yes, it is also true that the range of the multiplication opera-
tor~:~#--~A# given byM~(~):~ is closed. Is the fact that~ ~
has closed range necessary for a "yes" answer? (In the case~#=b ,
if ~ has closed range, then the PROBLEM has answer yes, as can be
shown by the techniques of [4] ). However, the main interest in the
PROBLEM is to find if (2) must hold with no additional assumptions
On P ,
REFERENCES
1. Jl e o H T B e B A.$. 0 CBO~OTBS~ nOC~Ie~OBaTe~BHOCTe~ J21He~HNX
arperaTOB, cxo~jn~Exc~ B o6~acT~,r~e n o p o ~ xHHe~H~e arpera-
TU C~CTeMa ~HE~ He ~eTc~ nox~ofi. - YcnexH MaTeM.HayE, I956,
II, ~ 5, 26-37. 2. E h r e n p r e i s L. Fourier Analysis in Several Complex
Variables. New York, Wiley-Interscience, 1970.
3. H a a a M 0 ~ 0 B B.H. ~e~H~e ~E~epeHn~a~HHe onepaTopu c
nOCTOm~U~ Eo~x~eHTa~m, M., HayEa, I967. 4. E h r e n p r e i s L., M a 1 1 i a v i n P. Invertible ope-
rators and interpolation in A ~ spaces. - J.Math.Pure Appl.
1974, 13, 165-182.
5. B 0 p ~ C e B ~ ~ A.H., ~ a n:z H r.H. 05 ZHTepnoxHpoBa2~
sexHx ~y~. - CE6.MaTeM.m., I968, 9, ~ 3, 522--529.
B.A, TAYLOR Mathematics Depa± ~ment
The University of Michigan
Ann Arbor, Michigan 48109
USA
568
CO~ENTARY
Papers [6], [7] contain useful information concerning the
Problem.
REFERENCES
6. B e r e n s t e i n C~A., T a y 1 o r B.A. A new look at inter-
polation theory for entire functions of one variable. - Adv.
Math., 1979, 33, N 2, 109-143.
7~ S q u i r e s W.A. Necessary conditions for universal interpo- g, lation in , - Canad. J. Math., 1981, 33, N 6, 1356-1364
(~R 83g: 3oo4o),
569
10.2. BASES OP REPRODUCING KERNELS AND EXPONENTIALS
I. Bases of exponential volynomials. For a non-negative integer-
valued function ~ (a divisor) in the complex plane ~ let us
denote by ~ (~) the family ~ ~A: ~ ~ } of exponential
polynomial subspaces ~A= I 26~Ax : p is a polynomial, ~ ~ < ~ (~) }.
QUESTION I, Pot what divisors ~ does the famil~ ~ (k) form an
unconditional basis in the space L~(0,@) , ~ > 0 ?
"Unconditional basis" is used in the usual sense and means the
existence, uniqueness and unconditional convergence of the expansion
for any function ~ ~( 0, ~) . It is clear that in this case
k~-0 off a countable discrete set ~= ~pp ~ and the starred ex-
pansion turns out to be a generalized Pourier series with respect to
the minimal family of subspaces ~(k)
The most interesting problem arises for ~<~ (i.e. k=~,the
characteristic function of ~ ) in which case the reader deals, as a
matter of fact, with the well-known problem on exponential bases on
intervals of the real axis.
Here is a bit of known information:
I) for K=~, 6~c~+ de~I~:~>0 } the Question has been
answered in [I]. Namely, ~ must be a (Carleson) interpolation subset
of ~+ and the function X ~ @~$eE~(x) must satisfy the Devi-
natz - Widom condition, where B~ is the Blaschke product with zero
set ~ and $ = ~xp (~@~) . In case st~p~<+~ the ans-
wer can be reformulated in terms of density of Y . Paper [I] contains
also exhausting historical remarks.
2) In the limit case @=oo (which implies 0~p~ cC~ ) no
~(k) forms a basis in ~(0, +e°) . The right analogue of the
problem in such a situatuon is to describe all divisors ~ for which
~(k) is an unconditional basis in the closed linear span of
~Qk} ~de~ 57@~ ~(~}) This problem has been solved in its complete
generality in [2] in terms of the generalized (multiple) Carleson con-
~ition.
3) It is mot hard to see that for o pp k c 6 + the Ques-
tion is equivalent to a kind of multiple free interpolation problem
for entire functions of exponential type @/~ (see [1], ~] for de-
tails).
570
2. Bases extendin~ a given basis. Exponential (or exponential-po-
lynomial) bases problem is a special case (@=*~(~@~)) of the
problem on reproducing bases in the model space
where ~+~ stands for the usual Hardy space in ~+ and @ is an in-
ner function. Denote by
the r e p r o d u o i l l g k e r n e l f o r K e for 6~c ~. •
QUESTION 2. Let ~@(~) be an unconditional bas is in ~p~w~(~9.
Is it true that there exist ttnoonditional base@ ~O( e z) in the
whole space K 8 contaimin~ ~O(~) (i.e. such that ~c ~""c., 6.~ ) .~
and put ~ o (~) = { ko ('' ~) : A ~ ~ }
QUESTION 2'. Let @~ 0 and let $(~) be an uncondit~o~! basis
i_.nn ~p~ ~(k) - !s %% true that there exist u/iconditional Rases
~(k f) in the whole space L~(0, ~ contaimi~ ~(~) (i.e~ such
that k~ ~z) ? Is it possible to choose such a ~r multiplici-
ty-free (i.e. ~r= ~er ) provided K-----~ ?
The second part of Question 2' is a special case of Question 2
0 = C~(~) ) • The answer to this part of Question is
known to be positive (V.l.Vasytulln, S.A.Vinogradov) under some addi-
tiomal assumptions (i.e.a quantitative relation between ~5 ~ ~,
~----- St~ k and the interpolating constant of ~ , see [I]).
3. Existence of a basis.
QUEST!O~ 3. In which model space KO does there exist an uncgn-
ditional basis of the form ~@(~) ?
Each of the following two questions 3' and 3'' is equivalent to
Question 3 (see [I], ~] for the proofs). For which inner functions
@ does there exist an interpolating Blaschke product ~ such that
3' ) ~,~,~¢@,BI-I"~)<I, d / i~¢B, OH'~) < ' I ? o r
3' ' ) the Toeplitz operator T@~ ~ deU % ~. O~ ~ , ~ H ~ is inver-
571
tible in ~ a ?
It is proved in [4] that, @ being an inner function, there
exist interpolating Blaschke products ~, ~f such that~@]~- ]~ < II 1 It follows that the space ~8 can be "complemented" by the space ~B
with an unconditional basis of reproducting kernels ( == of ratiomal
fractions in this case) to the space ~@B ~ c~eG(~e+~B)' ~SN~B = ~
in such a way that ~@B has also an unconditional basis of the form
A limit case of the problem (the existence of o r t h o g o n a 1
bases of the form ~@(.,i), I~I= ~) is considered in [5].
REFERENCES
%, %,~*
I. Hruscev S.V., Nikol'skii N.K., Pavlov B.S. Unconditional bases of
exponentials and of reproducing kernels, Lect,Notes in Math., 1981,
v.864, p. 214-335.
2. BacmH~ B.~. Bes~caoB~o cxo~sm~ecs cne~paa~e pasaomeH~s m 3a~-
~m ~H~epnoas~s. - Tp~ maTem,a~-Ta ~M.B.A.CTezaoBa AH CCCP, I877,
130, c. 5-49. 3. H~Koa~cK~ H.K. ~ez~ o5 one~aTope o~ra. MooFma, Hs~m, 1980.
4. Jones P.W. Ratios of interpolating Blaschke products. - Pacific J.
Math., 1981, v.95, N 2, p.311-321.
5. Clark D.N. On i~terpolating sequences and the theory of Hankel and
Toeplitz matrices. - J.Ihmct.Amal., 1970, v.5, N 2, p.247-258.
N. K. Nikol ' 8kii
(H.K.H~Koa~c~H~)
CCCP, IgIOI I, ~eNMHrpa~
~o~a~Ka 27, ~0~4
572
10.3. MULTIPLICATIVE PROPERTIES OF ~F . old
Let ~ be the Banach space of all functions ~-~-~0 ~(K)~
holomorphic in the unit disc D and satisfying
algebra with respect to the pointwise multiplication of functions if
p=#~ . Therefore, when studying the multiplicative structure of 6P, P ~.e, f ? p p "'
the space ~A = ~ A : ~eSA' V~CCSA } becomes very important. R'ecali that M P ' = MI ~*~=4, 4<P~; ~ coin- . . " 9, '~ 4 r ~ -r ~ p - oo _<
cides with the Hardy class H ; MA: ~k, ~A c MAc H , I~-P~ % • The first conjectures of the paper are closely connected with the
theorem of L.Carleson [_I] on the interpolation by bounded analytic
functions. Given a subset E of D let ~ £ denote the restriction
operator onto E.
THEOREM [ I ] . ~E ( H ~ ) = $o0(E) i l ~
Note that (O), being necessary for ~E(M~)=~(E), ~ <P< %,
i s n o t s u f f i c i e n t * ) . On t h e o t h e r lu~Zd i t t u r n s o u t t o be s u f f i c i e n t
for ' ~ < p ~ if E satisfies the Stolz condition (i.e. E is con-
tained in a finite union of domains ~(~) ~ {~:l~-~I<~ ~(~--I~I)} ,
where ~<~<~, ~T ), cf. [2] • Suppose that E satisfies the Stolz condition. Then it is easy
to check (see [4]) that
xb>0 & rcE) : .
I ~ 1 ~ 1 ~ 1
The conditions ~( ~ ) > 0 and ~ (E) < + ~ are important for
the problems of interpolation theory in ~P as well as in other spa-
CeSo [3]. Everything said above makes plausible the following conjec- ture.
*) The{ "(C) does not imply~E(~)= ~(E), ~<p'< ~ can be proved with help of [3].
573
CONJECTURE 1.6"( E)>0, ~" ( E)<÷~ => ' -~E( H~ ) ~ ~/,o( E ; .
Conjecture 1 i s r e l a t e d to
CONJECTURE 2. ~(~)<+~ ~---> BE~ N NPA where
stands for the Blasc~e product ~enerated
b y E .
CONJECTURE 1 follows from CONJECTURE 2. To see this it is suffi-
cient to apply the Earl theorem [5] about the interpolation by Blasch-
ke products. It is not hard to show that the zero set F of the cor-
responding Blaschke product can be chosen in this case satisfying
6"(E)P0, ~(E)<~ (see [6], §4 for details),
~=i~ ~p~°° ) that every follows fr= = M (v +
inner function I *~ ~ satisfies
for ~<p'~ ~. Therefore the proof of CONJECTURE 2 would give new non-trivial examp-
les of Blaschke products T with property (I). In this direction
at present, apparently, only the following is known. I . I~ ~oop z+~ does not satisfy (I) for I~p~$/3
(see [7], [8]). 2. For #<p~% ]~E satisfies (I) provided ~(E)<
and ~ satisfies the Stolz condition (see [2] ).
CONJECTURE 3. (a). Suppose ~ (E) < ~ T hen BE satis-
fies ( I ) , ~[ <p<%" (b) If ~(E)<.,o ~n~ ~ satisfies the Stolz condition
then B E satisfies (I)with p-~-~ .
Analogous conjectures can be formulated for multipliers of @@
REFERENCE~
I. C a r 1 e s o n L. An interpolation problem for bounded analytic
functions. - Amer.J.Math., 1958, 80, N 4, 921-930.
2 . B I H O r p a ~ o B C . A . ~y.~Tllnm~KaTopbl cTenelotl~X p~l~o~ c n o c ~ e -
~0BaTeJ~bHOCTb~ K0e~H~eHTOB MS ~P . - 3 a n . H a y ~ . c e M ~ . ~ I O M H ,
574
1974, 39, 30-40.
3. B H H O r p aA o B C.A. BasHcm Hs no~asaTe~H~x ~JH~ ~ CBO-
doAHa~ ~H~epnoasu, s B daHaxoB~x npocTpaHcTBaX C ~P- HopMoR. -
3an.Hay~H.ceMHH.~0~4, 1976, 65, I7-68.
4. B H H 0 r p a~ o B C.A., X a B H R B.G. CBodo~Ha~ HHTepnox~H2
B ~ H He~0Top~x ~tDyr,x ~accax SyH~u~. I. - 3an.Hay~H.ce~H.
~0E~, 1974, 47, I5-54.
5. E a r 1 J.P. On the interpolation of bounded sequences by boun-
ded analytic functions. - J.London Math.Soo., 1970, 2, N 2, 544-
548.
6. B ~ H O r p a~ o B C.A., X a B H H B.II. CBOdO~a~ ~HTepno~m~
B H @° H B He~oTopmx Apyr~x ~accax #yHKL~. II. - 3an.Hay~.ce~gH.
~10~I, 1976, 56, 12-58.
7. F y p a p H ~ B.l]. 0 ~aETopHsa~HN a6COJI~THO CXO~MXC~I p~OB Te~-
aopa M ~HTeI'pa~OB ~ypBe. - 3aN.Hay~.CeM~H.~I0~, 1972, 30, 15-32.
8. ~ ~ p o ~ o B H.A. He~oTop~e CBO~OTB& np~apH~( ~ea~o~ adco~-
Ho CxoA~m~Hxc~ p~OB Te~opa z 14HTerpa~oB ~ypBe. - 3al~.Hay~.ceMNH.
~I0~, 1974, 39, 149-161.
S • A. ¥1NOGRADOV
(C.A.B~0~A~) CCCP, 198904, /[eHzHrpa~, IIeTpo~Bope~, JIeR~Hrpa~icK~ rocy~apcTBeH~ yH~Bepcz-- TOT, MaTeMflTI~O--MexaHI~ecEE~ ~yJIBTeT
***
COM~ENT~Y BY T~ AUTHOR
Conjectures 2 and 3 are disproved in [9] (see Corollary I in [9]
disproving Conjecture 2 and Corollary 2 and Theorem 5 in [9] dispro-
ving Conjeoture 3> es, ts [8], [91 ead to the lowing question.
QUESTION. Is there a sin6~lar inner function ~n U ~ I~ L ,, ?
REFERENCES
9. B ~ H O r p a ~ o B C.A. My~TZna~xaT~Bm~e CBO~CTBa CTeHeHEaX
p~OB C noc~e~oBaTe~HocT~ XO~m~eHTOB ~8 ~P -. ~O~a.AH CCCP,
1980, 254, ~ 6, 1301-1306. (Sov.Math.Dokl., 1980, 22, N 2~ 560-565)
I0. B e p d z ~ E z i~ I~.3. 0 ~yx~TZn~EEaTopax IIpOCTpaHCTB ~ -
~rH~.aHa~ms zero npEJI., 1980, 14, BI~II.3, 67-68.
575
10.4. FREE INTERPOLATION IN REGULAR CLASSES
Let ~ denote the open unit disc in C and let X be a closed
subset of ~ . For 0 <%<I , let ~& denote the algebra of holo-
morphic functions in ~ satisfying a Lipschitz condition of order
. The set X is called an interpolation set for ~& if the
restriction map
A~ ~,,, L~p (~,X)
is onto. The interpolation sets for ~& , 0 < & < ~ (and also of
other classes of functions) were characterized by Dyn'kin in [3] as
those for which the following conditions hold:
The cond i t ion (K) I i f ~ ( K ) ~ { [ { - W ~ : W C ~ arcs I C T ,
, then for all
where IIl denotes the length of I.
The Carleson condition (C) : ~ must be a sequence
such that Cz~)
'~ ~ Iz~-z~l >0.
In the limit case ~ =~ ways of posing the problem:
I. We can simply ask when the restriction map A4-~
is onto, A~ being the class of holomorphic functions in
tisfying a Lipschitz condition of order I .
2. We can also consider the class
A~ = H(~) a c ~ (D)
and call ~ an interpolation set for
(the space of Whitney jets) such that in
A wi th , on X .
there are (at least) three different
Lip ('[, X) ~) sa-
A ~ if for all ~C ~(X) ~ 0 there exists {
576
3. Finally one can consider the Zygmund class version of the
problem. Let A, denote the class of holomorphic functions in h having continuous boundary values belonging to the Zygmund class of
. We say that ~ is an interpolation set for ~, if for any
~= in the Zygmund class of ~ there exists ~ in ~ such that
on X In [I] and [2], it has been shown that Dyn'kin's theorem also
holds for ~ -interpolation sets. For A 4 interpolation sets the
Carleson condition must be replaced by
(2C) ~ N~ is a union of two Carleson sequences.
Our PROBLE~ is the following: which are the interpolatiom sets
for the Zygmund class?
Considering the special nature of the Zygmund class, I am not
sure whether the condition describing the interpolatinn sets for A~
(one can simply think about the boundary interpolation, i.e. ~C~ )
should be different or not from condition (K). Recently I became
aware of the paper [4], where a description of the trace of Zygmund
class (of ~ ) on any compact set and a theorem of Whitney type are
given. These are two important technical steps in the proofs of the
results quoted above and so it seems possible to apply the same tech-
niques.
REFERENCES
I. B r u n a J. Boundary interpolation sets for holomorphic func-
tions smooth to the boundary and B~i0. - Trans.Amer.Eath.Soc.~1981,
264, N 2, 393-409~
2. B r u n a J., T u g o r e s F. Free interpolation for holo-
morphic functions regular up te the boundary.-to appear in Pacific
J. Math°
3. ~ U H ~ ~ ~ ~ E.M. ~[~o~ecTBa CBO60~O~ ~ e p n ~ ~z~ F~ac--
COB r~epa. - ~aTeM.c6opH., 1979, 109 (151), ~ I, 107-128
(Math.USSR Sbornik, 1980, 37, 97-117).
4. J o n s s o n A., W a 1 1 i n H. The trace to closed sets
of functions in ~ with second difference of order 0(~). - J.
Approx.theory~ 1979, 26, 159-184.
J.BRUNA Universitat autBnoma de Barcelona
Secci~ matematiques. Bellaterra (Barcelona)
Espa~a
577
o l d
L e t ~N b e t h e u n i t b a l l o f ~N ( ~>~ ) a n d d e n o t e b y I~
H~( ]~ N ) t h e a l g e b r a o f a l l b o u n d e d h o l o m o z ~ h i c f u n c t i o n s i n ]~ ,
An a i ~ a l y t i c s u b s e t ~ o f ]~1t i s s a i d t o be a z e r o - s e t
f o r H~(R N) (in symbois, E~EH'(R ~) ) if there exists a non-zero function # in H ' ( B N) with E=~-~(O) ; E i s sa id to be a n i n t e r p o I a t i o n s e t f o r ~o(~N)
(in symbols: E ~ I~(~ N) ) if for any bounded holomorphic func-
tion ~ on E there exists a function ~ in H °°(~N) with
~IE = ~ . The problem to describe the sets of classes ~H~(B N)
and I ~ao(~N) proves now to be very difficult. I would like to
propose some partial questions concerning this problem; the answers
could probably suggest conjectures in the general case.
Let A be a countable subset of ~ . Set
~ 0 ~ 1,.What are the sets ~ such that ~& ~H~(~ N ) ?
PROBLEM 2. What are the sets A such that T& El Ho~(~N) ?
It follows easily from results of G.N.Henkin [I] and classical
results concerning the unit disc that the following two conditions
are ~ecessary f o r 7~ ~ ZH®r.~BN)-
'"' ) ~e~ -I(~,O)F,O <~ , Os$ ~',IOl=l, ( i )
~, O-t~i~) ~ <oo.
These conditions do not seem however to be sufficient.
A necessary (insufficient) condition for T A ~ ~H°°(~ N)
can also be indicated. Namely, T A E!H~°(~ ~) implies that there
exists ~A >0 such that for every ~v~ A the set ~,
no ellipsoid ~A(~ I) with &t~ ~ , where
• , _ _ ÷ ~(I-ImF) 10,1 ~ ~-i~t ~
(2}
intersects
(3)
578
+ 4-I~I~o ~z (1~1~ l(z,~)l ~ '~ ~-I~I ~ \ - - ~ - F ~ / < 4 } '
If A lies in a "sufficiently compact" subset of ~N , it is
possible to give complete solutions to problems I and 2. Let
~@ ~(0~4) C>0 ; the (2, 0, ~) -wedge with the top at a ~oint ~'g ~B ~ is, by definition, the union of the ball {Z£~ :I~I<])
and the set ~(e=) ~[zE~N:II~(4-(~,e0~l .< c~e((-(Z, eo));
The scale of all ~, C,~) -wedges in equivalent (in a sense) to that of Fatou-Kor~nyi-Stein wedges [I]. The following theorem holds.
THEOREM. Le__~t A be a subset of a finite union of (2, C,~)-wed-
~es ~ith ~ < ~ . ~hen TA ~Z~(~ ~) ~f a~d on!,7 if
(~-I@[ ~) <c~ ; 7 A~I~'(~ ~) if and only if there ~gA exists [>0 such that T A intersects no one of the sets Q[(~')
with ~i @ ~ , ~ (~) being defined by (3).
In view of this theorem the following specializations of Prob-
lems I and 2 are of interest. Let A be a subset of a (2~O,~)-wedge, but in contrast to the theorem ~ can be an arbitrary number from
(o,0 •
PROBL~ 1'. Is i t true that T A ~ Z ~ ( ~ ~) implies
P~0BLE~;~ 2,. ~s it t~ue that T~ n Q~(~')=~ for all ~'~
implies T A eIH~(~ ~) ?
I.
REFERENCE
X e H E ~ H r.M. YpaB~eH~e P.~eB~ H ahab,s Ha nCeB~OBNFJE~OM
MHOIDO6pas~.- MaTeM.c6., 1977, 102, ~ It 71-108.
N.A.SHIROKOV
(H.A.E~POKOB) CCCP, 191011, ~eK~HI~0S~
~0HTaHEa, 27
~0~
579
10.6. old
REPRESENTATIONS OF F<E~CTIONS BY EXPONENTIAL SERIES
1. Let L be an entire function of exponential type with zero
divisor k- kL (k(~) is the zero multiplicity of L at the point
, ~c C ), and let ~ the Borel transform of L , namely
C
(X~C)
where the closed contour C embraces a closed set ~0 containing
all singularities of ~ . There exists a family{~K, ~" 0% k< k(~)]
of functions analytic in C \ ~0 and biorthogonal to the family
{:Zse xz: O-~s<k( ,k ) } , so that
,XZ ~at;i,
c
where ~
in [1], p.228). Any analytic function on
rier series
is the Kronecker delta (see the construction of ~K,~ can be expanded in ~ou-
kOgq
~ L L a.,x e ; a.,xo2 ,k, k(),)~ k=o C
(1)
The following uniqueness theorem is known ([I], p.255): if L has
infinitely many zeros and~ is a convex set then~K,k~ 0 ~ ~ 0°
The proof uses in an essential way the convexity of~) °
PROBLEM 1. Does the uniqueness theorem hold without the convexi-
ty assumption?
2. Let ~ be the closed convex envelope of the set of singular
points of ~ and suppose that L has simple zeros only (i.e.k(~)6~ ,
~ C ). The necessary and sufficient conditions for series (S)
to converge to ~ in the interior of ~ for any # analytic in
are the following:
[~(g)-~l~l
580
for any ~ > 0 ;
b) there exist numbers ~ >0 and ~k ' O< ~k t Oo such
that J L (~ ) I> e PIXI J~J= ~k k~{
Condition a) ensures the convergence of (I) in int ~ and
b) implies that the obtained sum equals # .
PROBLEMS 2. Is b) implied by a)?
The negative answer would mean that series (I) generated by
may converge to a function different from ~ .
3. Suppose that int ~ is an unbounded convex domain containing
(-OO, 0) . Suppose further that ~.~ ranges over the interval
(-go,~o), O< ~o ~ ~/~ when ~__.[~+~:~O0~+~54~g_k(~)=O}U,~ ranges through the set of all supporting lines of int ~) . The possi-
bility for ~ to be supporting lines is not excluded but in
such a case, evidently, the boundary ~ eventually coincides with
Let ~(~)=~) and let
c (2)
(may be with ~=± ~o in the above mentioned case). All zeros of L
are assumed to be simple. Let {~A: k(~)>0} be the biorthogonal
family to [~ : k(~)> O} ,
ooe
L;( ) ~-~-e,_ O.,t, , I f f l<~o, k (X )>O. 0
Condition (2) implies that qX are analytic outside ~ , conti-
nuous up to the boundary and bounded (by the constants, which may de-
pend on ~ ).
Let B(~) be the class of all functions ~ analytic in
int ~ , continuous in ~ and such that
0 4
Putting C =~, ~K,X----~A , k(~)>0 in (I), associate with
581
every function ~ ~<~) its Fourier series. In this section it is
convenient to enumerate the zeros of L , counted with multiplici-
ties: [~V}~4 "
We shall be concerned with the convergence of (I) to ~ in int ~ . Suppose that L satisfies the folloving additional requi-
rement. There is a family of closed contours ~k<k~) and a fa- mily of curvilinear annuli containing these contours
e =u 0 t e r k K ' K - K t~
satisfying
a) for a l l o v , #>0 , and 6 , 6 > 0 ,
,Ej~,H(X)= oo where H(X)--- k
E~ ILO,)I
for k>k([, 6) the function M is greater than ~(9)-6 on ~k , greater than ~(~o)-6 on ~ , greater than
/// i
~C-~, + ~)- 8 on ~k where r k is the ,part of F k lying in the complement of the angle J~J < ~o + ; II' ~ i/1 ~ the part of lying in the angle J~I< ~o-~ , ~k and @o k are the parts of
lying in the small angles Ig-~oJ%~ , J~+~oJ%[ correspon- dingly; / I! I#
b) if the boundary of the curvilinear half-annulus ~kU~KU~k is divided into the parts ~j and ¢:/ by < (moreover let
CII K be inside r ), then the lengths of the curves , CK , C~ are ~ [0(~)~ K] when k --,-oo ;
c) L has only one zero in the annulus between ~k and ~k+1 ,
namely kK " Under these contitions it has been proved in [2] that if ;~<~)
and F is a compact subset of int ~ , then
~,~ -~o%~ I#(~)-~ Ave l<e 5o=#o(E)>0 ~>~o (~e E)
S O
582
It was shown in [2] and [3] how the general case (i.e. the
case of an arbitrary ~ analytic in int ~ ) can be be reduced to
the case ~ ~(~0) •
PROBLEM 3. Show that for any domain int ~ there exists a func-
tion ~ with the properties (1), a), b), c).
REFERENCE S
I. ~ e o H T ~ e B A.$. P~j~ SECnOHeHT. M., Hs~Ea, 1976. 2. .[ e o H T ~ e B A.~. K Bonpocy 0 npe~cTa2~e~E aHaJn~THqeoENX
~yHELG~ B 5ecEoHe~HO~ B~LUyF~O~ odaaCTH p~aM~ ~pHx~e. - ~OE~.
AH CCCP, 1975, 225, ~ 5, I013-i015.
3. ~ e o H T B e B A.~. 06 O~HOM r@e~CTaB~eHH~ aH~T~ecKo~
#y~Eu~ B 6ecEoHeqHo2 B~my~o~ o6aacT~. -- Anal.Math.s 1976,
2, 125-148.
A. F. LEONT iEV
(A.$.~EOHTBEB) CCCP, 450057, Y~a y~. TyEaeBa, 50 ~Hpcz~ ~s~ AH CCCP
583
10.7. RESTRICTIONS OF THE LIPSCHITZ SPACES TO CLOSED SETS
The Lipschitz space
of the semi-norm
K A~ (~) is defined by the finiteness
~ ~Cl~l)
K k Here as usual A~ = (~- ~) and %~ #(G)) = ~(SC+~). The majorant
: ~÷ --~ ~+ is non-decreasing and~,GJ ~+0) = 0 . Without loss
generality one can suppose that ~0($~$K is non-increasing. of
Let X ~ ) be the closure of the set Co °° in Aw(~ ) . This no te dea l s wi th some problems connected wi th the space of
t~oes At (E) -- At, (RT I~ and with its separable subspace ~,(F) -~, where~F~ %" is an arbitrary closed set. Among ~
spaees~u~der consideration there are well-known classes C , C
and A~+4 whose importanCen ~is~indubitable" Recall that C &@ ~-th consists of all functions ~U with derivatives satis-
fying HSlder comdition of order @ . Replacing here H'~Ider condi-
tion by Zygmund condition K) we obtain the definition of the class
A g+t CONJECTURE 1. There exists a linear continuous extension op~ra-
M k tor ~' A~(F)-~A~(~ ).
A nonlinear operator of this type exists by Michael's theorem of
continuous selection [I]. The lineartiy requirement complicates the
matter considerably. Let us review results confirming our Conjecture I. Existence of
a linear extension operator for the space of jets ~,oL(F) connec-
~ -CL,~(-'I~ ~) is proved in the classical_ ___W~hitney th\eorem --[2J . ted with
But the method of Whitney does not work for A~+~(~) ° Recently the author and P.A.Shwartzman have found a new extension process proving
Conjecture I for ~=~ (the case ~-----~ is well-known, see for examp-
le E3 ] ). The method is closely connected with the ideology of the local approximation theory [4] ).
The following version of Conjecture I is intresting in connec-
tion with the problem of interpolation of operators in Lipschitz
spaces.
~ction $ satisfies Zy~d condition if IA~I=00~I)
584
CONJECTURE 2. Let c0~, ~ =t,~ , be majerants. There exists a K
linear exstension operator ~ :CCF~CC~ ~) mapping ~C F) K
into A~C~), ~=~,~. The above mentioned extension operators do not possess the re-
quired property. If Conjecture 2 turns out to be right we would be
able to reduce the problem of calculation of interpolation spaces K for the pair A~(F) , $ : ~, ~ , to a similar problem for ~.
PROBLEM. Pind condition necessary and sufficient for a ~iven
function # ~CCK) to be extendable to a ~E K n
~ C~ ~) }. ~n other words we ask for a description of the K
restriction ~C~)IF (or X~Cg~)IF ). The problem was solved by Whitney for the space CKC~) in
1934 (see [~)° In 1980 P.A.Shwartzman solved the problem for the
space A%C~ ) and in the same year A.Jonsson got (independently)
a solution for the space A~+~C~) (see [6,~ ). The situation is
much more complicated in higher dimensions; there is nothing but a
non-effective description of functions from the space A~C~)
involving a continual family of polynomials, connected by an infi-
nite chain of inequalities (see [8] for power majorant; general case
is considered in [9] in another way). Analysis of the articles [5-7]
makes possible the following
CONJECTURE 3. Let N=N(K,~,F) be the least integer with the
following propert~ ~): if the restriction of a function ~EC(F)
on any subset H~F with card H ~ N is extendable to a K
function ~H ~ At (~) and HS ~ I~H I~ < co , then ~ belongs K
to A~CF). Define N(k,n) by the formula
N (K,~) =,s~p NCK,~o, F).
Then the number N(K,t~) is finite. It is obvious that N(4 ~)=~ ; the calculation of N (K,~) for k > I is a very complicated prob-
lem. P.A.Shwartzman has proved recently that NC~,~) = S' ~-4
~) One can prove that Nck,~I~F)< O0,
585
and using this result has obtained a characteristic of functions
from ~F) , Fc~ ~ by means of interpolation polynomials
(see [6]). When ~>~ the number N(~,~) is too large and the
possibility ef such a description is dubioms.
In conclusion we note the connection of the considered problems
with a number of other interesting problems in analysis (spectral
synthesis of ideals in algebras of differentiable function, HP
space theory etc.)
REFERENCES
I. M i c h a e I E. Continuous selections. - Ann.Math., 1956, 63,
361-382.
2. W h i t n e y H. Analytic extensions of differentiable func-
tions defined in closed sets. - Trans.Amer.Math.Soc., 1934, 36,
63-89.
3. D a n z e r L., G r u n b a u m B., K 1 e e V. Helly's
theorem and its relatives. - Proc.Symp.pure math., VIII, 1963.
4. B p y ~ H H 2 D.A. Epoc~paHcTBa, onpe~ea~eM~e c noMom~m aoEax~-
H~x npE6am~eH~. -Tpy~ ~0, I97I, 24, 69-I32. 5. w h i t n e y H. Differentiable functions defined in closed sets,
I. -Trans.Amer.Math.Soc., 1934, 36, 369-387.
6. ~ B a p ~ M a H H.A. 0 cae~ax ~yHE~HR ~Byx nepeMemmx, y~oBaeT-
Bopam~x ycaoBm0 8~r~ys~a. - B c6."Hccae~oBaHm~ no Teop~ ~mu~
~moz~x Be~ecTBeHm~ nepeMeHH~X". - HpocaaBa~, I982, I45-
- 168.
7. J o n s s o n A. The trace of the Zygmund class AK(~) to
closed sets and interpolating polynomials. - Dept.Math.Ume~,1980,
7.
8. J o n s s o n A., W a i I i n H. Local polynomial approxima-
tion and Lipschitz type condition on general closed sets. - Dept.
Math.Ume~, 1980, I.
9. Bp y~ H H ~ D.A., m B ap n M a H H.A. 0n~caH~e c~e~a
~yHEIU~ ES 0606~eHHO~O rfpOOTpaHCTBa ~rmm~a Ha npoEsBO~H~i EOM--
naET. - B c6."Hccae~oBaH~ no ~eopHH ~yHEn~ MHO~HX Be~ecTBeH~mX
nepeMemmx". Hpocaa2a~ I982, I6-24.
Yu.A.BRUDNYI
(~.A.BPY~)
CCCP, 150000, HpooJza~l.~, HpooJIaBCEH~ IDcy~pCTBeHH~
yH~Bepc~eT
586
10.8. MULTIPLIERS, INTERPOLATION, AND A(p) SETS
Let G be a locally compact Abelian group, with dual group F .
An operator T'Ip (G)--~p(~) will be called a multipliAer provided
there exists a function T ~ I Qo~P) so that T(~)A--T~ , for all
integrable simple functions ~ . The space of multipliers on }?(G)
is denoted by Mp(G) . ~et CMp(@=IT~Mp(G):T~C(V) }. In response to a question of J.Peetre, the author has recently shown
that for the classical groups, CMp(G) is not an interpolation
space between M~(G)=M(G) and CM~(G)= L~(P)~C(P) .
cifioall~, we obtained the following theorem (see [2] )°
THEOREM I. Let G denote one of the groups ~ ~, ~
Then there exists an operator T so that
(a) T is a bounded ope,z~tor on L~(F)NC(F). (b) TIM(G) is a bounded operator on NCG).
(c) T ICMp(G) i_~ n o t a bo~de~ operator on CMp(@,p~t@.
Observe that T is i n d ep e n d e n % of p , ~<p <g .
Our method of construction makes essential use of certain results
concerning ~(~) sets. Recall that a set E~ Z is said to be of
type A(~) (~ <~ < oo) i f whenever ~L~(T) and ~(~)=0 for
all ~9~E , we have ~EI%~T) . We used the following elegant re- sult of W.RUdd.u [1].
THEOREM 2. Let S >~ be an integer, let N be a prime with
N >5 , and let M =5 S-IN s-~ . Then there exists a set F~_
{0, I , 2 , . . . , M } so tha t
(a) F contains exactl,y N points,, an d
(b) II ~ ~25 ~< cIl ~ II 2 , fo r evex 7 t r i ~onomet r i c vo!yaomial ~ , A
wlth ~(~)=0 for ~I,E¢: F (Suoh ~ are ca l led ~-pol~momials,).
(Here C is i n d e p e n d e n t of N ).
AS a consequence of Theorem 2, Rudin showed that there exist
sets of type A(~s) which are not of ~rpe A(25+6) , for all 6>0
(see [I] )o
An obvious conjecture arisimg from Theorem I is the following:
CONJECTURE I. Let 4 ~< P4 < P2 ~ ~ . Then %her e exists an ope-
More spe-
, or Z n.
587
r a t o r T so t h a t
(a) T is a bounded operator o n CMp~ •
(b) T ICH~ is a bounds¢ operator on CHpt •
(c) T l C H p i_~s n o t b o ~ e ~ operator on CHp , f o r a l l
P ~ C Pt, P2) I t i s n a t u r a l t o a t temp t to ana l yze t h i s c o n j e c t u r e by means o f
t he t echn iques used to o b t a i n Theorem 1. But i t soon becomes e v i d e n t
t h a t such an a n a l y s i s r e q u i r e s a deep e x t e n s i o n o f R u d i n ' s theorem.
Specifically, we require a result of the following form:
OONJECTURE 2. (The A(~) Problem) . Le t ~< ~ < oo . Then t h e r e
~ x i s t s a s e t o f %.ype A(~) which i s no t o f t ~ e A(p+~) , f o r a l l
S>O •
This conjecture (which was essentially posed by Rudin) has re-
mained unresolved for nearly a quarter of a century, and is one of
the fundamental open questions in harmonic analysis. Its solution
will undoubtedly require very subtle new ideas involving estimation
in Lp . Conjecture I may be just one of the manifold consequences
of the ~ (p) problem.
Let us attempt to briefly outline one possible approach to the
study of Conjecture 2. Let p= 25/~ where S>~ and ~ are in-
tegers, and S > ~ . Let F =F N be the set of Theorem 2, and let
F~,N denote the ~-fold sum F+...+F . In essence, the
"piecing together" of the F~, N (for an infinity of N'5 ) provi-
des an example of a set which is not of type ~(p+6) , for all 8>0 .
The difficulty is in proving t3at F~,N is of type ~ (p) (with
all constants uniform in N ). One may seek to accomplish this by
w r i t i n g an ~F~, M -po lynomia l _~ in a J u d i c i o u s way as a sum of p r o d u c t s o f F - p o l y n o m i a l s , and c a r e f u l l y examining the r e s u l t a n t r e p r e s e n t a t i o n o f ~ . However, new e s t i m a t i o n t e c h n i q u e s f o r ~p norms would s t i l l be v e r y much a n e c e s s i t y i n o r d e r t o c a r r y out t h i s program.
REFERENCES
I. R u d i n W. Trigonometric series with gaps. - J.~ath.Mech.,
1960, 9, 203-227.
588
2. Z a f r a n M. Interpolation of Multiplier Spaces, Amer.J.
Nath., to appear.
MISHA ZAFRAN Department of ~thematics
University of Washington
Seattle, WA 98195
USA
CHAPTER 11
ENTIRE, MEROMORPHIC AND SUBHA~ONIC I~UNCTIONS
This mid and ramified theory, the oldest one among those presen-
ted in this collection, hardly needs any preface. By the same reason
ten papers constituting the Chapter cannot reflect all tendencies
existim~ in the field. But even a brief acquaintance with the contents
of the problems shows that the main tendency remains invariable as
though more than a quarter of the century, which passed since the
appearance of the book by B°Ya.Levin "Zeros of Entire Functions",
has shrunk up to an instaut~We reproduce here the first paragraph of
the preface to this book:
"One of the most important problems in the theory of entire fumc-
tions is the problem of connection between the growth of an entire
function and the distribution of its zeros Nmny other problems in
fields close to complex function theory lead to this problem"~
The only discrepancy between then and now, apparently, consists
in more deep and indirect study of this problem
A good illustration to the above observation is provided by
Problem 11.6. It deals with description of zero-sets of sine-type
functions and is important for the purposes of Operator Theory.
Problem 11.2 is, probably, "the most classical" one in the Chap-
ter. The questions posed there look very attractively because their
formulations are so simple.
590
The theory of subharmonic functions is presented by Problems
11.7, 11.8 .
Problems 11.3 and 11.4 deal with exceptional values in the spi-
rit of R.Nevanlinna Theory.
Problem 11.10 concerns the limit behavi~trof entire functions.
An important class of entire functions of completely regular
growth is the subject of Problem 11.5.
"Old" Problem 11.9 by B. Ya.Levin includes three questions on
functions in the Laguerre-P61ya class.
Problem 11.1 is rather a problem of approximation theory
The problems 11.1, 11.5, 11.6, 11.8, 11.9 are "old" and the rest
a r e new.
591
11.1. old
THE INVERSE PROBLEM 0F BEST APPROXIMATION 0F BOUNDED
UNIFOR/KLY CONTINUOUS FUNCTIONS BY ENTIRE FUNCTIONS 0P
EXPONENTIAL TYPE, AND RELATED QUESTIONS
Let E be a separable infinite-dimensional Banach space, let
~ c E , C ... be a chain of its finite dimensional subspaces such
t~t ~l"4, Em= kt, and U ~ is dense in ~ . For ~£~ we
define the sequence of "deviations" from ~ by
e4 { II - iII: S , . . . .
S,N,Bernstein [I] (see also [2~ has proved, that for every sequence
{ ~ 0 of non-negative numbers such that ~n ~0 there
exists ~ £ E , with
This is a (positive) solution of the inverse problem of best approxi-
mation in a separable space in the case of finite dimensional sub-
spaces. Strictly speaking S.N.Bernstein has treated only the case of
E =C [ ~] , E~ being the subspace of all polynomials of
degree 46-4 , but his solution may be reproduced in general case
without any change.
Now let ~ (~) be the Banach space of all bounded uniformly
continuous functions on ~ with the sup-norm; let B~ be its
closed subspace consisting of entire functions of exponential type
% ~ (or, to be more precise, of their restrictions to ~ ).
S.N.Bernstein has shown [3] that many results concerning the best
approximation of continuous functions by polynomials have natural ana-
logues in the theory of best approximation in B(~) by elements of
We define the deviation of " ~ from ~ by
A function ~ being fixed, the function A(~,~) has the follow-
ing properties: 1. A(~, ~) ~ A (I:~) for ? < ~ .
592
PROBLEM 1. Let a bounded function 6 ~- ~ F(~) (0 4~<oo)
s a t i e f ~ c o n d i t i o n s 1- 3 . I s there a f u n c t i o n ~ , ~eS(~) such
PROBLEM 2. Let 3 ~-0 be the c l o s u r e of ~ 2 . S~
B~_ 0 iS a propgr subspace of B~ . What is itsf~+codimension in
PROBLEM 3. L e t ~ be a Boh z almostrp~@~ function. Is
A(I,~) necessarily a ,jump function?
PROBLEM 4. Let A (I,~') be a ~ump funotion. Is ~ almost-pe-
rlodic?
REFERENCES
I. B e p H m T e ~ H C.H. 0d odpaTHO~ ss~a~e Teop~ Ham~y~mero
np~d~e~ Henpep~_BH~X ~JHmq~. - B EH. : Codp.co~., T.2, M.,
MS~-BO AH CCCP, 1954, 292-294. 2. H a T a H c o ~ H.H. EOHOTpyET~BHa~ Teop~ ~ys~, M.-~.,
I~TT~, 1949. S. B e p ~ m • e ~ ~ C.H. 0 Harem np~d~m~e~ Henpep~Bm~x
~ ~a Bce~ Be~ecTBem~o~ oc~ np~ nOMO~ nexm( ~y~E~ ~am~o~
cTeneH~. -B EH.: Codp.co~., T.2, M., MS~-BO AH CCCP, I954, 371-395.
M. I. KADEC (M.~.~M)
CCCP, 3IO0(E, Xap~EOB Xap~EOBCE~ ~CTHTyT ~xeHepoB x o ~ s ~ I o r o CTpO~Te~OTBa
OC#n"I~AItY
Problems 1, 2 and 4 have been so lved by A.Gordon (A.H.rop~o~) who k i n k y s u p p l i e d us w i t h the f o l l o w i n g i n fo rm~t ion .
~ W W 1. (A, Gordon). Le__~t ~ sat is f~ 1N ~ above. Than theze
A ~ , e) ~(~) , ~ 0 .
593
PROOF. P i c k a dense sequence {~K}~>~o i n (O, e o o ) end c o n s i - d e r a monotone sequence o f p o s i t i v e numbers { ~ } ~>~o s u c h t h a t
K Let { tR} t e n d t o ~r ~o so f a s t t h a t the intervals T K d,,e~
K~> 0 . . = ~ e ~,: I~-~K < ~ - ~ } do not overlap, St~c~srd a r g e n t s show
~ o ~ ( t - ~ z ) for each } e [1% and
Here E.
(~)
--~--- 0 . (2)
~E s t a n d s a s u s u a l f o r t h e o h a l ~ o t e r i s t i c f u n c t i o n o f a s e t
~iveu )~ ~ ~ > 0 l e t
C l e a r l y ~ ~ ~ . The d e s i r e d ~ i s d e f i n e d a s f o l l o w s :
K~O
The s e r i e s c o n v e r g e s a b s o l u t e l y and , m ~ f o r m l y on compact s u b s e t s o f ( s e e ( 1 ) } , w h i c h i m p l i e s ~ e B ( ~ ) . F i x 5 > 0 end c o n s i d e r
~ ) d ~ ~ p(~).~(~, ~,~_~) . AK~
S i n c e ~ ~ i s c l o e e d u n d e r bounded and p o i n t ~ s e conve rgence on ~ At i s c l e a r t h a t ~¢~ B~. On t h e o t h e r hand ( I ) i ~ l i e s t h e i n e q u a l i t y
Suppose mew A(;~,o~) ~ ( f f ) . Then t h e r e e x i s t ~: '6", ~ ( ~ , f f ) <
of i n t e g e r s such t h a t ~ K ~ } ~ . t ~ o p e r t y 2 o f P end (2) imply t h a t
We may assume w i t h o u t l o s s o f ~ e n e r a l i t y t h a t t h e sequence
594
I t f o l l o w s d i s t ( ~ t ~ , ~ g ) ~ ~ i n c o n t r a d i c t i o n t o [3] - The -
r e f o r e A(.-~6") ~ ~I(.6") . @
THEO~ 2. (A.Gordon). ~ k ~ (B~/B~_o)=+ =
mOOF, Let ~ K ] and t'~K} be the sequences cons t ruc ted as above f o r t h e c o n s t a n t s e q u e n c e ~K------- 6~ . F o r e v e r y bounded s e q u e n -
c e ~ = l i S K ] K ~ O d e f i n e
~ ) d~ ~ ~ K ~ , ~,$_~) . k~0
Clearly ~t~ B~ and it is easy to check that
- K
Indeed, ~ r ~ ~ (~ -4 /~ , 8 ,~) = q ( ~ 6,~) i n B(~) . Therefore
Usimg t h e same argumen%s a s i n t h e p r o o f o f Theorem I , we o b t a i n
~ ( ~ K , B ~ ) ~ ~ I ~ I for ~ ~ ~ . i~ f o n o . , f r o . (~) t ~ t the f a c t o r - s p a c e ' ~ / ~ ~_ o contains a subspace isometr lo to ~ % and t he re fo re i s not separable. •
Note t h a t t h e f u n c t i o n ~ w i t h ~ p r o p e r l y o h o o se n ~ i v e s
t h e n e g a t i v e answer t o P rob lem 4. I n d e e d , i f ~ I ~ I ---~ ~ t h e n K
~) O , ~ ' ~ ~ •
595
11.2. SOME PROBLEMS ABOUT UNBOUNDED ANALYTIC FUNCTIONS
A famous theorem of Iversen [I, p.284a] says that if ~ is a
transcendental entire function then there is a path ~ along which
tends to ~e . Thinking about this theorem has led us to formu-
late the following four problems, which we would like to solve, but
cannot yet solve.
I. I_~f ~ is a transcendental entire function, must there exist a
~th ~ alon~ which e v e r y derivative O f ~ tends to oe ?
Short of that, how about just havin~ both and tend to infi-
~,ity alon~ ~?( I t renews, say, by w ~ - w l i r o n theory, that i f is a transcendental entire function, then there exists a s e q u -
e n c e (En) such that I (~) (Z~) ---~ as B--~ for each
k= ~,~,... , but to obtain a p a t h on which this hap-
pens seems much more difficult. )
2. If I is an unbounded anal,Ttic function in the open disc ~ ,
~ust there exist ~ sequence (Z~) of points of D such that for
e = (L) ev ~ k 0 , ~ , £ , . . . , ~ ( z ~ ) - ~ ~ ~ - - = ? Short of that , how about
just h a v i ~ ~ ( Z ~ ) - - ~ and ~t" (Z~) - ,~ : (~he authors have sho~ I' that one can always find (~) So that both (Z~)'~O@ and (Z~)-~.~
If ~ grows fast enough, i.e., if either
o r
=~, ( 2 )
then we can show that there exists a (~) with as
~-~ for ~=0,~,~,... , but we do not know hew necessary these
growth assumptions are. Note that conditions (I) and (2) are not
strictly comparable. The proof involving condition (I) quotes a theo-
rem of Valiron, while that involving (2) uses some Nevanlinna theo-
ry.)
3. Is there an eas 2 elementar~ proof I usin~neither Wiman-Vali-
ron theor~nor advanced Nevanlinnatheor~thatif ~ is a transcen-
596
dental entire function. . then there exists a .sequence (~) such that
~(k~(~) ---~ ~ a_ss ~--~ f0r ~ 0,4, ~ . . . ?
4, The following possibility is suggested by many examples: the
differential equation
where the ~ are polynomials in one variable, not all constants, has
no solutions ~(~) that are analytic and unbounded in the unit disc
. For example, the equation ~fz_m I can be solved explicit-
ly by means of the standard substitution~ V~-~ V~ V ~ and it is -'4
easily seen that it has no unbounded analytic solution in any disc,
lending support to the above hypothetical statement~ The QUESTION re-
mains whether that statement is true or not°
1. Tit c hma r s h
Oxford S 932.
REFERENCES
E.C. The Theory of Functions 2nd Edition,
JAMES LANGLEY
LEE AoRUBEL
Dept.Math,
University of Illinois
Urbana, IL 61801
USA
597
11.3. COMPARISON OF SETS OF EXCEPTIONAL VALUES IN THE SENSE
O!~ R.NEVANLINNA AND IN THE SENSE OF V.P.PETRENKO
Let ~ be a meromorphic function in C and put
where [@~ is the spherical distance between ~ and ~ . Demote
in the sense of V.P.Petx~nko an~ by E~C~)=IOv£~ :~(~,~)>0} . the set of aeZicie~t values o~ { . It'~s ~cl~ear that E'~(,,()C E#(_~). The set E~(~) is at ~ost countable if ~ is of finite order [I]. There are examples of ~ 's of finite order with E~({)~ E~(~) b - 41. , ,
PROBLEM I. Let E~C E~cC be arbitrary at most countable sets, Does there exist a meromorp..hic function I of f.inite order
with Eech=E,?
impl ies EN({)= E~({) [6 ] .
I~O:B:I~ 2. Let { be an,, ,entire f ~ c i ; i o n of f.i.mii;,e,,,, order,
REFERENCES
I. H e T p e H K o B.H. POCT MepoMop$s~X ~ , Xap~xOB,
"Bm~a m~oxa", 1978. ly 2. r p i~ m ~ H A,~. 0 cpaBHe,~ ~e~eETOB 05(@) . -- Teo-
p~ ~ys~, ~yHE~.a~a~. z ~x np~., X~KOB, 1976, • 25,
56-66. 3. r o ~ ~ ~ 6 e p ~ A.A. K Bonpocy O CB~S~ Me~y ~eSexTOM
oT~oHe~eM MepoMop~HO~ ~ym~mz. - Teop~ ~ , ~ymu~.aHs~. ~x np~., Xap~oB, I978, ~ 29, 31-35.
4. C o ~ ~ H M.~o 0 coo~ose~ M ~ MHOZeCT~ ~e~eETm~X
3~a~e~ s OTF~OHe~ ~ MepoMoI~HO~ ~ EOHe~O~O nop~a.
-C~6.MaTeM,~/pHa~, I98I, 22, ~ 2. I98-206.
598
. E p e M e H E 0 A.3. 0 ~e~eETax ~ O~OHeK~qX MepoMop~x
$ym~m~ EoHe~oro nops~a (B nepal).
A.A. GOL ' DBERG
(A.A.IY~hEBEPr)
A. E. EREMENK0
(A. ~.EP~EHE0)
CCCP, 290602, JI~BOB JL~BOBC~ rocy~apcTBe~ y~epc~TeT
CCCP, 310164, Xap~oB np.JleH~Ha, 47, ~sm~o-Te~ec~ ~HCTH~yT ~S~HX ~eMnepaTyp AH YCCP
599
11.4. VALIRON EXCEPTIONAL VALUES OF ENTIRE FUNCTIONS
OP COMPLETELY REGULAR GROWTH
Let E# be the class of all entire functions of order p , 4 ° < 9 < ~ , and let E9 be its ~ubcla~s oZ entire functions of
completely regular growth in the sense of B.Ja.Levin and A.Pfluger
D]. ~et E~(~) ,n~ E~(~) be the sets of e~ception~l ~l~es o~ a
function ~ in the sense of R.Nevanlinna and of G.Valiron respecti- @
Indeed, for every function ~ 6 ~ we have COJ~ _|~ (~) .< [ ~2]*,
where [~]*--~kgZ:~<~}, [2]. on the other hand there
exists ~a E# with the property C~% EN(~)=oo ,[3].
PROBLEM. Is it true that ~Ev(~ : ~E~}={Ev(~):~E~}? There are ex mple 6E I for which the set EV( ) has
the cardinality of the continuum [4].
REFERENCES
I. ~ e ~ z m B. ~. Pacnpe~eleHze Kop~e~ ~exHx ~y~w~. Moc~Ba, I~, I9~ (English translation: Levin B.Ja. Distribution of
zeros of entire functions. AIMS, New York, 1980.) 2. 0 u m K i - C h o u 1. Bounds for the number of deficient va-
lues of entire functions whose zeros have angular densities. - Pacif.J.Math. 1969, 29, No.S, 187-202.
3. A p a ~ e x ~ ~ H.Y. ~e~e ~yR~ NoHeq~oPo Hop~Na C 6ec~oHeq- m~M ~o~eCT~OM ~e~eETHNX saa~esw~. - ~o~.AH CCCP, 1966, 170,
2, 999-1002.
4. rox~6 ep~ A.A., Ep~Me~Eo A.B., 0cTpoB- C E ~ ~ H.B. 0 c~e ~ex~x ~yH~ ~noxme pe~J~pHoro pocTa. - ~o~x.AH YCCP, cop"A", 1982, ~ 2, 8-11.
A.A. GOL' DBERG
(A.A. F0~%~EPF)
A.E.EREMENKO
(A. B.EP~)
I. V. OSTROVSKI!
(M. B. 0CTPOBCK~ )
CCCP, 290602, ~BOB, a~BOBC~ rocy~apcTBesm~ yH~B epc ~ eT
CCCP, 310164, Xap~o~, ~s~o-~ec~ ~aCT~TyT s~s~x Te~epaTyp AH YCCP
CCCP, 310077, X&p~os,
XapbKOBCE~ ~ocy~apCTSeHHM~ y~Hsepc~eT
8OO
11.5. old
OPERATORS PRESERVING THE COMPLEXLY REGULAR
GROWTH
We suppose the reader is familiar with notions and theorems of
the theory of entire functions of completely regular growth (c.r.g.)
(e.g. as presented in chapters II and III of [I] ).
In [2] derivatives and integrals of the function ~ of c.r.g.
were considered and the following result was obtained: I) { I has
all e i
c.r.g, on rays ~/~=O except maybe for with ~2(8)=0
2) the integral ~(~)=~ zl ~(l) i~ ~ has c.r.g.T in ~ . 0
Now consider instead of the operator ~=~ a more general ope-
rator q(~) , where q is an entire function~ of exponential
type. Is there a result similar to that of [2] in this case? ~or
entire functions ~ of c.r.g, and of order # , # <~ , the an-
swer is given by the following theorem.
THEOREM. Let ~ be an entire function of c,r.~, and of order
~, ~ <~ , and ~ be an entire function of exponential type.
Then: I~ the function
except for ra,ys with
equation ~ (0) ~= {
has c.r,~, in ~ ,
The theorem follows from the result of [2] stated above and
from the following lemma.
be an entire function of order 2 ' '
be an entire function~ Then the asymptotic equalit,y holds
,
where H¢/ i s the m u l t i p l i c i t ~ of zero of ~ a__tt ~ = 0 , and Z - 2 ~ o o
~) In ~2] there is a more explicit and complete chataoterization
of the exceptional set of rays; ~I is the indicator function of { .
~) The set of solutions of this sort is nonempty, see A.O.Gel-
fond [3], p. 359.
~C~)~ has c.r.~, on all =ys.maybe
~CO}--O ; 2) eve r~olution F Of the
in the class of entire functions of order~
601
~o means that ~--~outside some -set.
For the proof of the lemma one needs the following assertion
which is an easy corollary of theorem 2 of [4]. Let ~ be a mero-
morphic function of order S' S < 4' ~ be a fixed number, ~>0 ;
then ~I(~+~) > 0 (~ o . co) uniformly with respect to E ,
and moreover, the exceptional -set does not de-
pend on ~ .
So this assertion and the equality
imply that tm i fo rmly with respect to ~, i~l<~
( e s s e n t i a l l y t h e same was o b t a i n e d a l s o i n [ 4 ] , p . 414) .
Now l e t ~ , ~ be a s i n t h e lemm~° Denote by ~ t h e B o r e l
transform of ~ and denote by ~ a circle surrounding all singula-
rities of ~ . From the equality
( i )
F and from the fac t tha t ~(K) (0) : 0 ( K- 0 , 4 , . . . ) tha t
g
- C -O! r
K (2 )
it is clear
r
602
where -
(~-0!
But we have
0
So applying (I) (with ~C~) =$(~)C ~) ) we obtain, that
~ CE, ~) ~ ~/~! (E o -oo) uniformly with respect to ~6~ •
Taking (2) into accotunt we see that the lemma is proved.
For entire functions ~ of order ~ , ~ ~ , one cannot
expect an analog of [2] as simple as in the case # <~ . It is seen
from the following example. Let ~({)={~-I , ~W~ and let ~ be
an entire function of exponential type ~ , ~ < ~ , which is not
a c.r.g, function. Denote by ~ the solution of the equation
~(~)~ = ~ in the class of entire functions of exponential type
(the set of the solutions is nonempty by the theorem IV of [3],
= ~- . Then ~ is an K=0
entire function of exponential type and of c.r.g, with the positive
indicator whereas ~ = ~C~) $ is not of c.r.g.
For functions of exponential type the following conjecture
seems plausible.
CONJECTURE. Let ~ be an entire fun qtion of exponential ~ype
and of c.r.~., and ~ has no zeros at the ' points of the boundary
of conju~at ~ diagram of ~ which are common endpoints of two seg-
ments on the ' boundary of the diagram. Then ~(~){ has c.r.~. In
particular , if ~ has no zeros on the boundary of the conjugate
diagram of { , then q(~)~ has c.r.~.
For functions ~ which grow faster than functions of exponen-
tial type the answer must be still more complicated, as solutions
of the equation qC~) ~=0 in this class may be not of c.r.g.
The following QUESTION remains unsolved too. Let ~ be an en-
603
tire function of c.r.g, and of order 2 , ~ ~ , q be an entire
function of exponential type. Are there solutions of q(~)~=
which are the entire functions of c~r,s~ with respect to the same
prqx!~te order as the pr0xlmate order of ~ ? If q is a polyno-
mial the affirmative answer is an easy consequence of results in [2]
and of the integrel representation of the solution, however the ge-
neral case does not follow by passing to the limit
REFERENCES
I. ~ e B ~ H B.H. Pacnp~eaeHHe EopHe~ ~eaHx~yHE~, M., 1956. (Distribution of zeros of entire functions. Providence, AMS, 1964~
2. F o a ~ ~ 6 e p r A.A., 0 c T p O B C E ~ ~ M.B. 0 npO~SBO~--
HHX ~ nepBo0OpasH~X nexHx ~y~Eu~ BnoaHe pe~yaapHoro pocTa. --
Teopm~yHm/~, #~.aHaa. E Ex np~., Xsp~EOB, 1973, BNII.18,
70-81.
3. F e a ~ ~ o H ~ A.0. Hc~caeH~e EoHe~xpasHOCTe~. M., HayEa,
1967.
4. M a c i n t y r e A.J., W i I s o n R. The Logarithmic de-
rivatives and flat regions of analytic functions. - Proc.London
~ath.Soc., 1942, 47, 404-435.
I.V.OSTROVSKII
(H.B. OCTPOBCIG~)
CCCP, 310077, Xsp~EOB,
ya.~sepX~HCKOrO 4,
XSp~KOBCrm~ rocy~spcTBesma~
yH~BepCmTeT
COMMENTARY BY THE AUTHOR
A partial progress in the last question of the Problem has been
made in [5S.
Por any trigonometrically convex ~ -periodic function ~ consi-
der the class [~, ~] of all entire functions ~ of exponential type
with indicator ~# satisfying ~ ~ ~.
THEOREM. Let ~ be a function in ~, ~]
growth on ~= ~ : ~-----@} and let
of exponential t,ype. Then each solution
completely regular growth on ~@
of completely regular
be an2 entire function
o~f ~(~--- ~ is of
604
REFERENCE
5. E n ~ ~ a H O B 0.B. 0 COXI08HeH~ onepaTOpOM 0Be!0TI~ He BnOJIHe
peZ~SpH01"O pOCTa ~yHE~aH. Ca6.MaTeM.~ypHa~I, I979, 20, ~ 2, 420- 422.
605
1 1.6. ZERO-SETS OF SINE-TYPE FUNCTIONS old
An entire function ~ of exponential type S~ is called
a s i n e - t y p e f u n c t i o n (s.t.f.) if there exist
positive constants ~ , ~ , H such that
~<IS(~)le <M for I J ~ I ~ H .
The class of s.t.f, was introduced in [I]. It found applicati-
ons in the theory of interpolation by entire functions and for bases
of exponentials in ~ ~(-S~,S~) •
THEOREM I ([2]). Let ~ be a s.t.f. .... with simple zeros
{ ~n }~Z satisfying
and l e t {Q}"~2 the s e r i e s
]1 ~K- kjl>O
be az47 sequence in ~P , ~< p<+co
C~
( I )
• Then
~P conver~es in L P and defines an isomorphism of onto the space
of all enti f ctions 0f oneo iol
THEOREM 2 ( [ 1 ] , [ 3 ] ) . Le t
I, Then the system { e~ka~ } ~-Z
sa t i s f ~ the . c o n d i t i o n s o f theorem
f,o,rms a, Ri,e,s,,z ,basis in ~ C-~,~) °
I ~ had been shown i n [ 1 ] , that t • te~Xw0;) ~ Z forms a ba- s i s in ~ (-S~, ~) , and later it was proved in [3] that actually it
is a Riesz basis. V.E.Katsnelson [4] has essentially strengthened
Theorem 2 and his result can be also formulated in terms of zero-sets
of s.t.f. A series of other results in this field has been obtained
in [51. The conditions of simple zeros and (I) can be omitted but this
results in a more complicated statements of Theorems I and 2.
606
In connection with the above results the followimg PROBLEM
seems to be interesting: describe the zero-sets { ~}~2 of s,t.f~
This problem is, of course, equivalent to the problem of identifica-
tion of s.t.f, because
5(~)
for every s.t.f. ~ .
We ca~ot take risk of predicting concrete terms in which the pro-
blem could be solved but the desired solution should be given with
help of "independent parameters". To clarify the last requirement
consider an analogous problem for the M.G.Krein class. By the way,
this class may be connected with our problem.
An entire function ~ belongs %o the M.G.K r e i n
class if
A. A.)
with real coefficients C, A, [A~}, {~} satisfying IAn IX . < < CO , The Krein class, which has been introduced in [ 7], is important for Operator Theory and for the moment problem. It
turns out that [ ~} is a sequence of zeros of a function in
Krein's class iff it can be obtained by the following procedure [6].
Pick an arbitrary domain~ of the form
p<K~
where p, ~, k are integers ( it is allowed that p=-oo~=+oo ) and
0~< ~K < co , and map ~ conformally onto C+ by the mapping
satisfying ~Coo)=oo . The images of slits I k of /~ will
be disjoint segments [6~K, 6 K ] , p < K <~ of the real line. Every
choice ~k E [~k,6k] defines a function # in ~.G.Krein's class.
Here the numbers p, % , ~ . , ~KE [ ~ . , ~ . ] play a role of inde- pendent parameters.
Single out some results connected with our problem. There
exists a necessary and sufficient condition for [~]n~Z
607
I < H to be the zero-set of a s.t.function (D], p. 659):
d,t, <oo
Suppose that ~K= K÷C K , C~ =0(~) , IKl----°° . Then {~K} is the zero-set of a s.t.function iff there exists an entire function ~ of exponential type % S~ satisfying ~ I~($)I < oo~
~(K)=(-~)~C~ , K~2 (see [8], Appendix VI). This condition
cam be reformulated in terms of special functionals applied to {CK}
(see[SiP.591). Perhaps, this observation indicates the right w~y to
the solution ? Anyway, in general it is not true that the zero-set
{~} of s.t.f, satisfies ~ = ~ +0(~) • On the other hand
~---- ~ + 0(~I~kl) for every s.t.f.
In conclusion note that without loss of generality the sequence
[~}~E~ can be assumed to be real. A sequence { '~ '~}~zZ '
IS~I <m is the sequence of zeros of a s.t.f, iff {~} is (see [9]).
REFERENCES
I. ~ e B ~ H B.H. 0 6asHcax noEasaTe~HHx ~y~En~ B k~(-~,~) .
- 8an.~S.-MaTeM.~-~a XIV ~ Xap~E.Ma~eM.O6--Ba, 1961, 27, 39--48. 2. ~ e B E H B.H. MHTepn~ se~l~ ~yHEI~LRMH 8EO~OHeH~EaJ~a-
IIOrO T~Ila. -- Tpy~M ~TMHT AH YCCP, cep."MaTeM.~HsHEa E ~HEz~.aHa--
an~s", 1969, BHH.I, 186-146. 3. I ~ 0 Jl O B E H B.~,. 0 6EOpTOPOHa~H~EX pas~o~eHE~X B L- no
JfHHe~G£E~ EOM6HHa~I~ClM HoEasaTe~BH~X ~yHELU~. -- 8an.MeX.-MaTeM. ~-ma XI~ E Xap~E.MaTeM.od--Ba, 1964, 30, 18--29.
4. K a ~ H e ~ ~ c o H B.S. 0 6asEcax noEasaTeJL~H~X SyHEIn~ B
m 2 . -~.aHax, zero npE~., 1971, 5, ~ I, 37-47.
5. ~ e B ~ H B.H., £ ~ 6 a p c E E ~ D.M. EHTepn~ se~M~
~ H ~ one~a~BH~x F~aCCOB E CB~SaHHNe C He~ pas~o~eHE~ B p~-- sECnOHeHT. -- HSB.AH CCCP, cep.MameM., I975, 89, ~ 8, 657-702.
6. 0 c m p o B C E ~ ~ M.B. 06 O~HOM z~acce sexax ~yHEL~. --
~OF~.AH CCCP, I976, 229, ~ I, 39-41.
7. K p e ~ H M.F. K Teop~ ~e~x ~yHE£~ ~ECnOHeH~a~Horo TEa.
-HSB.AH CCCP, cep.MaTeM., 1947, II, ~ 4, 309--326.
608
8. /[ e B E H B.~I. Pacnpe~exeHEe EopHe~ ne~ ~ysx~, M., roc~ex-
~S~aT, 1956,
L e v i n B.Ja. Destributions of zeros of entire functions, Pro-
vidence, Rhode Island, AMS, Translations of Math.Monographs~
v. 5, 1964.
9. Ji e B ~ H B.~{., 0 C T p 0 B C E ~ ~ 14.B. 0 M ~ BOSMy~e--
HEHX ~o~eOTBa EOpHe~ ~yHELU~M TEHa O~K~yoa. -- HsR.AH CCCP, cep.
MaTeM. , 1979, 43, #~I, 87-II0.
B.Ya,LEVIN
I.V.OSTROVSKII
( .B.0CTPOBC )
CCCP, 310164, Xap~EOB,
~s~Eo-Tex~m~ecEm~ I~HCTMTyT
~sE~x TeMnepa~yp AHYCCP
CCCP, 310077, XapBEOB~ XapBEoBCEH~ rocy~apCTBeHHN~
yHEBepCHTeT
COMMENTARY
The problem seems to be still unsolved. New related informa-
tion may be found in
Hruscev S.V., Nikol'skii N°K., Pavlov B.S. Unconditional bases
of exponentials and of reproducing kernels. - Lect.Notes in Math.,
1981, 864, 214-335.
609
11 .7 , AN EXTRE~AL PROBLEM PRO~ THE THEORY OF SUBHARMONIC
FUNCTIONS
A closed subset E of ~ is said to be r e 1 a t i v e 1 y
d e n s e (in measure) if there exist positive numbers N and
such that every interval of length N contains a part of E of
measure at least ~ . In this case we write E~E(N,~) . sup-
pose in addition that all points of E are regular boundary points
of the domain Ck E . It was proved in [I] that there exists a har-
monic function ~ positive on C \ E and (continuously) vani-
shing on E . Such a function was constructed in [2] using some spe-
cial conformal mappings. It can be shown that if we require in addi-
tion that ~(~) : 0(~) then E determines ~ uniquely up to a
positive constant facter (cf.[3]; see also [4] for a more general
result).
If E is relatively dense then a positive limit
A =
exists. ~Tultiplying by a positive constant we may assume A=I .
This normalized ~ will be from now on denoted by GE •
It was proved in Eli and [2] that ~E is bounded on ~ by a
constant depending only on N and ~ provided E~E(N,~) i.e.
Ee E (N ,5 ) ::~ s ~ O'E(CC).<C(N,~ ).
PROBLEM. Find the best possible value of C(N,g)
N and ~ .
This problem is connected with the following
THEOREM. uS uppose that
a) ~ is subharmonic in C ;
b) U.(~).<O for ~ E E , E E E ( N , ~ ) u,(z)
°) ~=~ Is-U <°°" I~,l",-c~
610
Then ~(~)~d~Ek~)'~ z , , ~ C . Moreover, the equality aS
en~ point ~ ~C \E implies ~ d ~ E .
Hence for the class of subhalT~onio functions ~ satisfying b)
and c) we obtain the estimate
z t i s e a s i l y ~een t h a t C ( N , d ) = N C ( ~ , # / N ) , and we can assu~e N :-~1 and 0 < ~ < ~ without loss of generality.
It was proved in [4] that
~Elg E
where E = U [ 1 ¢ - ~ , ~¢+ ~'//2] I CONJECTURE that
and that ~ ~E (Do) attains this value for E = E only.
REI~ERENCE S
1. S c h a e f f e r A.C. Entire functions and trigonometrical
polynomials. -Duke M ath.J. 1953, 20, 77-88.
2. Ax~ e s e p H.M., ~ e B ~H ~oH. 0dodmeHHe HepaBeEcTBa
CoH.FepHmTe~Ha ~ npO~SBO~HHX OT ~e~ ~ . - B ~. : Hccae~o-
BaHm~ nO coBpeMe~ npo6aeMsa Teop~ ~ym~ EoMz~.nepeM. ,IWl~,
MOCEBa, I960, III-I65.
3. B e n e d i c k s M. Positive harmonic function vanishing on
the boundary of certain domains in ~ . - Arkiv for Math. 1980,
18, N 1, 53-72.
4. ~ e B H H B.Ho Cy6rapMoHH~ecK~e MsaopsaTH H ~X npHaozeHH~. Bce-
comsHa~ EOH~ep, no ~KH. Xap~EOB, I97I, II7-I20.
B. Yd. LEVIN
(B.~.XEBHH) CCCP, 810164, Xap~EoB
np.~eH~Ha 47
~THHT AH YCCP
611
1 I. 8, A PROBLEM ON EXACT MAJORANTS old
Let ~ be a domain on the complex plane (~=~ for example),
and let ~ be a positive function on G . Consider the class 5h.
and define a function H by
I ,
OUR PROBLEM is to find conditions on k necessary and suffici-
ent for the equalit[ k-H • if ~=H the function ~ will be called a n e x a c t m a j o r a n t (e.m.)
It is clear that for any e.m. k the function ~@ ~ is sub-
harmonic. But easy examples show that it is not a sufficient condi-
tion. On the other hand the equality ~= ~F~ , F being an analy-
tic function, implies that ~ is an e.m. But this condition is not
necessary. When trying to solve the PROBLEM one may impose some addi-
tional requirements on k , e.g. suppose that ~ is continuous in
G or even (as the first step) in ~ G . Theoretically one may
treat this problem using the concept of duality in the theory of
extremal problems (cf. e,g. [I]). But I didn't succeed to get a use-
ful information concerning the description of e.m. by this approach.
The fact that each e.m, ~ is also an e,m. in every subdomain of G
is likely to be useful in this approach.
Let Q be the class of e.m.'s for G that are continuous in
G (or even in ~ ). HERE IS ONE OF CONJECTURES concerning the description of Q :
~ Q ~===~ ( ~ !,,S, in the closure of functions of the form
I~41+'''4~' ~i bein~ analytic in G ), here the closure is
either in C(~4~G) (if Q consists of functions continuous in
~ ) or in the projective limit of the spaces C(~G~), where the domains G~ exhaust G ( i f Q consists of functions con- tinuous only in G ).
Using the approach of the convex analysis we can formulate the
DUAL VERSION OF OUR CONJECTURE: let ~ be a real Borel measure on
G ; does the condition I" '~I~ >z0 for all anal~tic in G func- tions ~ imply S. ~ ~ for ~Q ? We may try tO treat -the question investigating the ~easures in the Riesz representation of the subharmonic function '{~ (not ~ I~l !). The answers %o
612
the above questions may happen to yield an interesting contribution
~o the theory of extrema in spaces of analytic functions.
REFERENCE
1. X a B ~ H C 0 H C.}]. Teop~ SEcTpe~HHX s8~aq ~L~ OI~&_~wqeH-- HHX 8Ha~IZTZqecE~X ~FHEIS~, y~OBJIeTBOpH3~RX ~O]IOJ[H~TP~HHM yC~OB~--
2M BHyTpz O6;IaCTH. -- Ycnex~ MaTeM.HayE, I963, 18, ~ 2, 25--98.
S. Ya. HAVINSON
(C. .XA OOH) COOP, MOC~Ba, 121352, MOC~OBC~]~ ~w~eHepHo--cTpo~T~ ~CT~
~.B.A.Ky~6~m~eBa
CO~N~_ARY
The CONJECTURE has been DISPROVED by A.Gordon (A.H.rop~o~; private communication).
Denote by C(O) the space of all functions continuous in the domain 0 and by ~(0) the closure (with respect to the usual
compact convergence topology of C(O) ) of the set of all sums
I ~ I + ' ' " +I;~N I , ~} ~ ~ ( 0 ) , where ~ ( 0 ) stands for the space of all functions holomorphic in 0 • A. Gordon has shown that the function ,~C~I,~}~ ~ D ~ being obviously
an exact majorant, does not belong to ~-~ ~ . HERE IS THeE PROOF.
Set I~(~) ~ ~ ! ~ (%~@) ~ . We shall see that ifH~
and l~(H) ~--_ ~ for all small values of • (say, for~6[O,~]) then I~(H)= I for all $~[0, I) (whereas I$(~) = ~ for
<- > for ) . The inclusion H ~ ~ implies the existence of functions
I~}, ~-~-J~,-..~ k~4,~,..., N~ such that ~e ~(~)
and ~ ~,e#. ~ I l K J tend to H in ~(~) when~-~.
and we may assume (reno : , . ) Thus
N~
613
B ~ ~ ~ ~ and so
~ (1~ • ~) ~ Denote by hull (~) the closed convex hull
of J~I$ in ~'(]~" ~] ) The sets'J~, and ~u l l (~W,] are compact in C (~" ~ ) , elements of ~ being uniformly
bounded and uniformly Lipschitzian on every disc ~, 0<% ~ ~ .
We see from (*) that H~ ~ ( ~ ) It is convenient to rew-
rite this as follows:
(**)
for every ~(C(~" ~))*, j~ being a probability measure on ~Z
(see, e.g., Proposition 1.2 in the first chapter of [2]). Equality
(**) holds, in particular, for ~ ~ I~ , %6 (0,~) . But
I$(H)~- ~ (0~) andl~(~) increases with
if ~ ~ . Therefore (see (**)) ~ I@(H)~ I~(H}
K ----I0(~) j~-&.e, on ~ . But if Se~ then ~ is the mo-
dulus of a function in ~( ~. ~ ) and the last equality implies
is constant in ~" ~ . Using (**) with ~-~- I$ ,$~ (f,~)
we see I(H$) ~ ~ on ~0~). @
A.Gordon remarked that this proof can be slightly modified to
yield the following assertion: if ~4,"" ' ~ ~ ( ~ ) and
REPERENCE
2~ P h e I p s R.R. Lectures on Choquet's Theorem. van Nostraud,
Princeton, 1966.
614
1 1 . 9 . ENTIRE ~UNCTIONS OP LAGUEERE-POLYA CLASS old
Laguerre-Polya class ~ plays an important role in the theory
of entire functions. This class consists of functions of the form
Q~
~cz~ ~ - ~ - r ~ ' ~ z + ~ • Kl (~- -~:) 6 ~ ( ~ oo) ,
f~ whereT~O , : ~ m ~ = O ,~K~-- 0 , ~ I~I -$ <°° . It is
known (see ~I] ), that this class is the closure (in the sense of the
uniform convergence on compact sets) of polynomials with real roots.
It follows that ~(K) c ~ , ~: ~,~, . . . for le ~ . In 1914 Po-
lya proposed the following conjecture: a real entire fumctiom (i.e.
an entire ~ with ~ (~) c ~ ) such that ~ and all its derivatives have 1-o ~eros off ~ is in ~,.~o
There is a plemty of works devoted to this conjecture. The bib-
liography can be found in [3~, [4~. Not long ago S.Hellerstein amd
J.Williamsom solved this problem (in a preprint, see also their
works ~4~, ~5] ). They hav~ shown that a real entire function ~ with
all the zeros of I~ If, ~rr real, is in~.
PROBLEM I. Prove th~$ a real entire function wi~h all the zeros
of ~ and real is indic
In [3] it is shown only that loglog M(~,S)----0(~$)for a real en- tire function ~ such that ~ and Ifr have only real zeros.
Consider a well-kmown class HB of emtire functions ~ de-
fined by:
a) the zeros of 00 lie in the upper half-plane ~ ~ ~ 0 only;
b) if ~ < 0 then I ~(~)I ~ l~(~)I -
An arbitrary entire function 00 cam be represented as
oo-----~ 2 + ~Q ,
where 2 and Q are real. It is ~own ([I]) that ~ H-~ if ~.d
only if for am arbitrary pair of real numbers ~, ~ the function
~:? + FQ
has only real zeros.
Applying Hellerstein-Williamson's result we mow deduce that if
~(K~ ~ , K~O,I,... , then ~e~* , the class ~ ~ b e in g
615
defined by:
c) ~ ~E
,I
, , i th~/>~Oand ~la,,,I-~,~ . I f d) h o l d s , c ) i s e q u i v a l e n t t o t h e f o l l o w i n g c o n d i t i o n :
Im~ + ~Im (d~) ~ O, I~a,~ ~0. I
It is known that ~ is t h e closure of the set of polynomials
having all their zeros in I~ ~ 0 (see [I]. for e~mple). So we
have co~"~* for ~e~* .
PROBT.~ 2..Prove that o0~J?* if all zeros of co ~, ~-0,~,...
are in the upper half-plane l~t % ~ 0
A similar problem can be formulated for entire functions of se-
veral complex variables. For simplicity we assume ~ g . A polyno-
mial is called an ~ -polynomial if it has no zeros im ~
----~(~,w):Im~0,I,~w< 0 } The closure of the set of ~-poly-
nomials will be denoted by ~* (the information about ~B -poly-
n~ials ~n~ abo~t the olass ~ can be fo~d ~ F1], oh.9).
PROBLEM 3. Prove,. that am entire function o~ belongs to ¢* if
this function and a!~ ' its derivatives have no zeros in ~ .
REFERENCES
1. ~ e B ~ n B.H. Pacnpe~e~eB~e NopRe2 ~exHx ~yn~l~. M., r~,
I956.
2. P o 1 y a G. Sur une question ooncernant los fonctions on-
fibres. -C.R. Acad.Sci.Paris 1914, 158.
3. ~ e B ~ B B.H. , 0 c T p O B C ~ ~ ~ M.B. 0 saB~c~MOCT~
pocTa ~e~o~ ~r~zE OT pacnoxo~eHz~ EopHe~ ee npoE32o~R~x. - C~6.
MaTeM.mTpH.I960, I, ~ 3, 427-455. 4. H e 1 1 • r s t e i n S., W i 1 1 i a m s o n J. Deriva-
tives of entire functions and a question of Polya. - Trans.Amer.
Math.Soc., 1977, 227, 227-249.
5. H e 1 1 e r s t e i n S., W i 1 1 i a m s o n J. Deriva-
tives of entire functions and a question ef Polya~-Bull.Amer, Math.
616
Soc., 1975, 81, 453-45~.
B.Ya.LEVIN
(B.H. ~EB~H)
CCCP, 310164, Xap~EoB np.~esHHa 47,
$~S~Eo--TexRH~ec~z~ ERCTHTyT H~BEF~X TeM~epaTypAHYCCP
617
11 .10 . CLUSTER SETS AND A PROBLEM OF A,P.LEONT'EV
We use notations from [ t ] . In connection with some interpolation problems A.F.Leont'ev pro-
posed the following PROBLEM [2].
Let ~ Ai(s~; ~$(~)>0 , ~ [ O ~ 3 E ] . Suppose that the de- rivative completely regular growth on the set of zeros of
~7 i .e . fo r the sequence Z~ = ~K e$~K of a l l zeros of we have
(1)
(the zeros are supposed to be simple). Is it true that ~ is of com-
pletely regular growth ( ~E A $~ )? The following proposition is a corollary of the results in r3].
Suppose ~ satisfies (I). Denote ~(~)= 8~p~(Z): ~E~[~]~ ( ~ (~) = ~(e ) is the indicator of ~ ). Denote by ~J'V
the mass distribution associated with 17 . Then for every ~ [ ~]
we have
CONJECTURE 1. Let
that ,every If E F~ [ ~ ]
Fffv[ ~ ] be t..he.clus...ter set of ~ -
satisfies (2), and
suppose,
ze . (3)
Then F~ [ ~ ] cozm.ists of the single element ~ , i . e . ~f ~ A~e ~.
If the Conjecture is true then the Leont'ev's problem has a posi-
tive solution. Condition (3) is essential-. To see this denote by ~o(~) the in-
@
dicator of Mittag - Leffler function
f s f,o , I tl !
618
There exists a function ~o e A (~) w't ~-~ [ [o] l~ [*J. Every function We F'u[~o] s~tisfies ~ondition'(2) but" ~0~A~
because Ao contains not only ~ . Condition (3) is not satisAed. 0
The condition .V~g¢:% [~]" is also essential as the follo-
wing example (pointed out by M.L.Sodin) shows. Let ~/(~) "=]~[~ for I for I I>I. :t is easily _
verified that V is subharmonic in C . Denote A= {v:" I: is clear that every function ~Yt e A satisfies (2) with ~(:)==
$t~ {it t " ½e ( 0)°°)}=IV: ~. The set A satisfies all conditions of
theorem I from [I] except ~"v o fl ~ + ~ Thus the set A is
not the cluster set for any entire function,
~ERENCES
I. A z a r i n V.S. Two problems on asymptotic behaviour of entire
functions. - This book, S.9.
2. JI e o ~ T ~ e B A.~. 06 ycao~m~x paS~ZO~OCT~ a~ax~T~mec~x ~y~- ,pr~ B ~ /~pmxae. - Hss.AH CCCP, cep.~mTee., I972, 36, • 6, I282-I295.
3. F p ,, m ~ s A.~. 0 asosecTsaX pez"yx~p2OCT,, pocTa ueae~ ~)ymmmK. -
Teop. ~ysm~, ~ysm.mosaa~s. a2aa. ~ ,¢x npza., I983, Xap~,~OB, SMn. 40, 41.
4. A s a p z s B.C. 06 ac~m]ZTOTZqecz¢o~ I[oBeAes-~r cydrai0aommec~x
~ya~ ]~oae~aoI,o uol0~]~a. - MaTeL C60pS., I979, I08 (I50), ~ 2, I47-I67 (Engl. Transl. - math. USSR Sborn., 1980, 36, N 2, 135-154).
V. S. AZARIN
( .C.ASAP )
A. E. ERE~KO
A. F. GRISHIN
(A.*.rP~H)
CCCP, 310060, Xap~, Xap~-
EOBCI~ ZHCTHT~T ~aHep~8 ze~esso~opoEHo~o TpaHCHOpTa
CCCP, 3I 0164, )[ap~oB, Om- s~o--Texazqec~d ,,,CT]m, yT
~ms~mx Te~mepaTyp AH YCUP
CC~P, 3I 0077, Y~p~oB, Y~p~ ~ o B c ~ r o c y ~ p c ~ e . ~ ySZ-- BepczTeT
CHAPTER 12
Our stock of ~-problems being very poor, we just arrange them
in author's alphabetical order (see also 1.4, 1.7, 1.13, 1~14, 5.10,
6,5, 7.1-7.3, 7.14, 8,12, 8.14, 9.13, 10.5, S.I0).
620
12.1. POLYNOMIAI~Y CONVEX HULLS
We shall denote Hausdorff one-dimensional measure ("linear mea-
sure") by ~4" For X a compact subset of C . X will be its
polynomially convex hull: ~E~ ~ '.I 0(Z)l~<~tlp(~)l 9 ~EX} for all polynomials p in ~ . The unit ball in ~'~ will be de-
noted by B. As usual; ~(X) will be the uniform closure in C(~) of the polynomials.
In [I] it was shown that If~ X is (or lies in) a connected set
of finite linear measure, then \X is a one-dimensional analytic
~riety. Recently V.M.Golovln [4] claimed that the connectedness as-
sumption could be dropped. We find his argument unconvincing and
shall list a special case as a first question.
PROBLEM I. Does ~ (X)-0 imply that X is Dolynomlally con-
vex (i.e., X =X )? Known methods to solve this kind of problem involve the clas-
sical F. and T,~.Riesz theorem for Jordan domains with rectifiable
boundaries. One way to treat Problem I would be to generalize this.
Namely, let ~ be a bounded domain in C wi~h ~4(~) ~
(do n e t assume that ~ is even finitely connected). Suppose
that the o u t e r boundary ~ of ~ is a Jordan curve. Let
Z 0 C~ and let ~ be a Jonson measure for the algebra P(~) supported on ~ with respect to Z 0 .
PROB~M 2. I.~s ~l~ absolutely continuous with respect to ~tl~? The F. and M.Riesz theorem is the case ~= F • A variant is:
PROBLE~ 21 . Let ~ be subharmonic on ~ and u.s.c~ on ~ .
Let E C ~ with ~(E) >0 • If ~(~)--~-~ a..gs ~ ~ ~ ,
does it follow that ~-~ ?
Examples of non-polynomially convex sets X which are totally
disconnected have long been known; a recent example was given b~Vi-
tushkin [6]. It is known that such a set cannot lie in a torus n •
PROBLEM 3. Find a set X c ~ ~ which is tctall.y disconnected
such that 0 ~ ~ .
One possible approach is to approximate such a set X C 0 ~ b~
sets V(~ ~ where V is an analytic (or algebraic) curve in
passing through the origin. Then V ~ ~ would be required to have
621
arbitrarily small components. On the other hand, this will not be
possible if there is a lower bound on the size of these components
- it is known that the s u m of their lengths, ~(~V. ~ , is
at least ~@ .
PROBLEM 4. Is there ~ lower bound for {~i(~): ~ a connected
n
In [3] , V.K.Belo apka oonject ed, for replaced by " d i a m e t e r " , t h a t one ~ " a l o w e r bound. He showed t h a t i f ~ i s r e p -
l a c e d by the bidisc ~ there is some component of diameter at
least one, A
There exist sets X C ~ such that X \~ is a non-empty but
contains no analytic structure. This phenomenon was discovered by
Stolzenberg [5]. A recent example of such a set X has been given
by Wermer [7] with the additional property that X~TxD . ^
PROBT,~ 5. Find , a set ~C~ such that 0~X and ~ \X
contains no analytic structure.
One interesting.property of such a set would be that it could
be "reflected" in ~ n~ which would then become a "removaBle singula-
rity".
The Stolzenberg and Wormer sets both arise from limits of ana-
lytic varieties. A well-known question asks if this must necessarily
hold. Our last problem is a special case of this.
^ F PROBLEM 6. Let XCTXD with ( X \X) n non-empty. ,,
I_~s X \ X a limit of a nal2tic subvarieties of D ~ ?
A rather particular case of this was considered by Alexander -
We=er [2]
REFERENCES
I. A 1 e x a n d e r H. Polynomial approximation and hulls in
sets of finite linear measure in C ~ . - Amer.J.~ath.~1971, 93,
65-74.
2. A I e x a n d e r H., W e r m e r J. On the approximation
of singularity sets by analytic varieties. - Pacific J.Math.~
1983, 104, 263-268.
3. B e ~ o m a n K a B.K. 06 o~oM ~eTp~eczo~ CBO~CTBe aEax~T~e-
czHx ~oxecTB. -- ~SB.AH CCCP, cep.MaTeM., I9V6, 40, ~ 6, I409-
-I415.
622
.
e
6.
7. Wormer
- Arkiv for
r O ~ O B ~ ~ B.M. HOX~Ho~a~Ha~ B~OCT~ ~ ~ozecTBa ~O-- ~ A
~e~o~ ~e~o~ Mep~ B C ~ . - C~6.MaTeM.z~p~., 1979, 20, ~ 5,
990-996.
s t o i z e n b e r g G. A hull with no analytic structure. -
J. of Math.and Mech.v1963, 12, 103-112.
B H T y = E ~ s A.r. 06 o~oi sa~a~e B.~a. - ~oz~.AH CCCP,
1973, 213, 14-15.
J. Polynomially convex hulls and analyticity. -
mat.~1982, 20, 129-135.
H.ALEXANDER Department of Mathematics
University of Illinois at Chicago
P.O. Box 4348
Chicago, Illinois 60680
USA
623
12.2. THE EXTREME RAYS OF THE POSITIVE PLURIHARMONIC PUNCTIONS old
I. Let ~ 5 and consider the class ~(~) of all holomorphic
functions ~ on ~ such that ~6~ > 0 and ~(0)~--~ , where ~ is
the open unit ball in 6 ~ , Thus ~(~) is convex (and compact in
the compact open topology). We think that the structure of N(~)
is of interest and importance. Thus we ask:
What are the extreme points of ~(~) ?
Very little is known, Of course if ~-~ , and if
= c4+ 8-) Ic4-~), (1)
then ~ is extreme if and only if #(~) ----- C~ , where O~T . It
is proved in Eli that if S(~)~ $ ~ and if ~ is the Cayley
transform (I) of £ , then $~E£~) , where E(~) is the class
of all extreme points of N (~) .
Let K~- (~,..., K N) be a multi-index and consider monomials K
~(~) ~ C$ K in 6~ such that I~(~)I ~ ~ if ~ ~ . ThuslCl~([~[) "~
where by we mean K,+... let
~(~)=(~+OK~K)/(~--0K~K). It is proved in [2] that ~(~) if and
only if the components of k are relatively prime and positive.
2. We have ~ ~ ~(~) , however it is a corollary of the just
mentioned theorem of [2] that ~6 65~(~) , where the closure is
in the compact open topology. Thus E(~)=~= c~o~ (~) . (If ~-~ ,
then E ( ~ ) = c ~ E c ~) ) . It is also known that if ~ is an extreme point of ~(~) and
if (I) holds(that is to say if ~=(~-~)/(~÷I) ), then ~ is irre-
ducible. This is a special case of Theorem 1.2 of [3]. The term "ir-
reducible" is defined in ~]. If ~ ~ , then ~ is extreme if and
only if~ is irreducible. However for ~>/ ~ , the fact that ~ is
irreducible does not imply that ~ is extreme.
3. The extreme points ~ in section I and the extreme points
that can be obtained from them by letting ~$ (~) act on ~(~)
have the property that the Cayley transform I=(~-I)/(~+~) is
holomo~hic on ~ U 8 ~
Is this the cas,e for eve,ry ~ in E(~) ?
If the answer is yes, then it would follow (since ~ ) that
824
the F. and M.Riesz theorem holds for those Radon measures on a
whose Poisson integrals are pluriharmonic. In particular there would
be no singular Radon measures =~= 0 with this property, which in turn
would imply that there are no nsnoonstant inner functi6ns on 8 .
REDOES
I. P o r • I i i F. Measures whose Poisson integrals are pluri-
harmonic I!. -lllinois J.Math. ~ 1975, 19, 584-592.
2. F o r e I i i F. Some extreme rays of the positive pluriharmo-
nic functions. - Canad.Math.J., 1979, 31, 9-16.
3. F o r e i i i F. A necessary condition on the extreme points of
a class of holomorphic functions. -Pacific J. Nath.~1977, 73,
81-86.
4. A h e r n P., R u d i n W. Factorizations of bounded holomor-
phic functions. - Duke Math. J., 1972, 39, 767-777.
PRANK ~ORELLI University of Wisconsin,
Dept. of Math.,
Ma~son, Wisconsin 53106,
USA
COMMENTARY
The second question has been answered in the negative See Com-
mentary in S. 10.
625
12 • 3 . PROPER MAPPINGS OF CLASSICAL DOMAINS
d domaln ~C~ ~ A holomorphic mapping q;~ ~ ~ of a bounae
is called p r o p e r if ~(C~C~), ~l~"~O for eve~ se-
A biholomorphism (automorphism) of ~ is called a t r i v i -
a 1 p r o p e r m a p p i n g of ~ . If .0. is the l-dimen-
sional disc ~ the non-trivial proper holomorphic mappings
~:~-'~ do exist. They are called finite Blaschke products.
The existence of nontrivial proper holomorphic mappings seems
te be the characteristic property of the l-dimensional disc in the
class of all irreducible symmetric domains.
CONJECTURE I. For an irreducible bounded s,~etric domain
in C ~ , ~ # ~ , every proper holomor~hic mappin~ ~--~ is an
aut omorphiem.
According to the E.Cartan's classification there are six types
of irreducible bounded symmetric domains. The domain ~ p~ of the
f i r s t type is the set of co lex matrices Z , ,
such that the matrix I - -Z ~ Z is positive. The following beauti-
ful result of H.Alexander was the starting point for our conjecture.
T~o~M I (H.Ale~nder [I]). ~et ~ be ths ~it ball in C P ,
i.e. ~=~?,~ and let p>/~ . Then ' eve~ proper holomorphic map-
,~i~ (~:_qp,~ ~I'9. p,~ is an. autom.o.~ph.i~m of the ban. Denote by S the distinguished boundary (Bergman's boundary)
of the domain ~ . A proper holomorphic mapping @: ~ • ~ is
called s t r i c t i y i r o p e r if ~(~(~u)} ~)--~0
for every sequence ~ E with the property ~ ( ~ S)-~0.
The next result generalizing Alexander's theorem follows from [2]
and gives a convincing evidence in favour of CONJECTURE 1.
THEOREM 2 (G.M.Henkin, A.E.Tumanov [2] ). If' -0. is an irreducib-
le bounded symmetric domain in C ~ and ~ ~ D , then,, any strictly
,proper holomorphic mappin~ ~ :/i--'/~ i,~,,, ~,~, ,~,utomo,rp~!sm,
Only recently we managed to prove CONJECTURE I for some symmet-
ric domains different from the ball, i.e. when ~2 ~ S .
THEOEEM 3 (G.M.Re.'~in, R.~.~ovikov). Let, ~cC ~ , ~ ~ , be the classical domain of the 4-th type, i,e.
626
$I = { z:z.~', ((,~'~') ~- Izz'l ~ )V~ < ~},
where E=(~,...,E~) and Z l stands forthe transposed matrix
The n every prgperholomorphic mappin~ ~-~ is an automorphism.
Note that the domain I~,~ of the first type is equivalent
to a domain of the 4-th type. Hence Theorem 3 holds for ~L~,~ e
We present now the scheme of the proof of Theorem 3 which gives rise to more general conjectures on the mappings of classical do- mains.
The classical domain of the 4-th type is known to have a rea- lization as a tubular domain in C~ ~ ~ , over the round convex
Gone
T h • d i s t i n g u i s h e d boundary of this domain coincides with the space
R =t~C • ~= 0 } The boundary 92 contains together with each point Z ~ \ ~ the l-dimensional analytic component
0 ~ = ~ :~EC~ I~ > 0 } . The boundary ~ 0 ~ • ~f this component is the nil-line in the pseudoeuclidean metric ~5
=~o-~®,-... ~en the disti~uished bo~d~ry S . If ~ is a map satisfying the hypotheses of the theorem, an
appropriate generalization of H.Alexander's [1~ arguments yields that outside of a set of zero measure on ~ a boundary mapping (in the sense of nontangential limits) ~ : 9~ --~ ~ of finite multipli-
city is well-defined. This mapping POsseses the following ~ro~erty:
for almost every analytic component 0 Z the restriction q l~ is a holomorphic mapping of finite multiplicity of ~ into some
component ~W " ~urthermore, almost all points of ~ are mapped (in the sense of nontangential limits) into points of ~ .
It follows then from the classical Frostman's theorem that
~I~Z is a proper mapping the half-plane ~ into thehalf-plane
YvW o it follows that the boundary map ~ defined a.e. on the distinguished boundary ~ C ~ has the following properties:
a) ~ maps ~ into ~ outside a set of zero measure; b) ~ restricted on almost any nil-line ~E coincides (al-
most everywhere on ~Z ) with a piecewise continuous map of finite
627
multiplicity of the nil-line S~Z into some nil-line S~W-
With the help of A.D. Alexandrov' s paper [3] one can prove
that the mapping ~: ~---~ satisfying a), b) is a conformal mapping
with respect to the pseudoeucli:dean metric on S . It follows that
is an automorphism of the domain II
To follow this sort of arguments, say, for the domains /Ip,~
where p~ , one should prove a natural generalization of H.Alexan-
der's and O.Prostman's theorems.
Let us call a holomorphic mapping ~ of the ball l~p, 1 a 1 -
m o s t p r o p e r if ~ is of £inite multiplicity and for
almost all ~Sip, 4 we have ~(~)~ ~'~-p,4 , where ~(~)
is the nontangential limit of the mapping ~ defined almost every-
where on ~Ap, I CONJECTURE 2. Let ~ be an almost Droner maDpin~ of ~ip,1 and
p~ ~. Then ~ is an automorphism~
If we remove the words "9 is of finite multiplicity" from the
above definition, the conclusion of Conjecture 2 may fail, in vir-
t~e of a result of A.B.Aleksandrov [4]
Pi~lly we propose a generalization of Conjecture S-
CONJECTURE 3. Let ~_ be a s2mmetric domain in ~ different
from an~ product domain il Ix @, Gc ~-I and let % be its distingu-
ished boundar 2. Let ~r and ~zz be two domains in ~ intersectim~
.. a proper mappin~ such that for some se-
t r. ~ r
Then there exists an automorphism ~ of /~ such that
The v e r i f i c a t i o n of Con jec tu re 3 would l ead to a cons ide rab le strengthening of a result on local characterization of automorphisms
of classical domains obtained in [2]
One can see from the proof of Theorem 3 that Conjecture 3 holds
for classical domains of the fourth type. At the same time, it follows
from results of [I] and [5~ that Conjecture 3 holds also for the
balls /ip, 4
REMARK. After the paper had been submitted the authors became
aware of S.Bell's paper [6~ that enables, in combination with C2] ,
to prove Conjecture I
628
REFERENCES
1. A i e x a n d e r H. Proper holomorphic mappings in ~ o -
Indiana Univ.MathoJ., 1977, 26, 137-146.
2. Ty M a H o B A.E., X e H E E H LM. ~oKa~Ha~ xapaKTep~sa-
E~H aHaJ~TE~eCEEX aBTOMOp~ESMOB E~accENecEEX o6~aoTe~. - ~OE~.
AH CCCP, I982, 267, ~ 4, 796-799.
3. A ~ e E c a H ~ p o B A.~. K OCHOBa~ Teopz~ OTHOCETex~HOCT~. --
BecTH.I~V, I976, I9, 5-28.
4. A x e E c a H ~ p 0 B A.B. CymecTBoBaHEe BRyTpeHH~x~yREn~ B
mape. - MaTeM.c6., I982, II8, I47-I68.
5. R u d i n W. Holomorphic maps that extend to automorphism of a
ball. - Proc.Amer.Soc., 1981, 81, 429-432.
6. B e 1 1 S.R. Proper ho~omorphic mapping between circular domains.
- Comm.Math.Helv., 1982, 57, 532-538.
G.M.HENKIN
(r.M.XEHEHH) CCCP, 117418, MOCEBa,
~eHTpa~BHH~ BEOHOM~Ko--MaTeMaT~HecE~R
ZHCT~TyT AH CCCP, y~.Kpac~EoBa, 32
R.G.NOVIKOV
(P.F.HOB~KOB)
CCCP, 117234, MOCEBa,
~eH~HCEEe rope, MIV,
~ex.-~aT.~u'~yJIBTeT
629
12.4. ON BIHOLOMORPHY OF HOLOMORPHIC MAPPINGS
OP COMPLEX BANACH SPACES
Let ~ be a function holomorphic in a domain ~ ~ 3c ~.
It is well known that if { is univalent, then ~f(~) =~= 0 in ~ .
Holomorphio mappings # of domains ~ c ~ for 11, > ~ also possess
a similar property: if I : ~ ~ 6 ~ is holm, orphic and one-to-one
then ~Q~)=~= 0 at every point $ of~(~)------- (~)},K=I~" ~
is the Jacobi matrix), or equivalently the differential ~(~)~ ~---
~-~Q~)~ is an automorphism of ~ , and then, by the implicit
function theorem, ~ itself is biholomorphic.
Note that the continuity and the injective character of
immediately imply that ~ is a homeomorphism because ~ ~=
~_~<+oo . The proof of the diffeomorphic property depends essen-
tially on the holomorphic properties of ~ . Such a result, as is
known, for real spaces and mappings is wrong, which is clear from 3 3 ~
the example (0c~..., ~) ' (~... ~ ~) : ~
Let now X and Y be complex Banaoh spaces and ~ be a holo-
morphic mapping of the domain ~ c X into ~ , Remind that the
mapping ~ is called holomorphic in ~ if it is continuous and
weakly G~teaux differentiable, i.e. for any ~ and '~l, eX there exists
Then ~ (~)~ ~- ~/(~)£ is a linear operator X ,,~ : . - Y . It is
proved that in the complex case (*) implies the strong Prechet dif-
ferentiability of ~ :
Y
,or = = = II IIx PROBLEM. Let ~ be a holomorphic o n e - t o - o n e mapuin~ of the ,,do-
main ~ C X onto the domain ~ % Y . Is the differential ~(X)
a~ lisomorphism (i,,el@ ' an in,iective and sur~ective mapping) of the
space X onto ~ at every point x~ ?
630
The positive answer and the implicit function theorem would
imply that all one-to-one holomorphic mappings of domains of complex
Banach spaces are diffeomorphisms.
Positive solution of the problem would allow to obtain, for in-
stance, some important corollaries in the geometric theory of functi- ons of a complex variable(in problems concerning univalenoy and quasi-
conformal extendability of holomorphic functions, characterization
of boundary properties of functions starting from the interior pro-
perties; all that can be reduced to the consideration of some Banach
spaces of holomorphic functions).
The author does not know any general result in this direction.
It seems likely that the problem in the general statement must have
a negative solution. The following conjecture can be formulated
(at least as a stimulus to refute it).
CONJECTURE. Suppose conditions of th~ Problem are fulfilled.
Then the mappin~ ~(~) (~) is in,~ectiv~ but there exist spaces
X , Y for which it is not sur,~ectiv e.
Then a QUESTION arises under what additional conditions of, may-
be, geometric character, concerning the structure of the spaces X
and ~ , the mapping ~ (x) is an isomorphism (for ~ satisfy-
ing assumptions of the Problem); will this be so at least for Hilbert
spaces or spaces possessing some special convexity properties, etc?
S. L. KRUSHKAL CCCP, 630072, HOBOOEdHpoE
~I-ICTETyT MaTeMaTEEH CO AH CCCP
CHAPTER 13
~ISCELLANEOUS PROBLEMS
632
13.1. BANACH ALGEBRAS 01~ FUNCTIONS GENERATED BY THE SET 01~ ALL ALMOST PERIODIC POLYN0~IALS WHOSE EXPONENTS BELONG
TO A GIVEN INTERVAL
1. Por any A~C0,co] l e t PA denote the l inear set of a l l almost periodic polynomials
LXK~ K
with exponents /~I<6C-A ,A) , endowed with the sup-norm II ~ U = = ~ { I~(t,)l : - ° ° < t < oo}.
Evidently every linear functional ~ on PA is completely de- fined by its ~-function(characteristic function): ~ CA) =
= ~(@A) <-A<k <A) , where eA(~)=~/~($~t)(-°° <~ < oo) The % -func- tion may be an a rb i t ra ry function from ( - A , A) to 6 •
Let us denote by ~A the set of all ~ : C-A A)~ c gene- rating linear continuous functionals ~ on ~A (~ ~ ~) " For
E~A we put II~ II A ---II~II. It is easy to see that ~A is a Banach algebra of functions. By ~A (~) we denote the Banach subalgebras consisting of all continuous (measurable) ~ E ~A "
The following proposition was proved in [1] for A =oo and in [2] fo r (O<)A< o0 .
THEOREM I. Let ~ be a fumction from (-A,A) to C . Then c
~ ~A iff it admits the representation
cx) = I ,xt co C-A, A), ¢1 - 0 0
where d is a complex measure on ~ of bounded variation V@t ~<co. C
Moreover every ~ ~ ~A admits representation (I) with V~ ~ =
=Jill A Clearly it follows from this theorem, that every function
II~'AA ( 0 <A < ~ ) aa~ts an e=~ension C ~ with II Iloo ~JJ~ .~On the other hand~ every ~c ~A (O<A <° ° ) aam~ts an exte~ion f ~ hA with II ~ I I ~ II~ll A by the ~ n B ~ o h theory .
"~ C 0 < A < co) admit an extensi- QUESTION 1. Does every ~E ~A ttb
633
has an extension a 2
~o tha~_
QUESTION II. If ever~
does there exist for this ~ such an extension
ULn =U UA.
QUESTION III. Does every ~ A (0<A~°°) admit a decomposi- C
%ion ~ =~C+ ~,% , wh~re ~CC ~A ' ~T~ ~ ~A and & equals zero
atet?
2. A functional ~ C PA is said to be real if it takes real values on real ~8 ~ ~ PA . A ~P; is real iff its %-func- tion ~ (~= ~£)~ is He~itian: ~ (-~) L ~(~) VA~ (-oo,oo) . Every functional ~ ~A admits a unique decomposition ~ =~2~ ~ ,where ~,~)~ ~;) are real. Therefore, it is easy to see, ~hat "~ Ques- tions I,II, III we may restrict ourselves to the case of Hermitian
only. , , Denote by ~A (0 < A ~ oo) the cone of all non-negative ~A"
Naturally, a ~9 (~ pA ) is said tO be non-negative if ~(~)>~0 ~S s o o n a s
Denote by ~A(0<A~ OO) the cone of all *-functions corresponding to the elements of ~A . The subcone of all conti-
nuons (measurable) ~ C PA will be denoted by ~A ( A ) "
it is easy to see that for every #C ~A we have: ~(0)=Jl~ll A For any Hermitian ~A there exists a decomposition ~=~+-~_ , where ~+~ ~A andJl~[l~ ~= ~+ (0)+ ~_(0).
To establish the last assertion it is sufficient %0 do this for A=oo . In this case p~ forms a linear dense set in the Banach space ~ of all almost periodic Bohr functions with sup- norm. The cone ~*Qo is dual to the cone ~ of all non-negative functions in 8 . As 8 may be identified with a Banach space of all continuous functions on a compact space, the existence of the required decomposition for Hermitian #~ ~oo follows. Moreover,this decomposition is unique and minimal in this case (A=oo) .
C If #~ 2A is Hermitian, then there exists a decomposition
c #=~+-~_ with ~+ PA and [I~UA----- ~+(0)+~_(0) . Indeed, for Hermi- tian {~2 we can obtain (I) with a real measure ~ , V@~ ~== =N~[[ A , admitting a unique decomposition g~-g+--g_ with nonnegative measures 6"± such that ~ 6~= V~b~ ~+ + V@% 6 ~ _ • This decomposition yields (via (~)) the required decomposition for ~ .
Does eve e=,ti n admit a deoom o-
634
It turns out, that the affirmative answer to this question im-
plies the same for Question III. This connection is due to a theorem
of [4] according to which everycfunction ~ ~ admits a decompo-
sition ~=~c + ~m ' where ~c~ , ~ ~ and ~ equals zero h
a.e. This theorem has been generalized recently in ~5].
It is plausible, that for any Hermitian #~ ~A in the decom-
position ~= ~+-~_ "~th ~±~:~A , II ~ IIA = ~+(0)+ ~_(0) (which always exists) automatically ~ +- ~ ~A "
3. Por better orientation we will indicate that A.P.Artemenko
[6,7] has obtained a general proposition which contains, in particu-
lar, the following characterization of functions # ~ ~A
THEOREM 2. Let 0 < A ~ co and let ~ be a function from
(-A,A) t_~o ~ . Then ~ iff for an~v ~0, A] and any
n
The necessity of this condition is trivial. A transparent proof
of it's sufficiency has been obtained by B.Ja.Levin [8].
Por a continuous function ~ from (-A, ~) to C the asser-
tion of theorem 2 (for A =oo ) is contained in the well-known theo-
rem of Bochner and for (0 <) A <°° in the author' s corresponding theorem [2].
A series of unsolved problems concerning extensions of functi-
ons ~ ~A ( ( 0 < A < oo) is formulated in [4]° In this connection we also mention [9].
REFERENCES
I. B o c h n e r S. A Theorem on Pourier-Stieltjes Integrals. -
Bull°Amer.Math.Soc., 1934, 40, N 4, 271-276.
2. E p e R H M.r. 0 npo6aeMe npo~o~e~ sp~ToBo no~o~e~x Henpep~m~x SyH~n~R. - ~o~.AH CCCP, I940, 26, ~ I, I7-2I.
8. E p e R H M.L 0 npe~c~aBxe~ Sys~z~i ~Terpa~ ~yp~e-CT~-
T~eca. - Y~e~e sanHc~ Ey~6~eBCEO~O r~, 1943, ~ 7, I28-I47.
4. E p e ~ H MoI'° 06 l~sMep~X ~pM~TOBO--noxo~ex~H~X ~ym~m~Xo - ~aTeM.SaMeTE~, 1978, 23, ~ I, 79--89.
635
5. L a n g e r H. On measurable Hermitian indefinite functions
with a finite number of negative squares. - Acta Sci.Math.Szeged,
1983 (to appear).
6. A p T ~ M e H ~ o A.H. 0 nOS~T~BHNX~He~n~x~yHEL~o~a~aXB npo-
~aHCTBe HOqT~ HepHoI~IeCEHX~FHEI~ H.Bohr'a . - Coo6~.Xap~E.
MaTeM.06-Ba, 1940, (4), 16, 111-119o
7. A p T ~ M e H E O A.H. 8p~TOBO nOXO~Te~HHe ~m~H ~ nOS~T~B--
HHe ~yHEL~OHa~H I. -- Teop~yHE~, ~JHEn.aHs~. ~xnp~., I983
(B neqaT~).
8. ~ e B ~ H B.H., 06 O~OM 0606meH~L~ Teope~$e~epa-P~cca. -~oE~.
AH CCCP, I946, 52, 291-294.
9. c r u m M.M. On positive definite functions, 1956. - Proc.
London Math.Soc., 1956, (3) 6, 548-560.
M. G.KREIN
(M.r.KP H) CCCP, 270057, 0~ecca,
y~.ApTeMa 14, EB.6
636
13.2. SUPPORT POINTS OP UNIVALENT FUNCTIONS
Let H(~) be the linear space of all functions analytic in the
unit disk ~ , endowed with the usual topology of uniform convergen-
ce on compact subsets. Set ~ be the class of functions i~m(~)
which are univalent and normalized by the conditions 4 ( 0 ) = 0
and I l (0)= 4 . Thus each I ~ ~ has an expansion of the form
l (z ) :z + ~ z s +~,s~,s,..., I~1 < 4.
Let LI be a complex-v~,lued continuous linear f u n c t i o n a l on H (~)
not co~tant on S . Beca,,,e D is a compact subset of H(~)) , the
functional Re { L} attains a maximum value on S • The extremal
functions are called s u p p o r t p o i n t s of S . In
view of the Krein-Milman theorem, the set of support points associat-
ed with each linear functional L must contain an extreme point of
S • It is NOT KNOWN whether every support point is an extreme
point, or whether ever~r ' extreme point is a support point.
The support points of S have a number of interesting proper-
ties. It is known that each support point ~ maps ~ onto the comp-
lement of an analytic arc ~ which extends with increasing modulus
from a point W 0 to CO , satisfying
w~ > 0 (I)
p r o p e r t y l ~ ( W ) l ~< ~ -~ , w ~ W o . ~he bound ~T/4 is best possible a n d i n f a c t t h e r e a r e s u p p o r t p o i n t s f o r w h i c h I~(Wo)l =~ . mt
is also ~o,~ t~t LC{~)~O , fro~ w~ch ±t fonows t~t F is asymptotic at infinity to the half-line
W=3L( ) (2)
An exposition of these properties, with further references to the
literature, may be found in m4]°
637
Evidence obtained from the study of special fnnotionals ~1,2,
6,31 suggests the CONJECTURE that the omitted arc ~ always has mono-
tonic ar~ent. This is true for point-evaluation functionals ~(~)=
=~(%)- , where ~ ~ ; for derivative functionals--~'~(~)=~v~l(~)~
for coefficient functionals L(~) = 63 + ~ ~ , where ~ E C , and
of course for coefficient functionals ~(~)=~ with ~% ~4 ~ ,
where the Bieberbach conjecture has been proved. A STRONGER CONJECTU-
RE, supported by somewhat less evidence, is that the radial an~le
• (W) tends monotonically to zero as W ~o0 alon~ ~ .
The Bieberbach conjecture asserts that I@~I ~ , with strict
inequality..for all ~ unless ~ is a rotation of the Koebe functi-
on ~(E)=Z(~-H) -~ . A geometric reformulation is that the arc P corresponding to each coefficient functionB1 ~(~) =@~ is a radial
half-line. It is essentially equivalent to say that the asymptotic
half-line (2) is a trajectory of the quadratic differential (1).
A weak form of the Bieberbach conjecture is that for each coeffici-
ent functional ~(~) =@~ the asymptotic half-line is radial. It
is interesting to ask what relation this conjecture may bear to
other weak forms of the Bieberbach conjecture, such as the asympto-
tic Bieberbach conjecture and Littlewood's conjecture on omitted va-
lues, now known [5] to be equivalent.
REFERENCES
1. B r o w n J.E. Geometric properties of a class of support
points of univalent functions. - Trans.Amer.Math.Soc. 1979, 256,
371-382.
2. B r o w n J.E. Univalent functions maximizing RE ~3 + ~ .
- Illinois J.~ath. 1981, 25, 446-454.
3. D u r e n P.L. Arcs omitted by support points of univalent
functions. - Comment.Nath.Helv. 1981, 56, 352-365.
4. D u r e n P.L. Univalent Functions. Springer-Verlag, New York,
1983e
5. H a m i 1 t o n D.H. On Littlewoed's conjecture ~or univalent
functions. - Prec. Amer. ~ath. Soc. 1982, 86, 32-36.
6. P • a r c e K. New support points of S and extreme poimts of
HS. - Proc.Amer.Math.Soc. 1981, 81,425-428.
P.L.DUREN Department of Mathematics University of Nichigan
Ann Arbor, Michigan 48109 USA
638
13.3- MORE PROBLEMS BY ALBERT BAERNSTEIN
Let ~ be a simply connected domain in C and F a conformal
mapping from ~ onto ~ , normalized by IF ! (~)I =4 when F(~):0.
Hayman and Wu [I] proved that
for some constant A • A simpler proof has been given by Garnett, Gehring and Jones [2B. Is it true that
~ n ~ ~ ~ e " - -~ I~ for_ some constant ~. when ~<p<% . T~e e P~ #(Z):(~-~) , F(~) the inverse o~fT , shows that l I F ' ( ~ ) l ~ : ~
is possible when ~ % ~ < ~@ . IRn.tL Using the technique of [I] or [ 2] together with classical har-
monic measure estimates, it can be shown that (I) is true for
~ . The inverse function of ~(Z) : (~-~)-~ shows that
~!~i F:(e:)l~l~t~ :l?wouldi:o::::i::e~:eleOr~zPt~o ! :~e inequality,
(2) below, is true. Let ~li~ be a Whitney type decomposition of
~2 as described in ~2 , § 3]. Denote by ~j the center of
Ii , ~i the length of L , L~ the vertical half line starting
f#om ~ ÷ ~4 , and let ~ be ~ the domain obtained by deleting
from~ A all ~the half lines ~i • Is it true that
for eve~ Z~ ~ and ~ ~ C ~th I ~ a-< 0 , where ~ and G denote the Green's functions of ~ and respcetivelE?
REFERENCES
I. H a y m a n W.K.,W u J.-M.G. Level sets of univalent functions.
-Comm.Nath.Helv. 1981, 56, 366-403.
2. G a r n e t t J.B., G e h r i n g F.W., J o n e s P.W. Con-
formally invariant length sums . -Indiana Univ.Math.J., to appear.
A. BAERNSTEIN Washington University
St.Louis, MO 63130, USA
639
13- 4- SOME EXTENSION PROBLEMS
Let K :G--~ K(G) and K : G--~R(G)associate with each open set GC~ " ~ a class of complex-valued functions on G • A set EcC will be reded negligible (K,R) i~, ~or eac~ open set G c { and ~ao~ ~ ~ K(G). , ~he existence of an ope~ s~t ~cG
such that {IG~-K(G) and I(G \ ~ ) c E implie~ that {~ KCG) • For the case when K-C (= sheaf of continuous functions) and K~A (= sheaf of holomorphic functions), negligibility of finite sets was established by T.Rad~ in [4]. P.Lelong showed in [3] that also all p o l a r se ts are negligible (C,A) •
PROBLEM I. What are necessa~f and sufficient condi,tlons for
Ec~ to be neFli~ible (C,~)?
For continuously differentiable functions some related results concerning harmonicity are known. If OJ ~ 0 is a continuous non- decreasing function on ~ with ~(~))0 for %~0 , we deno- te by C~(G) the class of all functions ~ on G satisfying the condition
I~(~)- I(v)l= O ( m ( l . - v l ) ) as I ~ - V l ~ 0
W locally in Q ; C, (G) will stand for the subclass of all
6 0W(G) enjoying the property
locally in G • F~ther we denote by (G) and ~ (~) the c l asses o f a l l c o n t i n u o u s l y d i f f e r e n t i a b l e r e a l - v a l u e d f u n c t i o n s
whose first order partial derivatives belong to C qa(~) and C~(G) , respectively. If H(~) denotes the class of all real-valued functi- ons harmonic on G , then the following result holds (cf. [2]).
THEOREM. A set EC~ is negligible [C~ H) if (and also
onl.y if in case ~J(0) m 0 ) the H~sdorff measure co rrespondin ~
to the measure function ~ vanishes ' on all compact subsets of E •
A necessary and sufficient condition for E c R to be n eRli~ible
(C4~ H ) consists in ~ -finiteness of the Hausdorff me a-
640
sure correspondin6 to ~ on al! cgmpact subsets of ~ .
For subharmonic functions similar question seems to be open.
(Of course, necessity of the corresponding condition follows from
the above theorem.) Let S(G) denote the class of all subharmonic
functions on ~ .
CONJECTURE. A~7 set ~C~ wit h vanishing Hausdorff measure
correspondin~ to the measure function ~ is negligible (C~-- "~ -~) .
l_~f E C~ has 6 -finite Haus.d.orff measure correspondin~ to ~ ,
then ~ is ne61i6ible (~,
PROBLEM 2. What are necessar~nd sufficient condition s for
E CC to be negligible (C~ A) or (C~ A) ?
Similar questions may be posed for various classes of functions
in more general spaces (compare ~SB).
REFERENCES
I. C e g r e I i U. Removable singularities for plurisubharmonic
functions and related problem. - Proc.London Nath.Soc.~I978,XXXVI,
310-336.
2. K r ~ i J. Some extension results concerning harmonic functi-
ons, to appear in J.LondonMath.Soc.~1983.
3. L e 1 o n g P. Ensembles singuliers impropres des fonctions
plurisousharmoniques - J.Math.Pures Appl.~1957, 36, 263-303.
4. R a d ~ T. ~ber eine nicht fortsetzbare Riemannsche Nannig-
faltigkeit - Math.Z. 1924, 20, I-6.
JOS~ m~L Matematicky ~stav ~SAV
11567 Praha I
Czechoslovakia
641
13.5. PARTITION OF SINGULARITIES OF ANALYTIC FUNCTIONS
Let S be a closed set in ~ , andl~ let A (S~ be the cl~) of functions ~ , holomorphic in W = ~ \ ~ , such tha t ) ! , . . . , , can be extended continuously to W ' - W U ~ (~:0,~,j~,...
Let S---~ 4 US$, each ~ being closed, and moreover ~--~; U ~ , 0
where S~ is the interior of S~ relative to ~ °
In this situation IT IS NATURAL TO GUESS
A'c,S ~--A'(.,~) +/~'<.,S~), ( , , :o ,4 ,~ ,s , . . . ) . (~)
To explain the d~fficulties involved in ~1), we suppose that ~ is
continuous in ~ and use the operator ~ defined by ~S-=S/9~TL~/SU.
Following the classical method, we choose a function (PEC~'(~&~ I 0 _<. 0 1 o " such that O ~ ( ~ l and (~=0 on ~ \ S 4 , = 1 on S\~2, . S~ppose that ~ k = ~ 9 ~ ~ i n the ~ense of d istr ibut ions, or the Cauohy- ~rsen fo~ula>. T;,en ~ ( ~ l - l ~ = ~ , a~d therefore F = ~ - k be- 0 I " / | l - - " 0 longs to ~ (S4) ; s imi lar ly ~ - F = ( ~ - ~ ) { + ~ belongs to ~(S,). Therefore (1) is true for Ht:O (a classical observation., to be
sure) but the reasoning seems to fail when ~-~ since { is gene-
rally not C (or even Holder-continuous) on R .
If (I) were true (for some ~/4 ) it would imply that the tri-
viality of A~(S)is a local property of S . (Triviality of A~(S)
means of course that all of its elements are restrictions to ,W of
entire functions. ) Even this much is unknown.
R. KAU~'~&~" Dept.Math.
Univ. of Illinois
Altegeld Hall
Urbana, Illinois 61801 USA
642
13.6. REARRANGE~ENT-INVARIANT HULLS O~ SETS
Let (S ,~-" : - , f f ) be a non-atomic finite measure space. Denote by
J~E ($>O)a family consisting of all ~-preserving invertible
transformations 60:8 S such that ~CCS : OJ(~¢)~~.
Each OO~ E generates a linear operator ~- W : ~ --'- ~ - (where
denotes the space of all measurable functions on ~ ) by the
formula ]-to ~ (~) ~(W(~))~ ~8, ~ ~ . The elements of a set
~e~ {%. GJ ~ ~E } are called the ~-rearrangements. Each
Toa preserves the distribution of a function, hence the integrabili-
ty properties of functions are also preserved.
Given a subset A of ~ define the r e a r r a n g e -
m e n t - i n v a r i a n t h u 1 1 s of A as follows:
RHo(A) ~ N R~ (A), g~O
RH(A) = UoR~(A).
The general problem of characterization of such hulls for a given
concrete set A has been posed by O.Cereteli. We refer to [5] for
the contribution of O.Cereteli to the solution in some concrete
cases.
The following results have been obtained in [2] and [3]- Consider
for the simplicity the case when ~ is [0,{] equipped with the usu-
al Lebesgue measure.
a) Let ~= {~}~ ~ be a family of bounded functions such
that for any ~ ~L °°
Z ! c,~ ( ~; qb)t ~ ~ oo~s~ II f I1~
with a constant independent of
For a non-negative sequence { ~;}
~t--~ co , define a class
• Here O ~ ( f ~ l : : ~ ) - ~ I { ~ t ~ , ~ - ) ~ . , ~] ~ such that ~---0 when
where C~' ~)* denotes the non-increasing rearra~eme~t of ~ . ~hen RHo(A)= RH(A)= IJ [2].
functions in U . For given p, ~ p < 2 , define a class
643
Am~ { f~ kt • I~,- (S, p l l~ - - o } ,
where S~ ~ denotes the ~ -th partial sum of the Fourier series of
with respect to (~ . Then ~@(A)=~H(A)=L 4 [3],
i.e. any complete orthonormal family of bounded functions is in some
sense a basis in ~ ~ ~ p < ~ ~ modulo rearrangements.
A different effect occurs for the class A = [ ~ ~ ~4}, 4
where ~ is taken over the unit circle T and ~ denotes the con-
jugate function of a function ~ . In that case [4, 5]
I T
Here
o , i f(~?l~ ~ .
The class
~,) d,, I < ~, }
arising in (1) (in [4, 5 ] this class is denoted by ~ ) coincides~
on non-negative functions ;with the class m ~+J. . Moreover,
L ~+ L c M ~+M. In addition to (1) it has been proved in [3] that if
I~I-Io (A) = RH (A)= M I~FJ M ,
i.e. any function from M ~ hA~F~÷ivl can be rearranged on a set of
small measure so that the obtained function has m r -convergent con-
jugate trigonometrical series.
For any p, p~ | , define a class m P over (S~,~) as follows
@ S
644
I t is clear that LPC M P and M P coincides with L P on non-ne- gative functions. The class M P in comparison with L P takes into
account not only the degree of integrability of function but also the degree of cancellation of the positive and negative values of the
function. It is known [6] that M ~+M is linear. As f o r M~ it has been proved in [3] that 1) rv~P~ p>2 __is non-linear, more- over, there exists ~ M P such that ~+4¢ MP~ 2) MP+L'c M P, ~< p-<2.
PROBLEM 1. Is the class M P linear for 4 <p~ 2 ?
Consider a family C~-= { ~} such as in b). Denote by S* the
maximal operator for the sequence of partial sum operators { ~} with respect to ~D , i.e. S @~ = $~p I S~l,~ 6 LI The problem
of finding the rearrangement invariant hulls of a set A = ={~C~ :~}is not solved even for classical families ~ . Some partial results have been obtained in [3]. For the trigonometrical system { e~p %~} -oo<~< +oo on T the following inclusion holds:
MPc RH.(A), p>I.
PROBLEM 2. Find R H. (A) and rical system.
Very interesting is the case when polynomials { L~} on the interval that
MPc RH0(A),
R H (A) ,~,r the trigonomet-
~D is the family of Legendre [-~,+4] . Tt is true [3]
p > _8._ • (2)
PROBLEM 3- F,ind ~ N o(A) and ~H(A) f o r the family of Le-
gen~e polynomials.
We pose also t~o easier problems related to Problem 3.
PROBLEM 3'. Is the inclusion L '/' C R Ho (A) true?
PROBLEM 3''. Is the inqlusion M '/~ C ~ Ho (A) true?
The number ~ in (2) appears from the general theorem proved in [3]. The theorem states that if a sequence of integral operators
[~-~}, ~ ~ ~ has a localization property in I ~ , and the maxi- mal operator T '~ ~ :" s ~ p I T n ~1 has a weL t=e (p,p) ~th some
p > 4 , then
645
MNrc R Ho(A), (3)
where A={ ~ ~ ~ . ~ i~ .¢ ~ ) . It is not known, whether the
power ~ in (3)is sharp on the whole class of the operators
under consideration. The maximal operator with respect to the
Legendre pol~lomialo system has weak~type (e, p), ~/3< e < ~ [I].
The number ~ is the value of ~--~ at p= •
The problems analogous to Problems 3, 3', 3'' can also be formu-
lated for Jacobi polynomial systems.
REEERENCES
I. B a ~ K o B B.M. CXOAH~OCTb B cpe~He~ H nOrTH BCD~Vp~I~OB ~ypbe
no MHOPOq2eHaM, OpTOPOHaJlBHB~ Ha oTpe3Ke.- 2~aTeM. c60pH., 1974, 95, ~ 2, 229-262.
2. ~ y Ji H C a • B H ~ H A.B. 06 Oco6eHHOCT~D( CyMMHpyeM~x g~y~K~. - San. Hays. ceMHH. Z0~, 1981, I13, 76-96.
3. F y a H c a m B H ~ H A.B. HepecTaHOB~H, paCCTaHOBNM 8HaEOB H CXO-
~HMOCTb Hoc~eAOBaTe2bHOCTe~ onepaTopoB.- 3an. Hay~H.CeMHH. ~0~,
1982, 107, 46-70. 4. ~ e p • T e ~ M 0.~. 0 coHp~eHHb~ ~yMEMH~X. - MaTcH. SaMeTK~,
1977, 22, ~ 5, 771-783. 5. ~ e p e T e ~ H 0.~. 0 conp~eHH~O( ~y~KUH~X. - ~O~TOpCEa~ AHccep-
TaL~M2, T6Ha~c~, 1976. 6. i e p e T e a H 0.A. 0~ OAHOM cayqae cyM~HpyeMocTH coHp~OHHNX
Sy~u,~.- TpyAm T6Ha~cc~oro MaTcH. HH-TaAH Fpys.CCP, 1968, 34, 156-159.
A.B. GULISASHVILI
(A.B.FYJIHCAI~Mf~i)
CCCP, 380093, T6H~HCH,
y~.Pyxa~3e MaTeMaT~c~ HHCTHT>~
AH Fpys~cKo~ CCP
646
13.7. NORMS AND EXTOLS OF CONVOLUTION OPERATORS
ON SPACES OF ENTIRE FUNCTIONS
Given a compact subset ~C let B(~} be the Be=stein
class of all bounded functions ~ on ~he dual copy) with
Fourier transform f supported on ~ 0 In fact, every function
~e B(~) can be extended to an entire function of exponential
type on ~ . The linear space B(~) with the uniform norm on
is a Banach space (in fact, a dual Banach space).
EXAMPLE. Let K be the unit ball in ~ , i.e,,
if and only if the function ~
of an entire function on C
is a restriction
satisfying
for some constant C.
we shall consider operators ~-" B(K) -4" B(K)
form
CTI = I
of the
~ being a complex-valued regular Borel measure of bounded variati-
on on . In other words, ~-~ = ~ * ~ , The function
~ ~ ~ is said to be the s y m b o i of T=~-~ The representation ~ ~ ~-~& is not an isomorphism, but nevertheless
the symbol q~ is uniquely determined by 7- ° The spectrum ~f ~-
coincides with the range of ~ and its norm with the norm of the
functional ~ ~ CT~) (0) . if K is a set of spectral synthe-
sis then the symbol ~" .uniquely determines the corresponding opera-
tot T-- ~" C D), D = ~ ~/~ in B(K) . Moreover, in
this case
647
DEFINITION. A n o r m a 1 e x t r e m a 1 for T is any
element ~ BCK) such that II~II---4 7 (T~)(0)= ll~-II .
It can be easily shown that the normal extremals always exist
(and form a convex set). For example, in the case ~= ~ ~ K-- [-(~,~]
O>0 , and T~ = ~f the classical result asserts that ~TI=~
and all normal extremals have the following form o,e ;p ÷
l~ e~p (- i, Ox). A measure ~ is called e x t r e m a 1 for -~ if T~ =
and I ITI I" The set of extremal measures may be empty even in case of finite
K. Such problems as calculation of norms and discription of extremals
go back to the classical papers of S.N.Bern~tein A survey of results
obtained in the field up to the middle of 60-ies can be found in [I].
~or additional aspects of the topic see [2], which is~unfortunately,
flooded by misprints, so be careful.
A compact set K in ~ is said to be a s t a r if with eve-
ry ~ it contains ~ ~ for each p ~ [ o, 4] . Every star
is a set of spectral synthesis and B(K) contains sufficien-
tly many functions vanishing at infinity. If ~, ~K and
~ ~(~)[ =~(~o) == 4 then I~ q~(D)~= 4 (i,e. the norm of
(D) coincides with its spectral radius) if and only if the fun-
ction ~ --*-~F (~ ÷ ~,) admits a positive definite extension
to R ~ . The operator "~(D) has extremal measures. If ~(D)ll>|
then every extremal measure is supported on a proper analytic subset
of ~a and the extremal measure is unique provided ~----~ o For ~>| the uniqueness does not hold (example: K is the unit ball and~(D)
is the Laplace operator).
PROBLEMS IN THE ONE-DI~tENSIONAL CASE.
~,~ery polynomial (in one variable) is related to a wide stock of
positive definite functions. Suppose that the zeros of a polynomial
"~ are placed in the half-plane ~e~ ~ ~ 6 > 0 and
that ~(0) = 4 , Then the restriction of ~ to [ 0~O]
extends to a positive definite function on ~ . It follows that
for all linear
where BO ~-e~ B ( [ - O , C ~ ] ) . - - In t h i s case a l l normal extremals can be easily determined and there exists an extremal measure (at le-
ast one). At the same time for pol,~.omial s qY of de~ree 2... these
6 4 8
problems still do not have a full solution. The simplest operator is
provided by ~ ~ _~ll. ~ , ~C~ . For X ~ the pro-
blems are solved (see papers of Boas-Shaeffer, Ahiezer and Meiman).
For some complex ~ (in particular those for which the zeros of the
symbol ~ satisfy the above mentioned condition) ~ admits (after
a proper mormalization) a positive definite extension, so that the
norm of ~(~) coincides with its spectral radius+
Is it ~ossible to calculate the norm for al~ ~C~ ? How do
the "Eulerequations"lcok in this case?
Note that according to the Krein theorem the extremal measure is
unique provided K=[-O, O] and the spectral radius is less than the
norm. Of course, these problems remain open for polynomials of higher
de~ree+
The Bernsteln inequality for fractional derivatives leads to the [-+,4] following PROBLE~. Consider on the function ~(~)~(~--I~I)~
o(>0 + The problem is ~o find
If 06 ~ ~ then ~ is even and is convex on [0,~] , and the-
refore coincides on [-|~ ~] with a restriction of a positive definite
function by the Polya theorem. For O~ < ~ T becomes concave on
[0,|] and moreover ~ cannot be extended to a positive definite
function on ~ . Indeed, if ~ is positive definite then -~"
is a positive definite distribution. At the same time, --~" is non-
negative, locally integrable on a neighbourhood of zero and non, in-
tegrable on a left neighbourhood of the point ~ = | . A positive
definite function cannot satisfy this list of properties.
The best known estimate of the norm for O~ C (0,4) is 2(4+0()~ ~
It is evidently not exact but it is asymptotically exact for o~-+-0
and o~ +-~ | . It should be noticed that in the space of trigonometric
polynomials of degree ~ PP~ the norm of the operator of fractional
differentiation coincides with its spectral rsdius for o~ >I o~0 ,
where oC o~o~ o(P~)< ~. Another example is related to the family { ~ of functions
definite. Consider the family for o~ > + . Since every charac-
teristic function ~ satisfies the unequality l~(~)Im~ ~(~+I~(2~)I),
649
there are no positive definite extensions for c~ 2
Consider now the case ~ <o(< 2 ° The following idea has be-
en suggested by A.V.Romanov Extend ~(~) to (|,2) by the for-
mula
v(1 ÷
Extend now the obtained function on (0,~o an even periodic fun-
ction of period 4 keeping the same notation ~ for this function.
We have
k ~ ,t
where the sum is taken over odd positive integers. It is easily veri-
fied (integration by parts) that ~K and
0 are of the same sign. Clearly ~I > 0 and
Hence ~ is positive definite if
@
for k s.
(cf. [4], Ch.V, Sec.2,29). The ~unction ~(c&) decreases on (4 ,2)and ~(4)=oo ~(Z)< 0 . Therefore the equation ~(OC)~0 has
a unique solution oCo ~( ~, ~) (Romanov's number). The functi-
on ~ is positive definite on [-I,~] if C~oC 0 . At the
same time slightly modifying the arguments from [5], Tho4.5.2 one can
easily show that for 0<oC I <oC~ < Z the function
4-
is positive definite on R . Hence ~4 is positive definite on
[-4~4] if so is~ z . The "separation point" ~0 is clearly ~o~
Is it true that ~o >o~@ ? For C~>~ the above problems re-
main open,
650
PROBLemS IN THE MULTIDIMENSIONAL CASE. For ~ ,except the case when the norm of ~(])) coincides
with its spectral radius, very few cases of exact calculation of
~(D)~ and discription of extremals are known. The GENERAL PRO-
BLEMhere is to obtain proper generalizations of Boas-Schaeffer's and
Ahiezer-Meiman's theorems, i e, to obtain "Euler's equations" at least
for real functionals. Our problems concern concrete particular cases;
however it seems that the solution of these problems may throw a
light on the problem as a whole.
If ~ : ~-'~ ~ is a linear form then the operator with
the symbol ~ I ~ is hermitian on B(~) and hence its norm coin-
cides with the spectral radius. This again will be the case for some
operators with the symbol of the form (p o ~)I K where p is a
polynomial. The following simple converse statement is true . If
is a polynomial and
for every symmetric convex star K then ~ : p o~ a linear form and p is a polynomial.
Does the ~ similar converse statement hold when
the balls?
where ~ ia
ranges over
Let
where p~ | and ~ C ~ ) = - ~ ~ • The operator ~ ( ~ ) is obviously the Laplacian ~ .4 The norm of ~ coincides with the
spectral radius in the following cases: n: ~ ~: ~,p=|~ ~,p:°°.
The proof is based on the following well known fact: if ~ is a pro-
bability measure then {~ " I ~(~)~ = ~} is a subgroup. The case
e:£ turns out to be the most pathological and perhaps the most in-
~eresting. We have IIAIIB(K a : ~. In this case extremal
measure is not unique and it would be interesting, to describe all ex-
tremal measures (notice that they form a compact convex set). The pro-
blem of calculation of the norm can be reduced to the one-dimensional
case for operators of the form ~(~) in ~(~2) . It is possib-
le to calculate the norms by operators with linear symbol in the spa-
ce ~ (K2) explicitly, For example, the norm of Cauchy-Riema~u ope-
651
rator equals 2 and its normal extremal is unique. Namely,
However, for the operators of the second order the things are
more complicated. If the symbol *'(~) -- ( A ~,~) real quadratic form then II (D)II A radius of Z(D) coincides wi~h that of ~
are of the form
is a positive
, the spectral
and normal extremals
where 10~1 is the norm, ~ ~ ~ t~ ! ~
At the same time nothin~ i s known about ext.remals ...and norm of the
operator 0~+ "@== in B (Kz) f o r ~ = Z .
REFERENCES
I, A x a e s e p H.H. ~e~az no ~eopu annposcmMa~a. Moc~m, HaT-
rm, 1965. 2. r o p a H E,A. HepsBeHc~a BepHmTe2HS C TOq~Z spe~ Teopaa oue-
pSTOpOB. -- BeOTH.XspBE.yH-TS, ~ 205. III~FJ~S MaTeaaTz~m a Mexa-
HME8, m~n.~5. - XaD~EOB, B~8 m~oaa, HS~-BO XaI~E.~H--Ta, 1980,
77-105. 3. ro p ~ H E.A., H o!o B ~ ~a o C.~. 3KcTpe~ma~ HeEoToI~X
~i~eloe~z~a~x one~aTopoB. - ~o~ no TeopH onep~TOpOB B ~HE--
~oHaa~x nI~OTImHOTBaX, MZ~OE, 4-11 ~as 1982. Tesac~ ~oz~.,
48-49. 4. Zygmund
1959. 5. Lukacs
London, 1970.
E.A. GOB.IN
(E.A.rOPm~)
A., Trigonometric Series, vol.l. Cambr Univ. Press,
E . Characteristic functions, 2 nded., Griffin,
CCCP, 117288, MOCEBa,
MezaEz~o-~a Tema T~e C~a~ ~ a ~ T e T
652
13.8. old
ALGEBRAIC EQUATIONS WITH COEFPICIENTS IN COMMUTATIVE
BANACH ALGEBRAS AND SOME RELATED PROBLEMS
The proposed questions have arisen on the seminar of V.Ya.Lin
and the author on Banach Algebras and Complex Analysis at the Moscow
State University.
In what follows A is a commutative Banach algebra (over C )
with unity and connected maximal ideal space M A . ~or ~ ~ ,
denotes the Gelfand transform of • .
A polynomial p(~ = ~ + ~ ~'~ + • + ~ ~ ~ £
is said to be s e p a r a b I e if its discriminant ~ is in-
vertible(i.e, for every ~ in M~ the roots of ~ + ~4 (~) +
+ +~(~) are simple); ~ is said to be c o m p 1 e t e -
I e r e d u c i b 1 e if it can be expanded into a product of po-
lynomials of degree one. The algebra is called w e a k 1 y a 1 -
g e b r a i c a 1 1 y c 1 o s e d if all separable polynomials
of degree greater than one are reducible over it.
In many cases there exist simple (necessary and sufficient)
criteria for all separable polynomials of a fixed degree Wv to be
completely reducible. A criterion for A = C(~) , with a finite
cell complex ~ , consists in triviality of all homomorphiams
~4(X) , B(~), B(~) being the At%in braid grQup with
threads [I]. If (and only if) ~.< 4 this is equivalent to
~ ( ~, ~ ) = 0 (which is formally weaker). The criterion fits
aS a s u f f i c i e n t one for arbitrary arcwise connected
locally arcwise connected spaces ~ .
It can be deduced from the implicit function theorem for commu-
tative Banach algebras that if the polynomial with coefficients ~
is reducible over ~(~A) then the same holds for the original poly-
nomlal p over ~ . On the other hand (cf, [2], [3]) for arbitrary
integers ~ , ~ , 4< k % ~ < co there exists a pair of uniform
algebras A c B , with the same maximal ideal space, such that
~/~ = 4 , all separable polynomials of degree % ~ are
reducible over A , but there exists an irreducible (over ~ ) se-
parable polynomial of degree ~ .
WE INDICATE A CONSTRUCTION OP SUCH A PAIR. Let G k be the col-
lection of all separable polynomials ~(~)= A k +~4~ k'~ + ,.. t~ k
with complex coefficients ~ ' I~ ' ' " ~ E k , endowed with the complex
structure induced by the natur~l embedding into ~k ~ (~4 .... ,~k)"
Define ~ as the intersection of Gk , the submsmifold {E 4 = 0 ,
653
is a finite complex. The algebra ~ is the uniform closure on
of polynomials in Z~,,,,Z k and ~ consists of all functions
in B with an appropriate directional derivative at an appropriate
point equal to zero. With the parameters properly chosen, (A,B) is
a pair we are looking for (the proof uses the fact that the set of
holomorphic functions on an algebraic manifold which do not take
values 0 and S is finite, as well as some elementary facts of Morse
theory and Montel theory of normal families that enable to control
the Galois group).
Do there exist examples of the same nature with A weakl~al-
~ebraicall~ closed? We do not even know any example in which A
is weakly algebraically closed and C(M A) is not. A refinement
of the construction described in ~4~ and [ 5B may turn out to be suf-
ficient.
If X is an arbitrary compact space such that the division by
6 is possible in H ~ (~Z) then all separable polynomials of deg-
ree 3 are reducible over C(~) . The situation is more complicated
for polynomials of degree 4: there exists a metrizable compact space
of dimension two such that ~ ~ ~) = 0 but some separable
polynomial of degree 4 is irreducible over C (~) [6~. on the
other hand, the condition that all elements of ~ (~,~) are divi-
sible by ~! is necessary and sufficient for all separable polynomi-
als of degree ~ ~ to be completely reducible, provided ~ is a
homogeneous space of a connected compact group (and in some other ca-
ses). These type's results are of interest, e.g., for the investiga-
tion of polynomials with almost periodic coefficients.
Is it possible to describe "al!" spaces ~ (mot necessaril Y
~om~ct) for which the problem of compl~$e reducibi~t~ oTer C(X)
of the separable pol.ynomials can be solved in terms of one-dimensio-
nal cohomolo~ies? In particular, is the condition ~ (~,2) =0
sufficient in the case of a (com~act~ hqmogeneous space of a oommec-
ted Lie ~roup? (Note that the answer is affirmative for the homo-
geneous spaces of c o m p 1 e x Lie groups and for the polynomi-
als with h o 1 o m o r p h i c coefficients ~9~).
Though the question of complete reducibility of separable pmly-
nemials in its full generality seems to be transcendental, there is
654
an encouraging classical model, i.e. the polynomials with holomorphic
coefficients on Stein (in particular algebraic) manifolds. Note that
the kmown sufficient conditions t9] for holomorphic polynomials are
essentally weaker than in general case.
The peculiarity of holomorphic function algebras is revealed in
a very simple situation. Consider the union of ~ copies of the an-
nualus ~ Z : ~-4 < I ~ ~ < ~ 1 identified at the point ~ = ~ . It can
be shown that a separable polynomial of prime degree ~ with coeffi-
cients holomorphic on these space, and with discriminant ~=~ is
reducible if ~ >, ~0 (~, ~) , primarity of Yv being essential
for Y~>/~ [I0]. If ~=~ , ~v can be arbitrary [2], and we
denote by ~0(~) the corresponding least possible constant. Now if
is even then ~o(~) = ~ , and so the holomorphity assumpti-
on is superfluous. However ~o (~) ~ C(~ ~ if ~ and ~ are
odd, with C(k)~ for k~5 . At the same time ~o(~) -< C ~
for all • . These results, as well as the fact that ~o(?) ~/P--.~
as p tends to infinity along the set of prime numbers, have been
proved in [I0~. Nevertheless the exact asymptotic of ~o(p) re--
mains ~own, it is ur...known even whethe r ~@ (p)--~ oo a..ss ~-~oo .
If ~ is a finite cell complex with H4(~,~) = 0 then each
completely reducible separable polynomial over C (~) is homotopic
in the class of all such polynomials to one with constant coeffini-
e n t s ( the reason is that ~'~$CG):0 for ~>~ ) . Let X:ivIA and consider a polynomial completely reducible over ~ .
Is it p ossib!e to realize the homotop.y within the class of pQly-
nomials over ~ ?
Such a possibility is equivalent, as a matter of fact, to the 13d
holomorphic contractibility of the universal covering space ~
for ~ . It is known [li] that ~ : C ~ ~ V ~-'k , ~-~ being a
bounded domain of holomorphy in C~ homeomorphic to a cell [12].
In ~ there are contractible but non-holomorphically contractible
domains [12~, though examples of bounded domains of such a sort seem
to be ,~n~own (that mi~t be an additional reason to study the above
question). Evidently ~5 = ~£ x ~ is holomorphically contractible.
Is the same true for ~ with ~ ?
There are some reasons to consider also transcendental equati-
ons ~W)=0 , where IRA-* ~ is a Lorch holomorphic mapping
(i.e. ~ is Fr~chet differentiable and its derivative is an opera-
655
tor of multiplication by an element of A ). In [13] the cases
when equations of this form reduce to albebraic ones have been treat-
ed (in this sence the standard implicit function theorem is nothing
but a reduction to a linear equation). A systematic investigation of
such trancendental equations is likely to be important. This might
require to invent various classes of Artin braids with an infinite
set of threads.
REFERENCES
I. r o p ~ R E.A., ~ E H B.A. Aaredpa~ecE~e ypaBHeHY~I C He-
npepm~G~ Eos~eHTaM~ E HeEoTopHe Bonpocw aare6pam~ecEo~ Teo-
pn~ Eoc. -MaTeM.cd., I969, 78, 4, 579-610. 2. r o p ~ H E.A., ~ ~ H B.A. 0 cenapadex~m~x no~oMax Ha~
Eom~yTaTZBm~M~ daaaxoB~M~ axre6paM~. -~oEa.AH CCCP, 1974, 218,
3, 505-508. 3. r o p n H E.A. ro~oMop~HHe ~y~EL~ ~a ax2edpa~ecEoM MHOrOOd--
pas~ ~ IIp~IBO~MOCT]~ ceHapade~H~x n~n~OMOB Ha~ HeEoTOp~M~ EOM-- ~.~yTaTI~ daHaxoB~ a~redps~. - B EH.: Tes~cH ~OF~.7-~ Bce-
CO~BHO~ TOH.I{OH~., MI~HCE, 1977, 55. 4. r o p E H E.A., K a p a x a H ~ H M.H. HecEoa~Eo saMeqa-
HN~ od ~re6pax HelIpepRBHMX ~yHI~ Ha ~OK~HO CMSHOM EOMIIaETe.
-B m~.: Tes~cH ~oF~. 7-~ Bceco~sHO~ TOII.KOH~., MHHCE, 1977, 56.
5. K a p a x a H a H ~.H. 0d a~edpax Henpep~BHMX ~y~EL~ Ha ~O--
ESJIBHO CMSHOM EOMIIaETe. -- ~HEI~.aHaJI. E ero np~., 1978, 12, 2, 93-94.
6. J~ ~ H B.A. 0 IIOX~HOMaX ~eTBepTo~ CTelIeHI~ Ha~ a~redpo~ Henpe-
p~mm~x @ym~n~. - ~ m : ~ . a ~ s . ~ . ~ ero r r p ~ . , 1974, 8, 4, 89-90. 7. 3 ~o s E H D.B. A~e6pa~ec~'~e ypa~Rem~:~ c Henpep~mm~m EOS~H--
LV~eHTaM~ Ha O~H0pO~H~X npocTpaHcTBaX.- BecTHnE M~Y, oep.MaT.Mex.,
1972, ~ I, 51-53.
8. 8 ~ s ~ H D.B., ~I ~ H B,~, HepasBe~B~eHH~e axredpa~ecE~e
pac~peH~ EOMMyTaTEBHRX daHaXOBMX a~redp. - f~aTeM.c6., 1973, 91,
3, 402,-420. 9. /l ~ H B.~I. AJmeOpo~m~e (~yHEs~a H roaoMop~m~e SaeMeHTH ZDMO--
mon~ecE~x r10ynn EOM~e~c~oro M~o~oodpas~. -~oEx.AH CCCP, 1971,
201, I, 28-31. I0. 8 I0 3 I~ H ~0.B. HenpHBo~w~e cenapade~H~e nom~HOM~ c rO~OMOIX~--
HBMH EOS~I~eHTaMI~ Ha HeEoTOpOM I¢~lacce EOMII~eEOHRX IIpOCTpaHOTB°
-MaTeM.Cd., 1977, 102, 4, 159--591.
656
II. K a ~ ~ M a H W.H. ro~oMop~Ha~ yH~epca:~aa~ H ~ H B ~ npo-
cTpaHCTBa nO~G~IOMOB des EpaTHRX EOpHe~. -- ~ . aHaJI. E ere
np~., 1975, 9, I, 71.
12. H i r c h o w i t z A. Apropos de principe d'0ka.- C.R.Aca~.
sci. Paris, 1971, 272, ATS2-A794.
IS. r o p ~ E.A., CaH~ e c Eapxoc @ep ~a~-
e c. 0 ~pa~c~e~eHm:~x ypaBHeRm~X B zo~aT~BH~X 6a~axoB~x
a~edpax. -~y~.aRa~. ~ ePo ~p~., I977, II, I, 63-64.
E.A.GORIN
(E,A.IDPMH)
CCCP, 117284, MOcEBa
~e~HcEEe ropH
MOCEOBCE~rocy~apCTBeH~
YHHBepCZTeT
Mexam~o--MaTeMaT~ecE~$BEy~TeT
COW~S~TARY BY THE AUTHOR
Bounded contractible but non-holomorphically contractible do-
main of holomorphy in C ~ have been constructed in K14]. All other
questions, including that of contractibility of the Teichmuller
space ~ , seem to rest open.
A aetailed exposition of a par~ of ~ 131 can be found in ~15].
REI~ERENCE S
14. 3 a ~ ~ e H 6 ep r M.r., Jl HH B.~I. 0 rOHOMOp~Ho He
CT~:r~BaeM~x o~a~m:eHH~X o6~aCTHX rO~OMOp(~HOCS .-- %oF~.AH CCCP,
1979, 249, ~ 2, 281-285. 15. F e r n ~ n d e z C. S a n c h e z , G o r i n E.A.
Variante del teorema de la funcio~n implicita en ~lgebras de
Banach conmutativas. - Revista Ciencias Matem~ticas (Univ.
de I~ Habana, Cuba), 1983, 3, N I, 77-89.
657
13.9. o l a
I. For any integer ~ ,=~>~+
consider the polynomial p(~)
~ (~) be the discriminant of p ,''',~ and the sets G~= { ~ :
HOLOMORPHIC MAPPINGS OF SO~ SPACES CONNECTED WITH
ALGEBRAIC ~VNCTIO~S
~4A~'~+ . • . + %~ , and let
. Then ~ is a polynomial in °
= uc N { z : z, = o } , 5G~, = { ~ : ~ = o , ~ ( z ) = t.}
are non-singular irreducible affine algebraic manifolds, oGp~ isomorphic %o G~ X ~ . The restriction ~ = ~ I ~ :
being
@
G~ " C* == C \ [0} is a locally trivial holomorphic fibering
with the fiber SG~ . These three manifolds play an important role
in the theories of algebraic functions and of algebraic equations
over function algebras. Each of the manifolds is ~ ( ~4, 4) for its
fundamental group ~4 , ~4 (G~) and ~4 CG~) being both iso-
morphic to the Artin braid group ~(~) with ~ threads and
~4 (SG~) i being isomorphic to the commutator subgroup of ~(~),
denoted B (~) ([I],[~). ~- and ~p -cohomologies of G~ are
k~own [I], [~,[~. However, our knowledge of analytic properties of
, ~ ~G~ essential for some problems of the theory of
algebraic functions is less than satisfactory (~]-[I~). We propose
several conjectures concerning holomorphic mappings of Go and ~G~
Some of them have arisen (and all have been discussed) on the Se-
minar of E.A.Gorin and the author on Banach Algebras and Analytic
Functions at the Moscow State University.
2. A group homomorphism H--~H~ is called a b e 1 i a n
(reap. i n t e g e r ) if its image is an abelian subgroup of H~
{reap. a subgroup isomo~hic to Z or {0} }. For comple~
spaces X and Y , C(X,~)__ Ho~ CX,Y) and H0~*CX,Y)
stand for the sets of, respectively, continuous, holomorphic and
n o n - c o n s t a n t holomorphic mappings from X to ~ . A mapping ~ C( o G~)is said to be s p 1 i t t a b 1 e if there
is ~ ~C ~, ~) such that is homotopie to ~°~° ~ ' ~, ~----~* being the standard mapping defined above; ~ is splittable
if and only if the induced homomorphism ~, :~(~) ~ ~4 <~)
G ° ~4 C ~) ~ ~(~) is integer. There exists a simple explicit "
description of splittable elements of H0~ , G~) [6].
658
CONJECTLhgJ~ I. Let ~>4 and ~@~ . Then (a) ever~
~EH0~CG ,~)is splittable; (b) H0~*(~, SE n ) = 2.
It is easy to see that (b) implies (a). Let ~ (~) be the union of four increasing arithmetic progres-
sions with the same difference ~ (~-~) and ~hose first members are According to [6], if
and F~(~) then all ~ o 0 in H~ CG~ , G~)_ are splittable. A complete description of all non-spllttable ~ in H0~(G; ,G~)
has been also given in [6]. If ~ >~ and ~<~ , there are only trivial homomorphisms from ~/(~) to ~(~) [11]. Thus for such
and ~ all elements of C ( ~ , G~) are splittable and all elements of g (S G~ ~ ~ G~) are contractible. The last assertion implies rather easily that H0~*(~G~ ~ ~G~)~ ~ . It is proved
in [10] that for ~ ~ ~ each # ~ Ho~G~, ~G~) is biho- lomorphic and has the form ~ (~,...~ ~)=~62~, 63~3,.. ,~) with 6~(~-0 = ~
C** ~ \ {0, ~} A useful technical device in the 3, Let = topic we are discussing is provided by explicit descriptions of all
E mo~ (X, C'*) for some algebraic manifolds functions ~ * associated with g~ , ~ and ~ ( [6], [8], [9], [10] ). This has led to the questions and results discussed in this section.
Let ~ be the class of all connected non-singular affine al- gebraic manifolds. For every X ~ the cardi~lity ~(X) of mo~ (X ~ C ) i s f i n i t (E.A.Gorin). Besides, if e "H~>'I¢I,G~ [~. (X) , ' lJ ("6(X) is the rank of the cohomology group H4(X,Z)) thenFlo~,'~(.X,l~\{~,...,}~}) "~ (~) for any d i S -
t i n c t points ~ ~ • . , , ~,14.,Ig;(~ . Using these two assertions, it is not difficult to prove that, given X~ and ~>,~ , the set m0~ ~ ~X, ~) is finite. In particular, for every ~,~>/~ the set Ho~(~G~, ~G~) is finite. Let Top,X) be the class of all y in ~ homeomorphic to X ; it is plausible that for an,~
~ ~ the function ~:To~ ( X ) - - ' ~ ' + is bounded. I even
do not know any example disproving the following stronger
CONJECTURE: there exists a function ~: ~+----~+
such that l ~ ( X ) ~ ~)('("(.X).) for all X {p..jI~ •
A function ~: 2+-" 2+ with ~<~) ~< ~4 <$<~)) for all
659
c u r v e s F ~ does exist. It has been proved in [14] that there exists a function V~: 2+ X 2+ --~ 2+ such that ~<X)~< ~< ~ (~4(X),$~(X)) for all manifolds X ~ of dimension two (here
$~ (X) = rank H ~ (X z) ~ = 4, ~) . For X ~ it is known that (~) if $(X)~<~ then ~(X)=0 ; (ii)if ~(X)=~ then ~(X) is o or 6; (iii) if ~(X)=~ then ~(X) is o, 6, 24 or
36 (all cases do occur), i
4. SG~ contains a curve~ ~=~O~.~ . 0o {~'. ~A. =.,. =~-o :0]
isomorphic to ~ = {(CC,~)E~ : ~ + ~-4 =4]. It can be proved that if ~ , ~g in Ho~ (SG~, SG~I,) agree on ~i then 24-------~g . Since H0~ ~ (SG~ , ~G~)= * provided ~>~ and ~<~, the following assertion admittedly implies conjecture ~ .
CONJECTURE 2. If ~>~>~ then Ho~*(r~,sG,) = ¢ .
The curve ~ can be obtained from a non-singular projective curve of genus (~-~)(~-~)/~ by removing a single point. It seems
plausible that H0~*(~(#), 5G~) = ~ for all ~>~ and all
cur_yes V(~ ~ of ~enus ~ < (~-~)(~-~)/~ . In any case the
following weaker conjecture is likely to be true (this is really the case if ~%~ or ~<~ , E.A.Gorin).
c o ~ c ~ U R ~ , ~. ~.et ~ > ~ , ~ ) 0 and ~ , . • . , ~ ' ~ C.
Then Hog (C',{~'~,...,Z;~}, S(G~,)) = ~.
5. Even the following weakened variant of CONJECTURE I would be useful for applications. Let X ~ , ~ X e= g ; ~>~ , ~>~.
° ~ ° CONJECTURE 4, (a) Let ~4 ~ Hog(G~,X¢), ~ HogCX ¢, G~). I f ~<~-~ then ~g" ~4 is .spl i t table,
(b) Let ~e Hog (sG~, X g) , ~ c Hog (X, 5G~) ; iX ~, ~-~ .th.en ~ o ~ is a constant ma~in~.
It follows from results of [6] ,[7], [10], [11] that the assertions 4(a) and 4(b) hold if either ~>~ , #$~< ~or ~>~ , ~=4 (of course, 4(a) is true for ~>~ and *~(~) ). ~ybe even the follo- wing sharpenings of 4(a) and 4(b) hold, though they look less pro- bable.
,
then the induced homomorphism ~* : ~4 (GI) --~ ~4 CX~) is abelian.
660
(b) , an,d Ho (SG , X #') t h e n
the induced homomorphism ~, : S~ 4 (~) ---*-~X g) is trivial.
If ~>~ and ~= ~ , ~, really has these properties. It can
be proved also that if ~ and ~ ~-~ then for any r a t i-
o n a ! ~ ~ Ho~ ~,~) the kernel of ~, is non-trivial. Con-
jecture 5 looks a little more realistic in, case when ~ is the com-
plement to an algebraic hypersurface in ~ and ~ is holomorphic
and rational.
6. We formulate here an assertion concerning algebraic functi-
ons. To prove this assertion it suffice to verify CONJECTURE 1 for
p o 1 y n o m i a 1 mappings from 8~ to Let ~(~)
be an algebraic function in ~ (~C ~) G~ " defined by the equation f'l, ,I,,t,-,l .t. + %~ ~ "'' + ~ = 0 and let ~ be the discriminant set of
this function, i.e. ~[~t~"~ {~: d'tl,(~)=O}"
CONJECTURE 6. ~or ~>~ there exists no entire al~ebraic func-
tion ~_-F(~) with the followin~ properties: (I) ~ is a compo-
sition of polEnomials~ and entire al~ebra%c functions in less then
~-~ variables~ (2) the d iscriminant set of E coincides with
~ ; (3) in some domain U ~ ~ the functions F and
have at least one ~oint irreducible branch.
Condition (2) means that ~ is forbidden to have "extra" bran-
ching points (compared with ~ ). It is known that CONJECTURE 6
becomes true if this condition is replaced by that of absence of
"extra branohs" (which is much stronger) [~5~
REFERENCES
I. A p H o x ~ ~ B.H. 0 HeEoTopHx TOnO~Or~qecFJ~X ~HBap~a2Tax a~-
redpa~ecK~x ~ym~m~. -Tpy~H MOCE.Ma~eM.O6--Ba, I970, 2I, 27--46.
2. r O p n H E.A., ~ ~ H B.H. A~TeOpsm~ecK~e ypaBHeH~ C He--
npep~BH~M~ EO~T~eHTaM~ ~ HeEoTop~e Bonpoc~ a~re6pa~ecEo~ Te- op~ ~oc. -MaTeM.cd., 1969, 78 (120), ~ 4, 579-610.
3. ~ y E C ~.B. KoroMo~or~ rpynnH ~oc no Mo~Jno 2. - ~ .
aHa~. ~ ero np~., 1970, 4, ~ 2, 62-73.
4. B a ~ H m T e ~ H ~.B. EOrOMO~Or~U~ rpynn Eoc. - ~-HE~.aaax.
ero np~., 1978, 12, ~ 2.
661
5. Jl ~ H B.fl. Axredpo~e ~ H ~oMop~e 8aeMeH~ roMo-
TonH~ecE~x rpynn EOM~eECHOrO M~OrOO6pasHa. -- ~OE~.AH CCCP, 1971,
201, ~ I, 28-81.
6. Jl ~ H B.~I. Axredpa~ecE~e ~yHELSm C yR~Be!0cax~S~M ~HcEp~MH--
HaH~ ~mOlX)odpas~eM. - ~yHEt~.aHa~. ~ ero npHx., 1972, 6, ~ I,
81-82.
7. ~ ~ ~ B.A. 0 cynepnosHr~Hx am~e6pa~xecE~x ~Ja~. - ~jm~.
aEax. E el~ np~., I972, 6, ~ 3, 77-78.
8. Eam~a~ m.H. rO~OMOIX~Ha~ yHHBepca~Ha~ Ea~p~Bam~a~ npocTps2cT- Ba nO~0MOB 6es EpaTH~X EopHe~. -- ~RE~.a~ax. H e~o np~., I975,
9, ~ I, 71. 9. E a ~ ~ M a ~ ]]l.M. roxo~p~a~ ym~Bepcax~Haa HaEpama~sx npo-
CTpaHoTBa IIO~HOMOB 6es EpaTH~X ~pHefi. -- Teop.(~, ~ym~. a~ax. ~ ~x npmao~., B~n.28, Xap~0B, 1977, 25-85.
I0. K a ~ ~ M a H M.H. ro~omop~m~e s~j~O~Ol~Sm~ ~oroo(~pas~
EOMS~eECH~X IIO~A~HOMOB O ~OER~NI~aHTOM I. - Ycn~x~ MaTeM.HayE,
1976, 31, ~ I, 251-252.
II. ~I ~ H B.H. 0 npe~cTaB~e~x rpynn~ Eoc nepecTaHoBEa~m. -
YclI~Xl MaTeM.HayE, 1972, 27, ~ 3, 192. 12. Jl ~ H B.fl. Hpe~cTaBxeH.m~ EOO IlepeoTaHoBEa~I~. - Yc~ex~ MaTeM.
HAYS, 1874, 29, • I, 173--174. 13. ~ m H B.H. Cy~epnos~m~ a~eOpa~ecE~x ~ym~. - ~nma~.~.
ero np~., 1876, I0, h I, 37-45. 14. B a H ~ ~ a H T.M. ro~o~p~H~e ~lmal~m 6es ~Byx sHa~em~l Ha
~HHHO~ IIOBepxHocT I~. -- BeCTHm~ MOCE.yHKB., cep.l, MaTeM. ,Mexa~.,
1980, ~ 4, 43-45. 15. ~I ~ H BjI. Eoc~ ApTm~a ~ cBasamme c m~m rpymm ~ npocTpa~c~-
Ba. -- B EH.: H~O~ HayE~ ~ TexHm~, cep."A~re6pa. Tono~or~a. reo-
Me~p~", MOC~Ba, 1979, ~.17, 159-227.
V.Ya.LIN CCCP, 117418, MOcEBa
yx. Epacm~oBa 32,
lie HTp. SEOH0t~ .-MaTeM .MHC TI~ TyT
AH CCCP
662
13.10. ON THE NUMBER 0P SINGULAR POINTS OF A PLANE AFFINE
ALGEBRAIC CURVE
Let p(~,~) be an irreducible polynomial~.~on C ~ . It has been
proved in ~I] that if the algebraic curve ~=~(~,~)E~:~(~,~)=0~
is simply connected then there exist a polynomial automorphism &
of the space C ~ and positive integers ~ , ~ with (k,~)=~
such that p(~(X,~))= ~k-~ . It follows from this theorem that
an irreducible simply connected algebraic curve in C ~ cannot have
more than one singular point. (Note that such a curve in ~$ may
have as many singularities as you like.)
In view of this result the following QUESTION arises:
does there exist a connection between the topology of an irreducible
~lane affine algebraic curve and the number of its irreducible sin-
gularities? Is it true, for example I that the number of irreducible
sin~arities of such a curve ~ does not exceed ~+~ , where
= H 4 (F, Z) ?
The above assertion on the singularities of the irreducible
simply connected curve may be reformulated as folYows: let ~ and
be polynomials in one variable E ~ C , such that for any dis-
tinct points ZI,Z ~ ~ ~ either ~(~4) ~ ~(~) or
~(Z 4) ~ ~(Z£) ; then the system of equations ~I(Z)=0 , Vt(Z)=0
has at most one solution. It would be very interesting to find a
proof of this statement not depending on the above theorem about the
normal form of a simply connected curve ~ . ~aybe such a proof
will shed some light onto the following question (which is a slight-
ly weaker form of the question about the irreducible singularities
of a plane affine algebraic curve). Let X be an open Riemann sur-
face of finite type (~9~) ( ~ is its genus and ~ is the num-
ber of punctures), and let ~ , ~ be regular functions on ~ (i.e.
rational functions on ~ with poles at the punctures only). Suppose
that the mapping ~:~---~ , ~(~) ~(~(~)~ ~(~))~ ~X~
is injective.
How man~ solutions (in ~ ) ma 2 have the system of equations
~ 0 ~a~=0 ~Here ~=~'~Z , where ~ is a holomorphic local
coordinate on ~ .)
663
REFERENCE
I. 8 a ~ X e H d e p r M.r., n i~ H B.H. HenpzBo~a~ O~HOCB2B--
Hall a~re6pazHecEa~ EpEBa~ B ~ SEBgBaJleHTHa ~as~o~opo~o~. -
~ o ~ a ~ AH CCCP, 198.3 ~ 271, ~5, 1048-1052.
¥.Ya.LIN
(B.H.2H) CCCP, 117418, MOCEBa,
yx.KpacEEoBa 32,
H~HTp.~OHOM.-MaTeM.MHCT~TyT
AHCCCP
M.G°ZAIDENBERG
(M.r. EPr) CCCP, 302015, 0pe~, KOMCOMOX~CEa~y~., I9 He;~aror~qecEE~ EHCT~TyT
SOLUTIONS
Under this title those "old" problems are collected which have
been completely solved (the "new" problem S.11 is an exception). All
are accompanied with commentary - except for S.9 where commentary by
the author is incorporated into the text. Problems S.1-S.10 follow
exactly the same order as in the first edition.
665
So1. old
ABSOLUTELY SUMMING OPERATORS PROM
TH~ DISC ALGEBRA
Let A denote the Disc Algebra i.e. the subspace of the Banach
space $(~) consisting of all functions which are boundary values
of uniformly continuous analytic functions in the open unit disc ~.
Let ~Io:I~I(~):I ~-----0 for every ~ , ~A } • Recall that a T bounded linear operator ~: X ~ ( X , 7 -
Banach spaces) is p - a b s o I u t e i y s u m m i n g
(0< ~< oo) if there is a constant ~-~-~(~) such that for every
finite sequence (~) ,
where the supremmn is extended over all 5" in the unit ball of the
dual of X . Finally by ~P we denote the Banach space of p-abso-
lutely summable complex sequences ( ~ e < oo) •
We would like to unders@and what differences and what simila-
rities there are between the properties of bounded linear operators
from %he Disc Algebra to Banach spaces and the operators from C(~)
-spaces. The results of Delbaen [I] and Kisliakov [2] characterizing
weakly compact operators and the results by PeEczy~ski-HitJagln E3~
that for ~ < p < :~ , e -absolutely summing operators from A into
a Banach space are p-integral (i.e. these operators extend to
p -absolutely summing operators from C (?)) are examples of simi-
lar properties while the existence of an absolutely summing surjec-
tlon from A onto $~ (cf E3~) indicates differences between A
and spaces of continuous functions.
The problems discussed below if they would have positive answers
will indicate luther similarities. Roughly speaking the positive ans-
wers would mean that properties of ~-absolutely summing operators
from A are the same as the properties of ~-absolutely summing ope-
rators from C( ~ )-spaces. The situation is clear for translation
invariant operators (Cfo [4]).
Let us consider the following statements:
(~) ~or every sequence ($~) in [J(~) such that
F_.., oo f o r every S, J.
666
there exists a sequence (~) in ~ I o such tha
~I!(~+~)~l~<oo for every~,~(~) ;
( ~ ) for every bounded linear operator i¢ : A ~ $¢
a finite non-negative Borel measure ~ on ~ such that
there exists
for every ~:, ;~e~ ;
( '~ ) for every sequences ~ ~ ) in ~4 (T) sa t is fy ing ( "t ) and.
5%L ~ < co > every sequence (~K) in A with epT I,,FK(~)I 2'
~¢ extends to a
into $~ is 2-abso-
( ~ ) every bounded linear operator ~:A •
bounded linear operator from C(~) into ~ ;
(E) every bounded linear operator from A
lu%ely mla..ing;
( ~ ) fo r every bounded l inear opera~ors I/ : ~ r A a n a u : ~ - ~ ~ the composition ~ : ~¢ ~ ~ belongs to the Hilber~-Schmidt class.
(a) Por every sequence ( ~ ) in L~(~) Such that
(2)
there exists a sequence ( ~ ) in ~ ~ such that
for every ,.~, ~ ~ C(.T) ;
(b) for every bounded linear operator ~: A ~4 ---~ there exists a
non-negative finite Borel measure ~ on ~ such %hat
11 ÷11 ' j i sr A; T
(c) for every sequence (~ ) e v e r y sequence (3~) in A with
667
i n ~(r~) ea t : l . s f y i~ ('2) and f o r
~ ""i
~" A--*~ ~ extends to a (d) every bounded linear operator
bounded linear operator from 0 (T) into ~ ;
(e) every bounded linear operator from A into ~i rely summing,
(f) every bounded linear operator from ~ into ly summing adJoint.
(A) Every bounded linear operator ~ : A r ~i is Hilber%ian x),
(B) For every sequence (~) in ~(T) s a t i s ~ ' ~ (2) and for every sequence (~k) in A with ~ ¢~ I#K(~)I <to@
K ~ ~ T
is 2-abslolu-
has absolute-
(.) the ~ a oe L'/~'o i s of oot~e 2 i . e . there i s a K ,K>O ,
such that for every posit ive integer ~, and every~,~ , , . . .~ ~m in
~=I S-eA
.here t~e s~ ~ e=tends for an seqoenoe. S=~%\, .ithS~----+-1 for ~=~,~,..., ~.
Using the standard technique of absolutely summing opera~ors one can prove
i ,iu ,
E) i.e. can be factored through a Hilber% space. - Ed.
668
PROPOSITION I. The followi~ implications hold
CA~--> ( B).
PROBLEH I. I..ss Ca) true?
PROBT, k~ 2. I ss (d) true?
PROBLEM 3. Is (A) true?
PROBLEM 4. I_~s (E) true?
>co)< > (÷)
>(I)<--> c~)<-->¢7~
REFERENCES
I. D e i b a e n P. Weakly compact operators on the disc algebra.
- Journ.cf Algebra, 1977, 45, N 2, 284-294.
2. E z c x ~ ~ o B C.B. 0d ycxoB~x ~a~op~a-HeTTzCa, nex~z~c~oro
m I~oTeH~m~a. - ~o~.AH CCCP, I975, 225, 6, I252-I255. 3. P e I c z y ~ s k i A. Banach spaces of analytic functions and
absolutely summing operators. CBMS, Regional Confer.Ser. in Nath.
N 30, AMS, Providence, Rhode Island 1977.
4. K w a p i e n S., P e E c z y ~ s k i A. Remarks on abso-
lutely summing translation invariant operators from the disc al-
gebra and its dual into a Hilbert space. - ~ich.Math.J. 1978, 25,
N 2, 173-181.
A. PELCZYNSKI Institute of Mathematics
Polish Academy of Sciences
~niadeckich 8,
00-950 Warsaw, Poland
COMMENTARY
J.BourgaLu has answered ALL QUESTIONS IN THE AFFIR~TIVE. A
summary of his main results on the subject with brief indications of
the proofs can be found in [5]. The proofs are to appear in " Acta
669
Mathamat ica".
Quite recently Bourgain obtained further improvements of his
results. So those who are interested in the question have to follow
his forthcoming publications. We review here some "Hard Analysis"
aspects of this new work.
First of all Bo~trgaln has proved that given a positive ~(~)
there exist W ~,~" , ,Ith ~ W .< ct~ , and a projection
P,IJ, tW) H (~W) satisfying the wsak t~e estimate
and which is bounded simultaneously in ~(~). This leads to a concep- tual simplification of the methods used in [5].
Further, Bourgain has proved that any operator mapping a ref- • 1 4 4 I I I 4 I I OO .
lexlve subspace of ~/~I to M admits an extension to an ope-
rator from ~/~' to --H °". This "result has an interesting applica-
~®_ there e+is+s F ~ H®(T + ) w i t h
I F(+,+)+ ++ +~(+)-~l+ (h, +iT, +:+,+, .... T
5. B o u r g a i n J.
- C.R. Acad. Sc. Paris,
REFERENCE
t 0perateurs sommants sur l'alg~bre du disque.
1981, 293, S~r I, 677-680.
670
S.2. GOLUBEV SERIES AND THE ANALYTICITY ON A CONTINUUM old
The collection of all open neighbourhoods of a compact set
~(~C) will be denoted by ~(K) . A function analytic on a
set belonging to ~(K) will be called analytic on K . It will
be called ~ - a n a 1 y t i c o n ~ (~>0) i£
~ ! ~ ~ $ for every t,teK.
DEFINITION. A compact set K ( ~ C ) i s r e g U I a r if
there exists a mapping ~K: ~+ ~ K ) enjoying the following
property: for every '5 > 0 and for every# S-analytic on
there exist a function ~ analytic in ~K ($) and a set W ,
W ~ u(K) such that
W ~ I~KC~,), #IW= odlW. The set
.-4 (1) S-- { j : j -- 4 , £ , . . . 1 U [ o ]
is not regular. Indeed, putting
0 {- (j-' < + (j+1) -I)
j-4,~,. ,.
fl
we see that U ~( is S-analytic on S for all ~ values of
and j but $) contains no set where a 1 1 ~ are ana-
lytic.
QUESTION. Is every plane ¢ontinuum(i,e. a q0mpact connected
s#t) regular?
671
This question related to the theory of analytic continuation
probably can be reformulated as a problem of the plane topology. Its
appearance in the chapter devoted to spaces of analytic functions
[the first edition of the collection is meant -Ed.] seems natural be-
cause of the following theorem, a by-product of a description of the
of the space ~ ( ~ ) of a l l fvalctions analytic on K . dual
THEOREM. Let ~ a reRular compact set and ~ a positive
Bore! measure on K such that C~(K\e) ~ K for ever~ e ,
e~K , with ~(e)= 0 • Then every function ~ anal2tic in A ~\ K i,s, representable b 2 th e followi~ formula
A
+ , o "*+ '
(~)~)0 beinR a sequence of
,,++,.,+ '/"" ,r,+,-,-.m II L+(~ ) = 0
L~(~)-functions and
This theorem was proved in [ I ] . The regularity of K leads to
a definition of the topology of ~(~) explicitly involving con-
vergence radii of germs of functions analytic on ~ .
Unfortunately, the regularity assumption was omitted in the sta-
tement of the Theorem as given in [1] (though this assumption was
essentially msed in the proof - see [I], the beginning of p.125).
The compact K was supposed to be nothing but a continuum. A psy-
chological ground (but not an excuse) of this omission is the prob-
lem the author was really interested in (and has solved in [I] ),
namely, the question put by V.V.Golubev ([2], p.111): is the formula
(2) valid for every function ~ analytic in ~\ ~ provided K
is a rectifiable simple arc and ~ is Lebesgue measure (the arc-
length) on K ? The regularity of a simple arc (and of every
1 o c a 1 1 y - c o n n e c t e d plane compact set) can be
proved very easily, see e.g., [~, p.146. The Theorem reappeared
in [4] and [~ and was generalized to a multidimension~l situation
in [~. It was used in [6] as an illustration of a principle in the theory of Hilbert scales.
672
We have not much to add to our QUESTION and to the Theorem. The
local-connectedness is not necessary for the regularity: the closure
of the graph of the function ~ - ~ ~ , t ~ ~0~ ~] is regular.
The definition of the regularity admits a natural multidimensional ge-
neralization. A non-regular continu~n in C ~ was constructed in K7].
The regularity is essential for the possibility to ~epresent functi-
ons by Golubev series (2): a function analytic in ~ \S (see (]))
and with a simple pole of residue one at every point j-~ (] = ~,~oo,)
is not representable by a series (2). Non-trivial examples of func-
tions analytic off an everywhere discontinuous plane compactum and
not representable by a Golubev series (2) were given in ~8].
REFERENCES
I. X a B ~ H B.H. 0~ a2a~or p~a ~opaHa. - B ~,: "Hcc~e~OB2~
no CoBpeMeHH~M npo6~eMaM Teop~ ~y~z~ EOMn~eKcHoro nepeMeHHoro ".
M., {~sMaT~s, I96I, I2I-I3I.
2. r o x y 6 e B B.B. 0~HosHa~e aHsJmT~ec~e ~yHzny~. ABTOMOp~--
~e ~yHKL~H. M., ~ESMaTI~g8, 1961.
S. T p y T H e B B.M. 06 O~HOM a~axore p~a ~opaHa ~ ~ MHOr~X Eo~eEc~x nepeMesH~x, rO~OMOp~ Ha C~BHO JG~He~Ho BH--
nyEm~K M~omec~Bax. - B C6.~rOJIOMOp~HNe ~yHEI~H~ ~g~OISLX EO~eECHRX
nepeMe~"o KpacHo~pcE, H~ CO AH CCCP, 1972, I39-152.
4. B a e r n s t e i n A. II. Representation of holomorphic func-
tions by boundary integrals.-Trans. Amer.Math. Soc., 1971, S 69,27-37.
8. B a e r n s t e i n A. If. A representation theorem for func-
tions holomorphic off the real axis. - ibid. ]972,165, 159-165.
6. M ~ T ~K P ~ H B.C., X e H ~ Z H r.M. ~[~He~e sa~a~ Eown-
~IeECHOI~O aH~sa. -Ycnexz Ma~eM.HayE, 1971, 26, 4, 93--152.
7. Z a m e R. Extendibility, boundedness and sequential conver-
gence in spaces of holomorphic functions. - Pacif.J.Math., 1975,
57, N 2, 619-628.
8. B ~ T y m z E H A.P. 06 o~Ho~ sa~a~e ~a~xya. - HSB.AH CCCP,
cep.MaTeM., 1964, 28, ~ 4, 745--756.
V. P. HAVIN
(B.n.XAB~H) CCCP, 198904, ~eE~Hrps~
HeTepro~, F~6xHoTe~Ha~ n~omaA&, 2 ~eHHHzpa~cE~A rocy~apcTBeEH~
yH~BepcxTeT, MaTe~aT~zo-~4exa~m~ e cE~
~aEy~TeT
673
* * *
CO~ENTARY BY THE AUTHOR
The answer to the above QUESTION is YES. It was given in [9] and
[10] . Thus the word "regular" in the statement of the Theorem can be
replaced by "connected" (as was asserted in 11] ).
REFERENCES
9, B a p ~ o ~ o M e e B A.JI. AHa~aTa~ecKoe npo~o~eH~e c KOHTgH~y~
H8 8I~ OK!08CTHOOTB. -- 381~CK~ HS~qH.CeM~H.~0~, 1981, I13, 27-
40.
10. R o g e r s J.T., Z a m e W.R. Extension of analytic functions
and the topology in spaces of analytic functions. - Indiana Univ.
Math.J., 1982, 31, N 6, 809-818.
674
8.3. old
Let A ~4cB ,
THE VANISHING INTERIOR OF THE SPECTRU~
and B be complex unital Banaoh algebras and let
then i% is well known that
6'ACx ) --, B,,B(x) an~ S~(~ ~ (~ ,
where ~(~; is the spectrum of ~ relative to ~ and ~gA(X)
is its boundary. Taking ~ %0 be the unital Banach algebra genera-
ted by X , in this context we say that x i s n o n-t r i v i -
a 1 i~ ~ ~(~ ~ * . ~ilov DJ has proved that if ~ ~. ~- is permanently singular in ~ (i.e. ~- X is not inver%ible in
any superalgebra ~ of ~ ) if, and only if, ~--x is an approximate
~ e ~ ~ s o r (AZ~ o~' A i.e. i~ ~ , ~ A , ~ ' ~ , l l = ' ~ , s u c h -~hat,8,,~(A-:~--,-O (I~, > ~ , ) .
Let ~ ~ ~ ~ 0 be a sequence in ~ with ~0~---1 and ~t.t~<~,~r,~ ~ ~ ~ 0. Then the power series algebra
(~ a~}) denotes a sequence of complex numbers) is a Banach algebra
under the norm N ~ ~ ~ ~ II ~--- E '@~ J ~ which i s generated by 0 ~ t~O
~ d ~ ( ~ i s a ~ isk o~ a r a ~ i ~ ~ . ~ i l o v [1] s h o . t ~ t ~or appropr ia te ~hoice of the sequence t~w } .$0 S $o such tha t 0</~o<$
~ i~ ~:~o~ ~ ~ ~ } , ~-~ is an A~Z in A • ~ ~ every ~u- peralgebra B this annulus is contained in YB (~; and we say that
CA(x) has a n o n- van i s h i n g i n t e r i o r .
If A is a uniform algebra then it is easy to show [5] that for
each non-trivial element X we can construct a superalgebra B such
that %~ ~(~)~ . If ~ is a subnormal operator on a Hilber%
space (i. e. T has a normal extension in a larger Hilbert space )
then %he algebra which it generates is a uniform algebra [ 2] hence
the same is true cf ~.
~ilov's theorem has been extended by Arens [3~ to commutative
Banach algebras which are not necessarily singly generated and
Bollebas ~4~ has shown that it is not, in general, possible to cons-
truct a superalgebr8 ~ of a Banach algebra A in which all the
675
elements which are not AZD's in ~ become simultaneously invertible.
QUESTIONS. Let A be ~enerated by the non-trivial element
euch that A- ~ is an AD~ in A if, and o~ it, ~8~ (~ . can
one construct a supera!gebra B such that ~B(X)~A(X) i.e.
superalgebra B in ~ch ~A(X) v~ni~es s!multaneousl~? !f~
is a non-trivial element of a C~-al~ebra ~ dges there exist a
superalgebra B of' A x ~uch that, ~B(X)-~-~Az (X) , where A~ is the , unital BanacAal~ebra zeneraSe 9 b~ ~ in A ?
REPERENCES
I. ~ ~ ~ O B F.E. 0 Bop~mpoBaRHHX Eox~nax c o2o~ oOpasymme~. -
~TeM.oO., 1947, 21 (63), 25-47.
2. B r a m J. Subnormal operators. - Duke Math.J.,1955, 22, 75-94.
3. A r e n s R. Inverse producing extensions of normed algebras.
- Trans.Amer.Math.Sec.~1958, 88, 536-548.
4. B o 1 1 o b a s B. Adjoining inverses to commutative Banach
algebras. - Trans.Amer.Math.Soc.~1973, 181, 165-174.
5. M u r p h y G.J., W e s t T.T. Removing the interior of the
spectrum. - Comment.Math.Univ.Carolin., 1980, 21, N 3, 421-431.
G.J.MURPHY 39 Trinity college
T.T.WEST Dublin 2
Ireland
COMMENTARY
The first problem has been completely solved by C.J.Read [5]
Moreover he has proved that for any commutative Banach algebra
and for an~ ~A there exists a suoeral~ebra 6 such that for ~ ~ C~
~-~ i~ ~ot invertible in 5 if ~ o~ if ~-~ is ~ ADZ in A This result solves the problem posed earlier by B.Bollobes in
The second question has a negative answer Indeed, let T be a
~ by Tek-~kek,where [ek~k~ weighted shift operator defined on ~Z and kk=4 + 4/~ if Z~< ~I is the standard orthogonal basis of
< k < ~ = 4 + It is easy to check that the
676
spectral radius of ~ equals I and~l{ ~ " ll'II T ~ =0 It fol- lows that in the algebra A T generated by ~ the spectrum,of. T. ..... ~I
• . g . ~f ~n ,, coincides w~th I~" I~I'~ ~ and T is an ADZ s~nce for~.~-~1 T /~T II
we have T ~ c A,~T.,~='t bu~ / ,~HTn , T ~ = 0 . ' . . . . .
REFERENCES
5. R e a d C.J. Inverse producing extension of a Banach algebra eli- minates the residual spectrum on one element. - Traus.Amer.Math. Soc. (to appear).
6. B o 1 1 o b ~ s B. Adjoining inverses to commutative Banach al-
gebras, Algebras in Analysis, Acad.Press 1975, edited by J.H.Wil- liamson, 256-257.
677
S.4°
old
A function ~ continuous on
if it satisfies the integral equation
@J
o
ON THE UNIQUENESS THEOREM FOR MEAN PERIODIC FUNCTIONS
is called~IOO -mean periodic
(1)
being a function of bounded variation with 0 and 00 as its
growth points. In the particular case of ~(~) ~ ~ (I) becomes
o
S(~+I~)%~ = O,
i. e°
~+~
I ~(~) ~ ~ 0,
which implies ~ (~ + ~)~ ~(~) , the usual periodicity.
An 0~ -periodic function vanishing on the "principal" period
~ E 0 ~ is identically zero. It is not hard to prove, using
Titchmarsh convolution theorem ~2S, that any ~J -mean periodic
function is also completely determined by its restriction onto ~ .
Put ~I ~e~ ~ _ E~]00 (~@~ is the largest integer~ ).
Suppose the set M, Mc ~ , satisfies {M}=A . Then an ar-
bitrary ~0-periodic function vanishing on ~ is identically
zero.
Is the same true for ~J-mea n periodic functi6ns?
1. D e i sa r t e
Math.Pures Appl.,
REFERENCES
J ° J. Les fonctions "moyenne-perlodiques'. - J.
1935, Set. 14, N 9, 409-453.
06 O~HOM z~acce i~Terpa~x ypaB~eH~. -
678
MaTeM.C6. 1956, 88, 188-202.
Yu.I.LYUBICH CCCP, 810077, XapBEOB n~.~sep~HCEOrO 4
XapREOBOEI~ rocy~apCTBeHHN~
yHEBepc~TeT
COMMENTARY
The answer is NO. P.P.Kargaev ~3 has constructed a non-zero
mean (~+8) -periodic function (for every ~ > O ) vanishing on
REFERENCE
3. E a p r a e B II.II. 0 Hyxsx ~y~, nepzo~ecEHx B cpe~HeM. -- BecTH~E ~UY (to appear )
679
S. 5. ~-BOUNDEDNESS OF THE OAUCHY INTEGRAL ON LIPSCHiTZ GRAPHS old
Let ~ be a real 0@~ -function defined on ~ , ~ the path
in the complex plane defined by the equation~(~)~+i@(~) (~G~),
and
(,X~9)(~) &¢~ v.p. I V(lh. (.'I+~/,pQ1b) ~,'I;
(the Cauchy integral of ~ taken along the graph of ~ ). I have
proved D]
II - I
where the finite positive function C is defined on an interval
E 0, ~), ~ being an absolute positive constant,+~t~0~(~ )~_~_ ~_+e~
(prodded ~p l ~ ~ ~ )
THE PROBT~M is to know whether
tion defined .... o n
i.e. whether the
Lipschitz graphs
in~ ).
C can be replaced by a func-
t h e w h o I e half-llne [%÷oo) ,
~-boundedness of ~T can be proved for a i I
(not only for those with the slope not exceed-
REFERENCE
I. C a i d e r 6 n A.P. Cauchy integrals on Lipschitz curves
and related operators. - Proc.Nat.Acad.Sci. USA, 1977, 74, N 4,
1324-I 327.
A. P. CALDER6N The University of Chicago
Department of Mathematics
5734 University avenue
Chicago, Tllinois 60637
USA
680
* * *
C0~TARY
The PROBL~ (coinciding with problem I of 6. I ) has been solved
in [3~ : the Cauch~ integral defines a bounded linear operator i n
o n .... e v e r ~ Lipschitz .~raph (for a~ 7 value of its slope). The
proof is based on an estimate of the operator ~ (see a) in ~roblem
III of 6.1) with AI~ ~@"(~) . It is proved in [3] that ~Aall~
c(~+ ~) ~ ~i II~ . using results of [2] Guy David found a lucid
geometrical description of the class ~ , ~ <p< ~ (we use the notati-
on from 6.2). Associate with every simple curve ~ a maximal functi-
on M F '~"~'(O,'t'~] , :
,
where I" I stands for the one dimensional Hausdorff measure. G.David
proved ( [4], [ 6]) that
A f t e r t h i s r e s u l t i t seems vez7 probable t ~ t the w e i ~ t OJp d lsous- sed i n 6.~ , i i i s connected w i t h ~ ? "
A lt is proved in [5] that the singular operator with the kernel
( z ) - A ( ~ ) 4 ~' e ® " FI ........ ~u I I~-~ is continuous in ~(~) whenever F ~ (~)
and A: %-~m satisfies the Lipschltz condition on ~ . The proof is based on the quoted estimate of the operator A~. The work [5 ] contains also a proof of the continuit~ of a singular Calder6n-Zyg-
round operator with odd kernel on ~ (~) , U being the graph of a Lipschitz function ~:~ > ~.
Articles [2] and [6~ show that the more the spaces H+(U) and
~t (0 are close to be orthogonal the more r is close to a straight line (and vice versa).
And now one more interesting
QUESTION. Which closed Jordan rectifiable curves ~ hav e the ~Oi! ,-
lowin~ DroDerty: all Cauchy integrals of measures on ~ be!on ~ to the
Nevanlinna class N~ i_nn ~ ~ (i. e. are quotients of bounded func-
tions analytic in I~ U )?
A.B.Aleksandrov has pointed out that no non-Smirnov curve enjoys this property. Moreover if ~ is not a Smirnov curve then there exist
i£ ~ (r) and a discrete measure ~ on r whose Cauchy integrals
681
do not belong to ~r ( ~ is found by a simple closed-graph argument,
the existence of~ uses some results from ~7]).
REPERENCES
2. C o i f m a n R.R., M e y e r Y. Une g4n~ralisation du th4o-
r~me de Calderon sur l' int4grale de Cauchy. Fourier Analysis, Proc.
Sem. E1 Escorial, Spain, June 1979, (Asociaci6n Matem~tica Espa~-
ola, Madrid, 1980).
3. C o i f m a n R.R., M c I n t o s h A., M e y e r Y. L'in-
t~grale de Cauchy d~finit un op~rateur born@ sur ~ pour les cour-
bes Lipschitziennes. - Ann.Math., 1982, 116, N 2, 361-388.
4. D a v i d G. L'integrale be Cauchy sur les courbes rectifiables.
Pr4publlcation Orsay, 1982, 05, N 527.
5. C o i f m a n R,R., D a v i d G., M e y e r Y. La solution • l
des conjectures de Calderon. Prepublication Orsay~ 1982, 04, N 526.
6. D a v i d G. Courbes corde-arc et espaces de Hardy g~n4ralis~s.-
Ann.Inst.2ourier, 1982, 32, N 3, 227-239.
7. A ~ e K o a H ~ p O B A.B. ~ asaxora TeOlOe~ M.l~cca o con!os-
xess~z ~s~sx ~ ~poczpascrs B.I~.CM~pSOBa E P , 0<p<~.
- B 06. "TeoloeS o~el0aTOl0OB e TeOpSS ~H~", 1983, ~ I, HS~-BO
~rY, 9-20.
682
s 6 SETS OP ~IQUE~ESS F0~ Q C old
By Q $ is meant the space of functions on ~ that belong to-
H gether with their complex conjugates to ~ C . Here, is the
space of boundary functions on ~ for bounded holomorphic functions
in ~ , and C denotes C(T) . It is well known [I 3 that ~ C
is a closed subalgebra of ~ (of Lebesgue measure on ~ ). Thus,
Q ~ is a 0 ~ -suhalgebra of ~ . The functions in ~ C are pre-
cisely those that are in ~ and have vanishing mean oscillation
~2~; see ~3~ for further properties.
A measurable subset ~ of T is called a s e t o f
u n i q u e n e s s for Q$ if only the zero function in ~6
vanishes identically on E . The PROBLEM I propose is that of
characterizing the sets of uniqueness for ~ C .
There are two extreme possibilities, neither of which can be
eliminated on elementary grounds:
I. The only sets of uniqueness are the sets of full measure;
2. A set meeting each arc of ~ in a set of positive measure
is a set of uniqueness.
If possibility I were the case then, in regard to sets of uni-
queness, ~C would resemble i~ , while if possibility 2 were the
case it would resemble C . One can, of coUrse, inquire about sets
of uniqueness for ~ and for ~oo+ $ . For ~ ~ the answer is classi-
cal: any set of positive measure is a set of uniqueness. In view of
this, it is quite surprising that, for ~ C , the first of the
two extreme possibilities listed above is the case. In fact, S.Axler
E4~ has shown that any nonnegative function in [,0° - in particular,
any characteristic function - is the modulus of a function in ~oo+ C .
Concerning ~ $ , I have been able to rule out only the second
of the two extreme possibilities: I can show that there are nonzero
functions in ~ $ that are supported by closed nowhere dense sub-
sets of T . The construction is too involved to be described here.
It suggests to me that the actual state of affairs lies somewhere
between the two extreme possibilities. However, I have not yet been
able to formulate a plausible conjecture.
REFERENCES
1. S a r a s o n D. Algebras of functions on the unit circle.
- Bull.Amer.Math. Soc. ~ 1973, 79, 286-299.
683
2. S a r a s o n D. l~unctions of van_ishingmean oscillation. -
Trans.Amer.MathoSec.~1975, 207, 391-405.
3. S a r a s o n D. Toeplitz operators with piecewlse quasi-
continuous symbols. - Indiana Univ.Math.J.,1977, 26, 817-838,
4. A x 1 e r S. ~actorization of ~ functions. - Ann. of Math.,
1977, 106, 567-572.
DONALD SARASON University of California,
Dept.Math., Berkeley,
California, 94720, USA
CO~NTARY BY THE AUTHOR
T.H.Wolff~has shown that the only sets of uniqueness for~C
are the sets of full measure. He did this by establishing the remark-
able result that every function in C can be multiplied into ~C by
an outer function in~C . The result says, roughly speaking, that
the discontinuities of an arbitrary ~ function form a very small
set. Wolff makes the preceding interpretation precise and gives other
interesting applications of his result in his paper.
REFERENCE
5. W o 1 f f T H Two algebras of bounded functions. - Duke Math.J.,
1982, 49, N 2, 321-328.
EDITORS' NOTE: S.V.Kisliakov has shown that for every set EcT
of positive length there exists a non-zero function ~ in VMO suppor-
ted on E and such that the Taylor series Z ~(~)~ con-
verges uniformly ~n the closed disc. Let g<fD) = ~ { I~I ~] co
Then go# ~VM00L =GC . However it is not clear if it is possib-
le to find a function ~ in ~C satisfying the mentioned condi-
tions (see C.B.FacaszoB. E~e pa3 o CBO6O~EO~ ~Tepnoamm~ ~ymumma
perya~pm~m BHe npe~n~csm~oro ~omecTBa. - 8an.H~.ce~.~0~,I982,
107, VI-88).
684
S . 7. ANOTHER PROBLEM BY R. KAUE~AN old
Let ~ be meromorphic in the disk ~ , and I~'I ~ I ~I
can be concluded about the growth of ~ ?
. V~at
R. KAU~T University of Illinois
at Urbana-Champaign
Department of Mathematics
Urbana, Illinois 61801
USA
COMMENTARY
The PROBLEM has been solved by A.A.Goldberg ~] with a later im-
provement of the proposer. The answer is "NOTHING", Namely, A.A. Gold-
berg had shown ~] that for every function(~($) tending to $~
as $--~ ~ there exists a function ~ meromorphic in ~ satisfying
l~zl ~ I ~ I in ~ and such that
~4~ T(~,}).~ I , (I) ~--~ ~P(~3
where T(~, #) denotes the Nev~u!inna characteristic of ~ . R~Kauf-
man has strengthened this. He has constructed an ~ [2] with
instead of (I). It is shown in [I S that for an ANALYTIC ~ satisfying
l~zl ~ I~I in ~ the following (precise) estimate holds:
(2)
5 ! This estimate is implied by a weaker assumption T (~,-~-)~ ~(~)~
In [2 S R.Kaufman has proposed A NEW PROBLEM for ftmotioms meromor-
phic in ~ . Suppose ~C~}~4 (~>0),~-~(~) ~ ~0 ( ~ ) .
685
What can ~e said about ~(~,~) and T(~,~) if
REFERENCES
I. F o x ~ ~ 6 e p r A.A. 0 pocTe MepoMop~HMX B Epy2e SyHEMH~ C or- paH~eHH2M~ Ha aorapH~HMec~y~ npozsBo~y~. - YKp.MaT.m., I980,
32, ~ 4, 456-462.
2. K a y $ M a H P. HeKoTop~e sa~e~a~ o6 ~HTepno~ aHa~THYec-
EHX ~yHKL~M~ H ~oPapM~MH~eCKHX I~p0HSBO~D(. - YEp.MaTeM.~., 1982,
34, 7~ 5, 616-617.
686
S.8. RATIONAL FUNCTIONS WITH GIVEN RA~IFiCATIONS old
, ~ be positive integers, and 9={q~ : 4.< i.<~} Let q
~:{~Ki " ~<k%9] ; ~$i%~} be two numerical systems satis-
fying
(1)
IXk~ k=4 o
(2)
We say that the problem ~[~,~,X] is solvable, if there exists
a rational function ~ of degree ~ and complex numbers ~K~ '
l.<k.<~ i , ~.<i,<% , so that 9~(z4.)=~(Z~])=...=g~Czg. i) ,
~'~ (" ~ , and the derivative ~! has a zero of order ~k3: at the point H~ ° Conditions (I) and (2) are necessary for the solvabi-
lity of the problem [~,9,~] ((I) is the well-known formula of
Ri emann-Hurwit z ).
PROBLEM. Find efficient criteria of non-solvabilit,7 of th~ prob-
.
It is known that the problem ~[~V~] is not always solvable.
For example, the problem with parameters ~=4 , $=~ , 94-9~=£ ,
95=4, A~=A~4=X~=A£~=4 , X4~=Z has no solutions ([I], p,468).
On the other hand, if all 9i=~ ' ~ ~< i 4~ , then the problem
~, {~}, X] is al~ys solvable ( [ I ] , p.469, th.4.1). A se- ries of sufficient conditions for solvability of the problem ~[&~V~k]
has been obtained by A.Hurwitz [2], [3]. The solution of the posed
problem should follow from one general result of H.Weyl [4], but that
result is formulated in a very inefficient form so that - according
to the author - it remains unclear how can one derive concrete coro-
llaries from it. B.L.Van der Waerden wrote on that result: "Leider
kann man nit der Schl~ssformel noch nicht viel anfangen".
687
REFERENCES
I. r o x ~ ~ 6 e p ~ A.A., 0 c Tp O ~ C ~ ~ ~ H.B. Pacnpe~exe-
~e s~a~e~epo~op$~x~. ~., Hay~a, I9V0. 2. H u ~ w i t z A. Ueber Riemann'sche Flachen mit gegebenen Verz-
weigungspunkten. - ~ath.Ar~u., 1891, 39, 1-61.
3. H u r w i t z A. U~ber die Anzahl der Riem~nm'schen Pl~chenmit
gegebenen Verzweigungspunkten. - Nath.Ann., 1902, 55, 53-66.
4. W e y 1 H. Ueber das Hurwitzsche Problem der Best~,~ung der
Anzahl Riemannscher ~lachen yon gegebener Verzweigungsart. -
Co~ent.math.helv., 1931, 3, 103-113.
A.A. GOL ' DBERG
(A.A.r0~E~Pr)
CCCP, 290602, ~BOB
~BoBc~rocy~apc~Be~ Y~zBepczTeT
CO~ENTARY BY THE AUTHOR
The problem has been solved by S.D Bronza and V G.Tairova ([5] -
17]). They proposed an effective algorithm which permits to decide
whether the problem ~ [~ , ~] is solvable and in case it is,
the algorithm permits to describe all solutions.
RE~ERENCES
5. B p o a s a C.~., T a K p o B a B.r. ~po~z~ p~asoBsx noBepx-
SooTed. - Teop~ Sym~z2, Sys~. aHaa. z zx ~pz~., Xap~om, 1980,
~ n , 33, 12-17. 6. Bp o H s a C.~., T a zp o B a B.r. EOeCTpyxpoBasxe pwa~oB~x
noBepxzocTe~ xmcca ~ . - ibid.,1983, BUll. 40 (to appear).
7. B p o ~ s a C.~., T a zp o B a B.r. No~oTpy~poBaa~e pmHo~zx
noBepxRocTe2 ~acca J.' . II. - ibid.,1984, B~II. 41 (to appear).
888
S. 9. TWO PROBL~S ON ASYMPTOTIC BEHAVIOUR OF ENTIRE FUNCTIONS old
1. Let SH(~) be the class of subharmonic functions in C of order ~ and of normal ~ type. Let V%, t ~ (0, oo) be the one-para-
meter group of rotations of ~ defined by
V,z : ~ i ; ~ , ,~ ~ R', P~: %V,.
Given ~ SH ~ ~) put
% {~) = ~ ( Pt ~) ~-~
Let be the space of Schwartz distributions.
It is known [I ] that the family { ~$} is compact in as ~-~ i.e. for each sequence ~ , co there is a subsequence
~{-~ c,o and a function i~ subharmonic in ~ such that ~%~-~ in . The set of all limits I~ is called the cluster set an~ is denoted by F~ [ ~&,V%] or F~ [t&] . it describes the asymp-
totic behaviour of ~ along the spirals &~={~-~P{e ~- t~(O,oo)} and, in particular, (whenoC=0 ) along the rays starting from the origin.
Let U ( ~, (3) be the class of subharmonic functions i) ~ satis-
fying v ( o ) = o ~ , v ' ( ; , ) ~ O t ~ l ~ ~;e ~) . ~he set F ~ [ ' ~ , ] is closed in ~i , invariant with respect to the transformations (.)~ ; further, F~ [t~]cU[~,O] and F%[t~] is connected in ~i
Let ~ ~ U [ ~,O ] . The simplest set with the mentioned
properties, which contains 1# , is
Let A(~) be the class of entire functions ~ of order no=al type. ~et F~[ ~ ] ~ F ~ [ ~ , I i ] . ~, --d of
PROBI,~ 1. Does there exist an entire function ~ ~ A (~) such
that F~ [~] - A (v) We ~enote by F~.[~], F~_ [v] the sets of all limits in ~)' of the f~milies { V~} as % -- 0 and ~--o~ respectively. ~he following theorem solves Problem 1.
THEOREM I [2 ]. A neces sar,y and sufficient comdition for the exi-
stence of a function ~ for Problem I is
689
The paper [2] contains examples which show that condition (2) may fail for some ~.
and the lower indicator respectively~ One of the possible (equivalent) definitions of the indicators is
(e iq ) L
The equality
s h o w s that ~ is a function of completely regular growth (cf.[3], p-139) on the ray { O~Z--- ~ ( ~ E A ~ ) . It is known [3], [I] that ~ e A~ implies the equalities
(5)
It is also known [4] that (5) "- (4).
PROBLEM 2. Prove that
(6)
(6) ~ (4), (7)
ANSWER. A necessary and sufficient condition for (6) to hold, ex- pressed in terms of F~ [~ ] (cf,[5]~ shows that (7) is not true. But if~ersuppose_~that (6) holds for ~ E c [ 0~] and the set
eiE ~'e~ ~e i~. ~ is thick at the point e~o then is true for ~= ~o"
REPERENCES
I . A s a p x ~ B.C. T e o p ~ pocTa c y ( J r a l : ~ o ~ e c ~ a z ~ m w ~ , 11, ~oa~-
690
next ~emm~, Xap~xo~, XFY, I982. 2. A s a p z H B.C., r z ~ e p B.B. 0 cTpoemmnpe~e~suxuaoxecTn
~e~x x cydrapmommecFax ~y~-~ .- Teop.~ymmj~, ~ymmaoga~. am~a.
Z ~X HpE~I.~BH~. 38, XapY~oB, 1982, 3-12. 3. JI e B z s B.H. Pacnpe~eaesxe ~opRell x~ea~x ~yRmm~. M., 1956
4. A s a p z ~ B.C. 0~ o~oM xapa~Tep~cT~qec~ou CBOi*C~e ~ySmUd~ mnoaae pex,/z~pHoro pocTa.-Teop. ~y~, #ym~J~ossmmH. a~aa. ~ ~x
-pza., B~n. 2, X~pr~to~ 1966, 55-66. 5. F ~ H e p B.B., Ho~omeB ~ . P . , C o ~ , s M.~., 0 cao~e,~
~zr~mx mum~aTopoB ~ea~x ~ymma~.-Teop.#ys~, ~y~o~aa~.asaa.
z ~x npza., :s,~-. 43, Xap~o~, 1984 (B neqaTz).
V. S. AZARIN
(B.C.~) OCCP, 310050, Xapt,~oB, Xap~.- ~asc~nll ~aCTZ~yT H~ZeHepO~ zeaesHo~opozsoro TpaHCIIOpTa
691
S. I0. THE INNER PUNCTIONPROBLEM IN BALLS old
The open (euclidean) unit ball in ~ (with ~ at least 2 )
is denoted by B . A n o n - c o n s t a n t bounded holomorphlc
function ~ with domain B is called i n n e r if its radial
where "a.e." refers to the rotation-lnvariant probability measure
on % o
CONJECTURE I. There are n 9 inner functions in B .
Here is some evidence in support of the conjecture:
D
(i) __.~ # is inner i n ~ , and if V is an open subset of
that interesects ~ , then S(E U V) is dense in the u~!t disc
@
PROOF. If not, then ~ contains one-dimensional discs
with ~D c ~ , such that S I 3 is a one-variable inner function
whose range in not dense in ~ , an impossibility. •
In other words, at every boundary point of ~ , the cluster
set of ~ is the whole closed unit disc. No inner function behaves
well at any boundary point.
(ii) l_~f I is i~er in ~ and if E is the set of all ~ ,
WE ~ , at whic h ~ II(~6)I~ ~, then E has no interior ~rela-
tlve to ~ ).
PROOF. If not, an application of Baire's theorem leads to a con-
trad/ction with (i). •
CONJECTURE I could be proved by proving it under some additional
hypotheses, for if there where an inner function~ in ~ , then there
would exist
(a) a zero-free inner function, namely ~C~ f~l-~)~+1 ~ ,
(b) an i er f ctlon with ) 0,
via Prostman's theorem; for almost all one-dimensional discs
through the origin, ~I~ would be a Blaschke product ;
(c) an inner function ~ that satisfies (b) and is not a pro-
duct of two inner functions i.e., ~ is irreducible, in the termino-
lo~ of [I].
692
(d) a non-constant bounded pluriharmonlc function ~ with
~*= ~ or 0 a.e. on ~ , namely ~-~(~o#) , where ~ is a con-
formal map of ~ onto the strip 0~o~ (i.e., there would be a
set E , Ec~, C( E)~---~ , whose characteristic function has a plurihar-
monlc Poisson integral) ;
(e) a function ~ ~÷8 _ ~ with ~ ~ > 0 in B but ~e ~*= 0
a . e . o n
This ~;~ would be the Poisson integral of a singular measure. Hence CONJECTURE I is equivalent to
CONJECTURE 11 . If ~ i sa positive measure on ~ whose Poisso n
integral is pluriharmonlc, then ~ cannot be singular with respect
t_~o ~ .
Porelli [3~, [4~ has partial results that support the following
conjecture (which obviously implies I z ):
CONJECTURE 2. If ~ ~s areal measure on ~ , with plurihar..m,,,, ~-
nlc Poisson inteKral, then ~<<6 ~ .
CONJECTURE 2 leads to some related HI-problems:
CONJECTURE 3. If # is holomorphic in B and ~C~ ~ 0 , then
CONJECTURE 3 [. There is ~ constant C , C< ~ (dependin~ on-
ly on the dimen:iQ such that all
~, ~eAcB) (~he ~II al~ebra).
co~JEcTm~ Y; . I_Zz ~ is....h.o.!omo.rphic in B , ~ = ~ + ~v , aua
Clear ly, 3 ; implies 3, and 3~#is a reformulat ion of 3 that might be easier to attack. Let N(~) be the Nevanlinua class in~
( ~ l ~ % l ~ Y ~ is bounded, as ~--*-~ ), and l e t ~ . ( ~ ) consist
o~ a l l ~ , ~ N ~ , ~orwhich ~ ~ ÷ 1 ~ 1 | i s u ~ o = = ~ y i n -
tegrable.
This would imply I z, hence I.
CONJECTURE I leads to the problem of finding the extreme points
of the unit ball of ~I(~) . (When ~ , these are exactly the
693
outer func t ions of norm 1.) Let ~ = { ~Ht(~) : ]1~11t~ ~ } .
CONJECTURE 5. Ever~ ~ , ~ 6 H' ( B ) , ~th II II, = I is
extreme point of ~ .
It is very easy to see that 5 implies 1. If ~A(~) it is
known (and easy to prove) that ~(~)---~ ~(B) . It is tempting to try
to extend this to ~(E) : Is it true for ever 2 I , i~oo(~) ,
that the essential range of ~* on ~ is equal to the closure of
An affirmative answer would of course be a much stronger result
than CONJECTURE I. To prove it, one would presumably need a more quan-
titative version of ~(~)~(~) . Per example: Does there exist O ,
0 > 0 (dependi~ onl~ on the dimension ~ ) such that
for every ~ , ~ A(E) , with ~(0)~-0 , I~I < ~ ? Finally, call
a holomorphio mapping c~ E-~ i n n e r if St~ cP(%w)~
for almost all ~0 , 00e~ .
CONJECTURE 6. If ~D is inner, then c~ is one-to.one and onto.
This ~mplies I, as well as the conjecture that every isometry
of ~P(B) into HP(~) is actually onto, when ~=4=~ . See [5]. If
"~nner" is replaced by "proper", then CONJECTURE 6 is true, as was
proved by Alexander [2].
REFERENCES
I. A h e r n P.R., R u d i n W. ~actorizations of bounded ho-
lomorphic functions.- Duke Math.J.,1972, 39, 767-777.
2. A I e x a n d e r H. Proper holomorphic mappings in ~. -
Indiana Univ.Math.J., 1977, 26, 137-146.
3. F o r e 1 1 i P. Measures whose Poisson integrals are pluri-
harmonic. - Ill.J.Math.. 1974, 18, 373-388.
4. P o r e 1 1 i P. Measures whose Poisson integrals are pluri-
harmonic II.- Ill.J.Math.~1975, 19, 584-592.
5. R u d i n W. hP-isometrics and equimeasurability. - Indiana
Univ.Na~h.J.~1976, 25, 215-228.
WALTER RUDIN Department of Mathematics University of
Wisconsin. 110 Marinette Trail
Madison, WI 53705, USA
694
COMMENTARY
The existence of non-constant inner functions in the ball of Cm was proved independently (and by different methods) by A.B.Aleksand- roy [7] and E.L~w [9] (see also [18]). Both articles are refinements of preceding papers by A.B. Aleksandrov [6] and M. Hakim - N. Sibomy [8] respectively, where the problem has been solved "up to 6 " (but in different senses ). Here are principal results of [7~, [8], [9~.
THEOREM I ([7~). Let ~ be a positive lowe r semicontinuous func-
tion on $ , ~ £ ~ (~) . There exists a sin~lar positive measure o_.nn $ such that 9 (~) = I ~ i~ , and the Poisson integral of~-9
S is pluriharmoni c.
THEOREM 2 ([8]). ~or ever~ continuou s positive function ~ o_~n
and for every positive number 5 there exist a compact set ~, ~C ~,
and function , { C S\ K) a H®(B) such that
~-~ ~<~I-<~ eye--here on ~\~ and ~(0)=0.
THEOREM 3 ([7], [9]). Let ~ be as in Th.1, [email protected](~) • Them there
is a function ~ ~?(~) such that I(I~ a.e. on ~ , and I(0)=0.
Th. I implies Th.3 and Th. 3 implies the existence of non-trivial inner functions in ~ . The last fact DISPROVES ALL CONJECTURES of the
PROBLEM and yields a negative answer to the second question of 12.2. Some of CONJECTURES of S. 10 have been disproved in ~6].
THEOREM 4 ([IO]). Let ~ be a positive lower semicontinuous func-
tion on 5 , 6~0. Then there exists a ~Ctil,O ~ ~ A(B) (= th_._ee
ball algebra) such that I~I~<~ everywhere on ~ and ~[I~I~] <6.
The following two theorems can be viewed as a multidimensional analogue of the Sohur theorem (on the approximation by inner ftmction)
and of the Nevanlinna - Pick interpolation theorem.
THEOR/'~ 5 ([12]). Let q be as in Th.4, ~ £ ~,P(~) , 4 ~<~%+oo,
i+ 4 = 4 Then the weak ~?~ ~?') -closure of the set
I~(B):l~i=q a.e.} coincides with {IC~P(~):I~ x<~ ~.
This theorem was proved in [7] for ~=-~ , and then, independently, proved in [18].
Denote by ~p.c ~) (see [19]) the set of all ~, ~a~(B)
695
such that ~ ~(Z) exists for almost all ~ ~ ~.In other words the
boun~v v~s of a H -f~ction ~ree a e. on ~ with a ~e-
mann integrable function.
THEOPD~ 6 (~12]). Let ~ be as ,in Th.~ and suppose ~,~E~;¢.(~),
~ 0 . Suppose ther@ exist s a function ~ E ~.¢. (~) suc h that
k@O, I~1 ÷lkl .< ~ a.e. on S . Then there ex is ts an F £ ~P(]~)
s=ch t~t IFI=~ a . e . a n d (F-I)~ "4aN,17~). This theorem has been proved in [7~ for ~=-~ . Some particular
cases (also for ~-~ ) have been independently rediscovered in [18]
and ~203. Th. 6 has been rediscovered (for ~ =- ~ ) in [19] • Natural analoEues of Theorems I-6 hold for strictly pseudoconvex
bounded domains with a ~t-boundary and for pseudoconvex bounded do-
mains with a C~-boundary. In these situations theorems 2,3,4 are
due to L~w ~11] . These results combined with those of [12~ imply ana-
logues of Theorems 1,5,6 as well. In ~12] analogues of Theorems I-6
for Siegel domains of the first and the second kind are established
(and in particular for bounded ~etric domains).
Th.4 implies the existence of ~ ~(B)~ J l~ '~ ~ , ~(0~ =0, { I ( l = ~} > O. G.M.HenEin has remarked (see also [133 ) that meth-
ods of N.Sibony [14] can be used to show that such a function cannot
satisfy the Lipschitz condition of order > ~ . Whether it can be lip-
schitzian of order ~ ~ remains unclear.
The follow!~ result by Tamm yields a precise sufficient conditl-
o
notes the set of all essential values of I(£ ~ (5)).
THF, O]~ 7 [15]. I f ~ E ~® (~) and llI-i1, ll~,(B):o((~-'~) ~/' ) ~hen
On the other hand....., . . _ i t is stated in [15~ that "a majority" of ele-
ments of the set do not satisfy
this equality whenever ~ ( ~/~.
A.B.Aleksandrov ~16] has proved analogues of Theorems 2,3,5 for
gradients of harmonic functions. Note also a very simple proof ~I 2]
of Theorem 4 based on the following nice result of J.Ryll and P,Wojtas-
zozyk: there exists a homogeneou@ polyaemlal PN (Z,...., ~) of degree N retch that IPNI.<I in3 and I lPwI~'~,~'>~'u4"; (0-<N<+oo,~4).
W.Rudin has proved in [18] that the linear span of inner functions
is not noxln-dense in H~ (~) . This gives a negative answer to the
question posed in [7] and shows there is no analogue of the ~arshall
696
theorem for ~ ~.
The following QUESTION has been put in [18]: does the ~-norm-
clo~e ' of the linear ,~pan of inner functions contain A(~) (or-what
is She S ame ([18]) - at least one non-constant element of ~))?
The results of H.Alexander [21] and of M.Hakim - N.Sibony [22]
show that A(~) and ~(~) -functions may have zero sets "as large
as functions of the Nevaulinna class ~) ". Recall that zeros of
~)-functions are completely described by a well-known G.LHenkin-
H.Skoda theorem. The work [21] uses Ryll - WoJtaszczyk polynomials
and [22 ] uses techniques of [8~.
REFERENCES
6.A ~ e z~ c s z[ ~ p o B A.B. E ~ o c - X a l ~ ~P a no~yBKyTpemzue ~y~-
~U B rope. - ~oTA.AH CCCP, 1982, 262, ,It 5, 1033-1036.
7.A ~ e x c a H ~ p o B A.B. CymecTBoBs~me B~y~pez~x ~ y ~ m B w-
~e. - MaTeU.c60~ZE.,I982, II8, ~ 2, I47-I63. 8.H a k i m M., S i b o n y N. Fonc%ions holomorphes born~es sur
la boule unit~ de ~. - Inv.math., 1982, 67, N 2, 213-222.
9. L ~ w E. A construction of inner functions on the unit ball in
C~ . - Inv.math., 1982, 67, N 2, 223-229.
10.A ~ e X O S H ~ p 0 m A.B. 0 r~a~qsmx sHsqez~sx ro~ouop~-ux B rope ~wxU~. - ~o~.AH CCCP, I983, 2VI, ~ 4.
W E. Inner functions end boundary values in ~)and 11. L
in smoothly bounded pseudoconvex domains. Dissertation. Princeton
University. June 1983.
12. A ~ e x c s ~ ~ p o B A.B. B~z~emme ~ymmJ~ ~m xoEsx~mx n~o-
CTlm-OT~ex.--~zz~.sHs~a~ a ez'o n~a.~. (to appear) . 13. R u d i n W. Function theory in the unit ball of ~. N.Y. -
Heidelberg - Berlin: Springer-Verlag, 1980.
14. S i b o n y N. Valeurs au bord de fonctions holomorphes et en-
sembles polynomlalement convexes. Lect,Notes Math., 1977, 578,
300-313.
15- T a m m M. Sur l'image par une fonction holomorphe borz~e du
bord d'un domaine pseudoconvex. - C.R.Ac.Sci., 1982, 294, S~r.I,
537-540.
16. A ~z e x c a ]z ~ p o ~ A.B. BHyT1)ez~ze ~yzzz~a ss ZZpOOZpa~CT~X
O~ZOI~ZOrO Tens . - 8azz.may~m.cez~e}z.~[0~, I983, 126,
7-14. 17. R y 1 1 J., W o J t a s z c z y k P. On homogeneous polyno-
697
mials on a complex ball.- Trans Amer Math Sock, 1983, 276, N 1,
107-116
18. R u d i n W. Inner functions in the unit ball of ~. - J.Funct.
Anal., 1983, 50, N I, 100-126.
19. H a k i m M., S i b o n y N. Valeurs au herd des modules de
fonctions holomorphes. Pr6publication Orsay. 1983, 06.
20. T o m a s z e w s k i B. The Schwarz lemma for inner functions
in the unit ball in ~. Preprint (Madison, WI, ) 1982.
21. A 1 e x a n d e r H. On zero sets for the ball algebra. - Proc.
Amer.MathoSoc., 1982, 86, N 1, 71-74~
22. H a k i m M., S i b o n ~ N. Ensemble des z@ros d'une fonc-
tion holomorphe bornge dans la boule unit~. -~ath.Anno,1982,260,
469-474.
S.11.
698
HOMOGENEOUS ~EASURES ON SUBSETS OP ~.
A locall~ finite positive measure j~ supported by a closed sub-
set ~ of ~ and satisfying "the doubling - condition"
i s c a l l e d h e m o g • n • o u s . H e r e ] 3 ( ~ , ~ d e n o t e s t h e b a l l
c e n t e r e d a t ~ w i t h r a d i u s $ . E v i d e n t l y , s u p p ~ - E . A s e t
supporting a homogeneous measure becomes a space of homogeneous
type in the sense of Ooifman and Weiss D-4]. The theory of Har~
spaces ~P , 0< p~ ~ , can be extended to such sets. On the other
hand, the existence of a homogeneous measure is important for the des-
cription of traces of smooth functions on ~ and for free interpola-
tion problems [4,5],
CONJECTURE. Each closed subset of ~ supports a homogeneous
measuree
Except for some evident examples of sets with constant dimension
(Lipschitz manifolds, Cantor sets), the existence of a homogeneous
measure has been proved (up to the present) only for subsets E of
satisfying the following condition:
R I for any interval I c [5] . This condition means that the di-
mension of E is in some sense less than I. Per general sets on the
line and for sets in ~ the problem is open.
Our conjecture has an interesting dual reformulation. Let
N N
: , N ,
699
i.e° ~% is the multiplicity of the covering ~ ~} at ~ , and
i s t h e = t±pliotty •
CONjECTURE. ~W~ ~/~ ~ K , where the constant K
d e ~ e n d s onl~ on 11, .
The e q u i v a l e n c e o f t h e s e two c o n j e c t u r e s f o l l o w s b y H a n h - B a n a c h
theorem.
Independently of the general conjecture, it is interesting to
connect the properties of a homogeneous measure ~ (if it exists)
with geometric characteristics of E . In particular, it is interes-
to estimate the growth of~(B(~,%)) in ~ in term of the ting
Lebesgue measure of & -neighbourhoods of E . a~
REFERENCES
1. C o i f m a n R.R., W e i s s G. Extensions of Hardy spaces
and their use in analysis. - Bull.Amer.Math~Soc.~ 1977, 83, 569-
645.
2. M a o i a s R.A., S • g o v i a C. A decomposition into atoms
of distributions on spaces of homogeneous type. - Adv.Hath., 1979,
33, 271-309.
3. P o I I a n d G.B., S t e i n E.M. Hardy spaces on homogeneous
groups. Princeton, 1972.
4. J o n s s o n A., S J ~ g r e n P., W a I I i n H. Hardy
and Lipschitz spaces on subsets of ~ . - Univ.Ume~ Dept.~ath.
Publ. , 1983, N 8.
5. ~ H H L K ~ H E.~. CBO60~a~ ~HTepn~ ~ C npOHSBO~--
Ho~ ~m ~ . - 3azmc~ ~a,7~.ce~.~0]~I, 1983, 126, 77-87.
E.N~ DYN' KIN CCCP, ~e~rpa~, 197022 Zem~rps~c~ sxeETpoTe~ecE~ ~HCTHTyT
* * *
COMMENTARY
Recently S.V.Konyagin and A.L.Vol'berg have proved that any closed
Ec~ ~ carries a probabilistic measure j~ satisfying~(~,K~))~
~(~)K~j~(B(~,%)) (X~ E, ~> 0, K> I). They proved also a more
precise assertion for E's of a lower (4 ~) dimension and a generali-
zation to metric spaces.
SUBJECT INDEX
a.b means Problem a.b, a.o means Prefaoe
to Chapter a, o.o means Preface
absolute contraction 4.25
absolutely continuous spectrum 4.1, 4.2,4.35,5.5
absolutely summing operator 1.0,1.2, 1.3,4.24,S.I
Adamian-Arov-Krein theorem 3.3,5.15
Ahlfors domain 6.2
Ahlfors-Schimizu theorem 7.3
algebraic curve 12.1,13.10
algebraic equation 13.8
algebraic function 8.11,13.9
algebraic manifold 13.8,13.9
almost normal operator 4.34
almost periodic function 5.9,13.1
analytically negligible curve 4.36
analytic capacity 4.36,8.0,8.15, 8.16,8.17,8.18,8.19
analytic curve 5.4,12.1
analytic disc 2.10
analytic family of operators 2.13
analytic functional 1.14
antisymmetry set 2.11,5.6
Apostol-Foia~ -Voiculescu theorem 4.34
approximable family of operators 4.22
approximate zero divizor S.3
approximation property 1.8
Artin braid group 13.8,13.9
automorphism of an algebra 4.39
backward shift 7.11
badly approximable function
Baire theorem S.10
ball algebra 1.6,S.I0
Banach lattice 1.9
8.13
Banach-Mazur distance 1.4
basis of exponentials 10.2,10.6
Beltrami equation 8.6
Bergman space 5.3,5.7,7.8,7.9,7.10, 7.14
Bernoulli convolution 2.7
Bernstein inequality 13.7
Besov class 3.2,4.24,6.15,8.1,8.21
Bessel potentials 8.1,8.21
best approximation 5.1
Betti numbers 6.18
Beurling-Carleson condition 4.5,7.8, 7.10,7.14,7.15,8.3,9.4
Beurling-Carleson theorem 9.3
Beurling-Malliavin theorem 9.9
Beurling's theorem 3.3,4.14,5.2,7.11, 7.17,8.8
Bieberbach conjecture 6.10,13.2
biharmonic operator 8.20
Billard's basis 1.5
Bishop's operator 4.37
bistochastic measure 3.5
Blaschke-Potapov factor 4.16
Blaschke product 2.3,4.9,4.10,5.2, 5.4,6.11,6.12,6.15,6.19,7.7,7.12, 7.15,10.2,10.3,12.3,8.10
Blaschke sequence 8.3
Bloch space 6.11,6.12
Borel transform 10.6
boundary value problem 5.15
bound state 9.12
Bradford law 0.0
Brelot - Choquet problem 6.18
Brown-Douglas-Fillmore theorem 4.34
Brownian motion 3.0
701
Calder6n-Zygmund kernel 6.5,6.8
Calder6n' s theorem 8.15
Calkin algebra 4.22,4.28,4.31,4.37
Cantor set S. 11
capacity 1.10,4.36,6.15,7.7,8.0,8.9, 8.10,8.11,8.15,8.19,8.20,8.21
Carleman class 9.10,9.11,9.12
Carleson interpolation condition 4. I0, 6.19,10.0,10.2,10.4
Carleson measure 7.13
Carleson±Newman theorem 4.39
Carleson set (see Beurling-Carleson condition )
Carleson theorem 9.3
carrier i. 14
Cartan domains 5. i0
Cartan' s theorem 7. I
Cartwright class 3.2
Cauchy-Fantappi~ formula I. 13
Cauchy formula 4.0
Cauchy-Green formula 13.5
Cauchy integral 6.1,6.2,6.3,8.9,8.15, 8.18,S.5
Cauchy problem 9.4
Cauchy-Riemann operator 4.37
Cayley transform 12.2
center of an algebra 5.6
characteristic function of an operator 3.5,4.0,4.9,4.10,4.11,4.13,4.22, 5.4
Choi-Effros theorem 4.34
cluster set ii.i0
Coburn ' s lemma 5.2,5.6
Cohen-Rudin theorem 2.0
cohomologies 7.2,13.9
commutator 6.1,6.3,6.8
commutator ideal 5.6
complemented subspace 1.5,1.7
complex interpolation 8.22
complex manifold 2.3
conformal mapping 5.4,5.15,6.2,8.0, 8.7,8.9,8.15,13.3
conjugate Fourier series 13.6
conjugate set 1.13
continuous analytic capacity 4.36, 8.17,8.18
continueus spectrum 4.33
continuum eigenfunction expansion 4.1
contraction 2.2,2.3,3.5,4.23,4.24, 4.25,4.26,7.19
convolution 3.0,4.14,9.0,13.7
corona conjecture 5.6
corona theorem 2.0,2.10,4.0,4.12, 6.18,7.0,7.13
cotype l.l,S.l
Coulomb problem 4.2
critical point 2.6
cyclic operator 4.37
cyclic vector 4.9,4.13,5.4,7.7,7.8, 7.9,7.10,7.11,7.19
defect numbers 4.10,5.15
defect space 4.15
Denjoy's conjecture 8.0,8.15,~.16, 8.19
derivation 4.37,4.38
determinant 4.9,4.10,4.17,4.30
deterministic process 3.5
Devinatz-Widom condition 10.2
diagonal operator 7.14
differential operator 7.0,7.3,7.6
dilation 3.5,4.2,4.13,4.25,5.4
Dirac equation 4.2
Dirichlet integral 6.13,7.8,8.20,9.3
Dirichlet problem 6.18,8.1,8.20
disc-algebra 1.0,i.I,I.2,1.4,1.5,2.0, 2.1,2.12,2.13,4.39,6.19,8.13,9.2, 12.2,S.I
discrete spectrum 4.3,4.4
discriminant 13.9
dissipative operator 4.7,4.11
distinguished homomorphism 2.10
divisor 7.0,7.4,10.2
divisorial subspace (sumbodule) 7.0, 7.4,7.7
Dixmier decomposition 2.10
Douglas algebra 3.2,6.14
Douglas conjecture 6.0
dominated contraction 4.25
Dragilev's class i.ii
Dvoretzky theorem 1.3
dynamical system 4.22
Dyn'kin's theorem 10.4
elliptic operator 7.2,8.0,8.20,9.8
endomorphism 4.39
entire function 3.0,7.6,8.7,9.1, I0. I, 11.0, ii .2, ii.3,11.4,11.9, S.9
entire function of completely regular growth ii.0,ii.4,11.5,11.I0
entire function of exponential type 3.1,3.2,4.17,7.5,9.9,10.2,11.1, 11.5,11.6,13.7
entropy 4.22
ergodic theory 4.22
essential norm 5.1
essentially normal operator 4.34, 5.3
essential spectrum 4.27,5.1,5.6
Euler equation 13.7
exact majorant 11.8
exceptional set 8.21,11.5
exceptional value 11.3,11.4
exponential series 10.6
extension of an operator 4.27
extension operator 10.7
extreme point 2.9,12.2,13.2
factorization of an operator 4.18
factorization of functions 5.9,7.0, 7.8
Fantappi~ indicator 1.13
Fatou-Kor~nyi-Stein wedge 10.5
Fefferman-Stein theorem 6.10
field theory 4.19
filter 3.0
702
finite operator 4.38
Fr~chet differential 4.21,12.4,13.8
Fr@chet space 1.11,1.12
Fredholm operator 2.4,4.0,4.29,4.30, 4.34,5.6,5.9,5.10,5.13,5.15
Friedrichs' s model 4.11,4.14
Frostman' s theorem 6.19,12.3,S. I0
Fuglede-Putnam theorem 4.37
functional calculus 4.0,4.22,5.4, 7.1
fundamental group 13.9
fundamental solution 8. I0
functional model 4.0
Galois group 13.8
Gateaux differential 4.21,12.4
Gaussian model 5.12
Gaussian noise 3.1
Gelfand transform 2.1,2.11,6.12, 13.8
generalized character 2.7
generator 4.9
Gevrey class 7.16,9.0,9.6,9.12
Gleason distance 2.12
Gleason part 2.6,6.11
Gleason-Whitney conjecture 5.6
Golubev series S.2
Green's function 1.10,6.18,8.9,13.3
Grothendieck problem 1.3
Grothendieck theorem 1.2,4.24
ground state 4.3
group algebra 2.8
Haar system 1.5
Hadamard lacunae 8.11
Hamiltonian form 4.3
Halmos-Lax theorem 4.0
Hankel operator 3.0,3.3,4.0,4.15, 4.24,5.0,5.1,5.2,5.3,5.15,6.6, 8.0
Hankel determinant 8.11
Hardy inequality 6.4
Hardy space 1.4,1.5,1.6,1.7,1.8, 3.0,3.1,4.0,4.9,4.14,5.6,5.7, 5.8,5.9,5.13,6.0,6.10,6.14,6.16, 6.18,7.9,7.11,7.14,8.3,8.21,10.2, I0.3,S.II
harmonic approximation 8.7,8.10
harmonic conjugation 6.11
harmonic function 5.7,6.18,8.0,8.7, 8.10,8.15,S.I0
harmonic measure 2.10,4.33,8.8,8.9, 13.3
harmonic synthesis (see spectral synthesis) 2.0,7.22
Hartman theorem 5.1
Hausdorff distance 4.32
Hausdorff measure 4.33,6.11,8.8, 8.9,8.14,8.15,8.16,8.17,8.21, 12.1,13.4,S.5
Hausdorff moment problem 9.1
Heisenberg equation 6.22
Helson-Szeg6 theorem 3.3,5.0,6.8, 6.9,6.10
Helton-Howe measure 4.34
Herglotz theorem 9.6
Hermite interpolation 10.0
hermitian element 4.21
Hilbert matrix 6.6
Hilbert scale I.I0,S.2
Hilbert-Schmidt operator 1.2,1.3, 4.34,4.37,5.1
Hilbert transform 4.6,6.1,6.4,6.8, 6.9,6.14,9.4
Hill's equation 4.3
H61der class 3.2,4.2,4.4,8.19,8.20, 9.0,9.6,9.9,10.4,10.7,13.5,s.i0
holomorphic bundles 4.20
holomorphic fibering 13.9
holomorphically convex set 8.14
homogeneous measure S.11
H6rmander theorem 7.1,7.2
hyperelliptlc curve 4.3
hyponormal operator 4.0,4.35,4.36
703
ideal 7.0,7..15,7.16,7.17,12.2
idempotent 2.6
implicit function theorem 12.4
index of an operator 4.30,4.31,5.4, 5.9,5.10
injeetive tensor product 4.24
indicator of an entire function II.5,11.I0,S.9
inner function 2.10,2.12,3.1,3.2, 3.3,4.5,4.9,4.13,4.14,5.2,5.4, 5.5,6. II,6.12,6.15,6.19,7.7, 7.8,7. II,7.15,S. I0
integral Fourier operator 4.0
interpolating Blaschke product 3.3, 5.2,6.9,6.19,10.2
interpolating sequence 5.2
interpolation 4.0,5.0,9.6,10.0,10.1, 10.3,10.4,11.5
interpolation of operators i0.0,10.8
interpolation set I. 1,10. 5
intertwining operator 4.13
invariant subspace 4.0,4.7,4.8,4.9, 4.17,4.22,4.29,5.2,5.4,5.5,7.0, 7.7,7.11,7.14,7.18
inverse spectral problem 4.2,5.0, 8.4
irreducible polynomial 13. I0
irreducible singularity 13. I0
irregular set 8.15,8.16
Ising model 5.12
isomorph I. 2
isometry 1.6,4.6,4.15,4.25,5.4
Jacobi polynomials 13.6
Jacobi variety 4.3
Jensen measure 12.1
John-Nirenberg inequality 6.0
Jordan curve 6.2,8.2,8.8,8.13,8.14, 12.1,S.5
Jordan domain 8.9,12.1
Jordan operator 9.4
K~hler metric 7.1
Kamowitz-Scheinberg theorem
Kantorovich distance 3.6
4.39
Kellog's theorem 8.3
Koebe theorem 6.10,8.8
Korteweg-de Vries equation
K6the duality 1.12
K6the space 1.11,1.12
Kramer's rule 7.4
Krein class 11.6
Krein-Milman theorem 13.2
Krein space 4.7,4.16,4.31
Krein theorem 13.7
4.3
Laguerre-Polya class 11.0,ii.9
Lam~-StoKes system 8.20
Laplace equation 9.4
law of large numbers 3.4
Legendre polynomials 5.12,13.6
Leray boundary 1.13
Liapunov curve 4.36,6.6
Lie group 13.8
linearly couvex set 1.13
Liouville theorem 4.0
Lipschitz condition 7.1 ,S. 5
Lipschitz domain 9.7
Lipschitz graph S. 5
Littlewood conjecture 6. I0,13.2
Lizorkin-Triebel space 8.21
local approximation I0.7
localization 5.13,7.0,7.2,7.4, 7.5
local operator 9.9
local ring 7.12
local Toeplitz operator 5.6
logarithmic capacity 2.10,6.11, 8.0,8.15,9.3,9.5
Lorentz space 1.3,1.9
lower triangular operator 4.18
marginal 3.6
~.~rkov operator 3.5
Markov process 3.0,3.5
Martin's boundary 6.18
matrix function 4.16,4.19,5.15
704
Matsaev ideal 4.8,4.18
Matsaev's conjecture 4.25,4.26
maximal ideal space 2.0,2.6,2.8,2.10, 2.11,4.39,5.6,7.17,7.19,8.14,12.2
maximal function 6.4,6.17,S.1
maximum principle 9.0
mean periodic function S.4
measure algebra 2.0,2.6,2.7
Mergelyan's theorem 8.13
meromorphic function 4.19,6.18,8.7, 9.1,11.0,11.3,11.5,S.7
Miller's conjecture 2.6
minimal family 10.2
Mittag-Leffler function
M6bius transformation
model space 10.2
module 12.2
modulus of an operator
modulus of continuity
11.10
4.39,6.10
3.3,4.35
4.4
modulus of quasitrianoularity 4.34
moment problem 5.0,5.12,9.0,9.1,9.2
Muckenhoupt condition 6.7,6.9
multiplier 2.6,4.0,4.24,4.25,6.16, 10.0,10.1,10.8
multi-valued function 8.11
MCmtz condition 9.10
Naimark theorem 3.3
Nehari's theorem 3.3,4.24,5.1,5.2
von Neumann's inequality 4.24,4.25, 4.26
Nevanlinna characteristic S.7
Nevanlinna class 4.0,7.7,7.9
Nevanlinna theory 9.11,11.0,11.2
normal extension 4.36
normal family 9.0,9.4,13.8
normal operator 3.4,4.0,4.31,4.32, 4.33,4.36,4.37,5.4
nuclear operator 3.2,4.4,4.5,4.9, 4.17,4.30,4.33,4.34,4.37
nuclear space 1.11,1.12
numerical range 4.38,5.8
705
Oka-Cartan theory 7.1
Oka's theorem 7.1
operator algebra 4.24,4.29
operator function 4.4,4.10,4.12, 4.15,4.20,5.8
operator K-theory 4.0,5.0
Orlicz space 1.6,1.9
orthogonal polynomials 5.0,5.12
Ostrovski lacunae 8.11
other hand 4.0,8.12,8.16,9.1,10.3, S.lO
outer function 3.1,3.2,3.3,4.13,5.5, 5.15,6.19,7.9,9.6,S.6
Pad~ approximation 8.0,8.11,8.12
Painlev~ null set 8.15,8.16
Painlev5 problem 8.17
Paley-Wiener theorem 9.1, i0.1
Parreau-Widom type surface 6.18
partial indices 4.20,5.8
Past and Future 3.0,3.2,33
peak point 2. Ii
peak set 2.6,4.36,9.0,9.6
Perron-Frobenius theorem 4.40
perturbation of spectrum 4.32,4.33
phase function 3. I
piecewise continuous function 5.13
pluriharmonic function 12.2 ,S. 10
plurisubharmonic function 2.3,7. I, 7.2
Poincar~-Beltrami operator 5.16
point spectrum 4.1,4.5,4.10,4.11, 4.35
polar decomposition 3.3,4.11,4.35
polar set 13.4
Polya theorem 13.7
polynomial approximation 8.3,8.9
polynomially bounded operator 4.24
polynomially convex set 8.14,12.1
positive definite kernel 4.12,13.1
positive definite sequence 5.11,13.1
potential 4.2
power bounded operator 4.24
power series space 1.10,1.11,1.12
prediction theory 5.0
premeasure 7.10
primary Banaeh space 1.5
prime ideal 7.12,7.17
projection method 5.14
projective tensor product 4.24
proper holomorphic mapping 12.3
pseudocontinuation 7.11
pseudoconvex set 7.3,8.14,S.I0
pseudodifferential operator 4.0
quasi-analytic class 4.3,4.8,7.16, 7.17,8.0,9.0,9.8,9.10,9.11,9.12
quasicommutator ideal 5.13
quasiconformal continuation 8.6
quasidiagonal operator 4.38
quasinilpotent element 2.5,4.28
quasinilpotent operator 4.0,4.22, 4.40,5.2
quasi-similarity 4.13
radical algebra 7.21
Raikov system 2.6
ramification S.8
rational approximation 3.3,8.0, 8.6,8.11,8.12,8.17
rationally convex set 8.14
rearr~lgement 13.6
rectifiable curve 4.36,6.2,6.7, 8.15,8.16,9.10,S.2
reducing subspace 4.36,4.38
reduction method for Toeplitz operators 5.13,5.14
re-expansion operator 4.6
reflection coefficient 4.3
regular domain I.I0
regular J-inner function 4.16
regular point 8.20,11.7
706
regular set 8.15,8.16
removable singularity 8.0,8.15,8.17, 8.19,12.1
representing measure 5.6
reproducing kernel 10.2
Riemann-Hurwitz formula S.8
Riemann metric 7.1
Riemann sphere 8.5,8.14
Riemann surface 1.10,2.10,6.18,8.0, 9.1,13.10
Riesz basis 5.2,10.2,11.6
Riesz (F. and M.) theorem 6.2,6.18, 12.1
Riesz-Herglotz theorem 5.11
Riesz kernel 6.16
Riesz operator 4.28
Riesz projection 4.13,5.8,5.9,6.5, 6.6,9.11
Riesz representation of subharmonic functions 11.8
ring of fractions 7.12
root vector 7.7
Runge's theorem 7.1
Sarason hull 4.33
saturated submodule 7.4
scattering operator 4.2
scattering theory 3.5,4.0,4.2,4.6, 4.33
Schatten-von Neumann classy. 3.2,4.25, 4.31,4.3~,4.37
Schauder basis 1.4,1.6,1.10,1.12
Schr~dinger equation 4.2
Schr6dinger operator 4. i i, 9.12
Schur -Nevanlinna-P ick interprolation problem 4.16, S. I 0
self-adjoint operator 4.0,4.1,4.2,4.4, 4.6,4.11,4.14,4.21,4.22,4.31,4.32, 4.35,5.2,5.5,9.12
semi-Fredholm operator 2.4,5.9
semigroup 2.4,2.6
separable polynomial 13.8
separating space of a homomorphism 2.5
spectral
spectral
spectral
spectral 8.0
sheaf 13.4
shift operator 3.3,4.13,4.22,4.25, 4.35,7.7,7.9,7.14
Shilov boundary 2.0,2.6,2.10,2.13, 4.39,5.10
Shilov's theorem S.3
Siegel domain S.I0
similarity of operators 4.11,4.23, 4.24,5.0,5.4
simple spectrum 4.3,4.9
sine-type function 11.0,11.6
singular integral operator 4.0, 4.6,4.30,5.0,5.14,6.0,6.2,6.3, 6.9,8.0
singular numbers of an operator 3.3,4.31,5.0,5.1,5.2,5.15,8.0
singular spectrum 4.1,4.4,4.5,4.11
Smirnov class 5.15,6.2,7.11
Smirnov curve S.5
Sobolev space 1.8,5.14,8.0,8.1, 8.2,8.9,8.22
decomposition 4.10
density of a process 3.2
inclusion theorem 5.6
(maximal) subspace 4.10,
spectral measure of an operator 4.15,4.22,9.4
spectral measure of a process 3.2, 3.3
spectral multiplicity 5.4
spectral operator 9.0
spectral radius 2.1,2.2,2.3,2.4, 13.7
spectral set 5.4
spectral synthesis 7.0,7.5,7.7, 7.18,7.22,8.0,8.1,9.0,10.7,13.7
spectrum 2.1,4.4,4.22,4.33,4.35, 5.0,5.2,5.4,5.6,5.7,7.0,7.6, 7.7,7.18,8.0,S.3
spline approximation 5.14
stable submodule 7.4
standard ideal 7.21
starlike domain 8.8
707
stationary Gaussian process 3.0,3.2, 3.3,3.4
statistical phy3ics 5.12
Stein manifold 1.10,7.1,13.8
Stolz condition 10.3
Stone-Weierstrass theorem 4.0
strong boundary point 2.6
strongly elliptic operator 5.14
strongly linear convex set 1.13
strong mixing condition 3.2
structure semigroup 2.6
subharmonic function 6.10,9.5,10.1, II.0,11.7,11.8,12.1,13.4,S.9
submodule 7.0,7.2,7.5
subnormal operator 4.0,4.36,4.37,7.9
sufficiently Euclidean space 1.3
support point 13.2
support set 2.11
Swiss cheese 8.6
symmetrically normed ideal 4.30
symmetric domain 12.3
symmetric measure 7.9
symmetric operator 5.0,8.9
symmetric space 1.6
Sseg6 condition 5.12
Szeg6 determinants 5.0
S~eg6 limit theorem 5.11
Szeg6's alternative 3.3
S~.-Nagy-Foia~ model 4.0,4.10,4.11, 5.0,5.4
tangential approximation 8.7
Tchebysh~v polynomials 5.12
Thue-Siegel-Roth theorem 5.12
Titchmarsh's theorem 4.14,7.19, S.4
Toeplitz operator 2.11,4.0,4.20,4.23, 5.0,5.2,5.3,5.4,5.5,5.6,5.7,5.8, 5.9,5.10,5.11,5.12,5.13,5.15,5.16, 6.6,10.2
Trudinger inequality 6.13
two-sided ideal 2.1,5.13,5.16
uncertainty principle 9.4
unconditional basis 1.7,4.10,10.0, 10.2
unilateral shift 5.2
uniform algebra 2.0,2.9,2.11, 2.13,4.39,S.3
uniformly convergent Fourier series 1.0,i.I,S.6
uniformly minimal family 4. i0
unimodular function 3.2,5.5
uniqueness set 7.15,7.16,8.0, 9.3,S.6
uniqueness theorem
unitary dilation 5.4
unitary extension
unitary operator 4.32,4.35
univalent function
upper triangular operator
9.0,9.13,10.6
3.5,4.2,4.13,
4.15
4.5,4.9,4.22,
6.0,6.10,13.2
4.18
ValiZon theorem 11.2
Vogt-Wagner's class 1.11
wave operators 4.2,4.6
weak generator 4.9
weakly invertible element 7.7,7.8, 7.9,7.10
weak type inequality 6~5
weighted automorphism 4.39
weigh ted shift 4.37,4.38,7.19, 7.21,S.3
white noise 3.0
Whitney decomposition 13.3
Whitney jets 10.4
Whitney theorem 10.7
Widom's theorem 5.6,5.7,7.17
Widom surface 2.10
Wiener algebra 5.16
Wiener condition 8.10
Wiener criterion 8.20
Wiener-Hop f operator 4.0,4.29,5.0
Wiener-Levy theorem 7.22
Wiener-Pitt phenomenon 2.6
Wiener's TaU berian theorem
Wiman-Valiron theory 11.2
winding number 5.4,5.6,8.13
Wold decomposition 5.4
7.0
Yang-Baxter equation 5.16
Yentsch's theorem 4.40
zero set 4.4,7.9,9.6,9.7,9.12, 10.1,10.2,10.5,11.0,ii.6,S.I0
Zygmund class 4.24,10.4,10.7
AF-algebra 4.22
AFI-algebra 4.22
A-support 1.14
BMO 1.0,1.8,3.3,4.24,6.0,6.7,6.8,6.9, 6.10,6.12,6.14,8.22
C*-algebra 2.1,2.5,3.5,4.0,4.21,4.22, 5.10,5.13,S.3,S.6
C.o, C. o, Cll-contraction 3.5,4.13, 5.5
C-support 1.14
-equation 7.2,7.3
f-propetry 6.12
H*-algebra 2.1
H ~ + BUC 5.1
H ~+ C 2.11,3.2,5.1,5.5,5.6,5.13, 6.0,S.6
J-dissipative operator 4.7
J-inner function 4.16
K-functor 4.22
K-propetry 6.12
K-spectral set 5.4
L-subalgebra 2.0,2.6,2.7
"tspace 1.2 n-circular domain 1.11
PC-support 1.14
p-semidiagonal operator 4.37
p-trivial space 1.3
708
pw-topology 4.25
QC 5.5,5.6,S.6
SR-algebra 2. I
s-space 7 . 7
U(£ )-set 7.23
VMO 6..0,6. I 1,6.12,6.14,S.6
~ -regular Gaussian process 3.2
~-sectoriality 5. I 1
-entropy 4.22
A( p ) -set 6.5,10.8
AUTHOR INDEX
a.b means Problem a.b, a.O means Preface to
Chapter a, Ack. means Acknowlegement.
Adams D.R. 8.20,8.21
Adamyan V.M. 7.8,7.9
Aharonov D. 7.8,7.9
Ahem P. 6.15,12.2,S.I0
Ahiezer N.I. 11.7,13.7
Ahlfors L.V. 6.2,8.15,8.17
Aizenberg L.A. Ack.,l.13
Akcoglu M.A. 4.25,4.26
Akermann C.A. 2.1
Aleksandrov A.B. Ack.,6.17,7.11, 9.3,12.3,S.5,S.I0
Aleksandrov A.D. 12.3
Alexander H. 8.14,12.1,12.3,S.I0
Allan C.R. 2.4
Alspach D. i. 5
Amar E. 6.9
Amick C. 8.2
Anderson J.H. 4.38
Anderson J.M. 6.11,6.12,8.5
Andersson J~E. 6.2
Ando T. 2.10
Antonevich A.B. 4.39
Aposto! C. 4.22,4.34
Apresyan S.A. 7.7
Arakelyan N.U. 8.0,8.5,8.7,11.4
Arazy Jo 4.25
Arens R. 8.14,S.3
Arnold D.N. 5.14
Arnold V.I. 13.9
Arocena R. 6.8
Aronszain N. 8.21
Arov D.Z. Ack.,3.3,4.15,4.16,5.1, 5.15
Art~menko A.P. 13.1
Arveson W. 4.12,4.34
Arzumanyan V.A. 4.22
Aupetit B. 2.5
Avron J. 4.1
Axler S. 2.11,5.1,5.3,5.6,S.6
Azarin V.S. Ack.,ll.10,S.9
Azizov T.Ya. 4.7
Azoff E. 4.32
Babenko K.I. 7.17
Bachelis G.F. 7.23
Bacher J.N. 7.19
Bade W.G. 7.19
Badkov V.M. 13.6
Baernstein A. 6.10,13.3,S.2
Baillette A. 9.10
Bagby T. 8.1,8.10
Baker GoA. 8.11,8.12
Banach S. I0.0
Bandman T.M. 13.9
Barnsley M. 5.12
Barth K.F. 8.5
Basor E. 5.11
Batikyan B.T. Ack.
Behrens M.F. 2.0
Belavin A.A. 5.16
Bell S.R. 12.3
Beloshapka V.K. 12.1
Belyi V.I. 6.2,8.6
Benedicks M. 11.7
Berenstein C.A. 1.12,10.1
Berezanskii Yu.M. 4.1,9.8
Bernstein S.N. 3.3,8.9,11.1,13.7
Berg B.I. 3.6
Berg Ch. Ack.,9.2
Berg I.D. 5.1
Besicovitch A. 8.15,8.16
Bessis D. 5.12
Beurling A. 3.1,7.9,7.17,8.1,8.9, 8.15,9.3,9.9
Bhatia R. 4.32
Bieberbach L. 8.11
Billard P. 1.5
Birkhoff G.D. 4.20
Birman M.S. Ack.,4.6,4.21,4.31
Bishop E. 2.11
Bj6rk J.-E. 1.14,2.12
Blanc - Lapierre A. 3.4
Blumenthal R. 2.12
Boas R.P. 13.7
Bochkar~v S.V. 1.4,1.5,1.7,
Bochner S. 13.1
Bognar J. 4.7
Boivin A. 8.7
Bollob&s B. 4.27,S.3
Borisevich A.I. i0.I
B6ttcher A. 5.13
Bourgain J. 1.1,1.2,1.4,6.5,S.I
Boyarskii B.V. 5.15
Bram J. S.3
de Branges L. Ack.,2.9,4.8,9.9
Brannan D.A. 8.5
Brelot M. Ack.,6.18
Brennan J.E. Ack.,7.9,8.3,8.8,8.9
Brodskii A.M. 4.2
Brodskii M.S. 4.17
Bronza S.D. S.8
Browder A. 2.1i
Browder F. 4.1
Brown G. 2.0,2.6
Brown J.E. 13.2
Brown L. 8.5
Brown L.G. 4.31,4.34
Bruna J. 7.16,10.4
Brudnyi Yu.A. 10.7
Bryskin I.B. 1.6
Bunce J.W. 4.38
Burenkov V.I. 8.10
Buslaev V.S. 4.2
710
Calder6n A.P. 6.0,6.1,6.3,6.16,8.0, 8.15,8.16,8.17,8.18,8.19,S.5
Carey R.W. 4.34
Carleman T. 8.7,9.9,9.12,10.0
Carleson L. 2.0,4.10,4.33,6.9,7.8, 7.13,8.8,8.9,8.16,9.2,10.3
Cartan H. 7.4
Casazza P.G. Ack.,l.5,6.19
Caughran J.G. 7.15
van Castern J. 4.11,4.24
Cegrell U. 13.4
Cereteli O.D. 13.6
Challifour J. 9.2
Chang S.-Y. 2.11,6.13,6.14
Charpentier P. 4.24
Chaumat J. 2.10
Chebotar~v G.N. 5.9
Chernyavskii A.G. Ack.,9.8
Chirka E.M. 8.14
Chisholm J.S.R. 8.12
Choquet G. 8.10
Clancey K.F. 4.34,4.35,4.36,5.4
Clark D.N. Ack.,4.5,4.23,5.2,5.4,
5.5
Clary S. 8.3
Clunie J, 6.11,6.12
Cnop I. 7.1
Coburn L. 4.22,5.10
Coifman R.R. 4.25,5.3,6.0,6.1,6.3 6.8,6.9,7.11,S.5,S.11
Cole B. 2.11
Connes A. 4.22
Cotlar M. 6.2,6.4,6.8
Courant R. 9.8
Couture R. 9.10
Cowen C.C. 5.4
Cowen M.J. 4.23
Crofton M.W. 8.15
Crum M.M. 13.1
Curtis P.C.Jr. 7.19
Cuyt A. 8.12
7t l
Dales H.G. 2.5,7.19
Daletskii Yu.L. 4.21
Danilyuk I.I. 6.2
Dang D.Q. 5.14
Danzer L. 10.7
David G. 6.2,6.7,S.5
Davie A.M. 1.5,4.22,4.24,4.36, 8.15,8.16,8.17,8.19
Davis Ch. 4.11,4.32
Davis W.J. 1.3
Deddens J.A. 4.38
Deift P. 4.3
Delbaen F. 1.2,1.5,S.1
Delsarte J. 7.0,S.4
Denjoy A. 8.15,8.16
Deny J. 8.1
Devinatz A. 5.8,5.11,9.2
Diaconis P. 9.2
Dixmier J. 4.22
Dixon M. 9.10
Djrbashyan M.M.
Djrbashyan A.E.
Dollard J. 4.2
4.9t7.17,8.3,8.8,9.1
9.3
Dolzhenko E.P. 4.36,8.18,8.19
Domar Y. 7.19,7.22,8.9
Donchev T. 8.20
Douglas R.G. 2.10,4.9,4.23,4.31, 4.34,5.0,5.6,5.10,6.0
Dovbysh L.N. Ack.
Dragilev M.M. I.i0
Drinfel'd V.G. 5.16
Dubinsky E. 1.3
Duffin R.J. 9.5
Dufresnoy A. 7.2
Duren P.L. 5.4,6.2,7.8,13.2
Dym H. 3.0,3.1,8.4
Dyn'kin E.M. Ack.,6.2,7.22,9.6, I0.4,S.II
Dzhvarsheishvily A.G. 6.2
Earl J.P. 10.3
Edelstein I. 1.9
Effros E.G. 4.22
Ehrenpreis L. 7.0,7.3,10.1
Elschner I. 5.14
Enflo P. 1.4,1.5,1.8,7.8
Epifanov O.V. 11.5
Eremenko A.E. 11.3,11.4,11.10
Erkamma T. 9.10
Erofeev V.P. 1 . 1 0
Erohin V.P. 1.10
Faddeev L.D. 4.2,4.3,4.4,4.19,9.6
Farforovskaya Yu.B. Ack.~.21
Federer H. 8.15
Fedchina I.P. 4.16
Fefferman Ch. 4.24,6.9,6.10,6.17
Fel'dman I.A. 4.20,4.29,4.30,5.13
Fern&ndes C.S~nchez 13.8
Ferrier J.-P. 7.0,7.1
Fiegel T. 1.3,1.5,5.14
Fillmore P.A. 4.31,4.34
Foia~ C. 3.5,4.10,4.11,4.12,4.15, 4.18,4.22,4.23,4.25,4.34,5.4,
5.5
Folland G.B. S.II
Forelli F. 7.12,12.2,S.I0
Frankfurt R. 4.9,7.9
Friedman J. 4.25
Frolov Yu.N. 7.6
Frostman O. 8.9,8.16
Fuglede B.V. 8. i ,9.2
Fuka J. 8.19
Fuks D.B. 13.9
Gamelin T.W. 2.6,2.10,4.36,6.18, 8.7,8.13,8.14,8.17
Ganelius T. 6.2
Gaposhkin V.F. 3.4
Garabedian P.R. 8.15 o
Garding L. 4.1
Gariepy R. 8.20
Garnett J. 2.10,4.36,6.7,7.13,10.0, 8.13,8.15,8.16,8.17,8.18,8.19, 13.3
Gauthier P. 8.7
712
Gehring F.W. 8.7,13.3
Gel' fand I.M. 1.12,2.11,3.2,4.1,7.17, 9.8
Gel'fond A.O. 11.5
Gilbert J.E. 7.23
Giner V.B. S.9
Ginsberg J. 4.9
Ginzburg Yu.P. 4.17
Glicksberg I. 2.0,2.11
Gohberg I.C. 3.3,4.6,4.8,4.17,4.18, 4.20,4.22,4.29,4.30,4.31,5.1, 5.13,5.14,6.6
Gol'dbero A.A. Ack.,11.3,11.4,11.5, S.7,S.8
Golinskii B.L. 5.11,5.12
Golovin V.D. 11.6
Golovin V.M. 12.1
Golubev V.V. S.2
Golusin G.M. 8.3
Gonchar A.A. 8.11,8.12
Goncharov A.P. 1.11
Gordon A.Ya. Ack.,11.1,11.8
Gorin E.A. Ack.,4.39,13.7,13.8,13.9
Gorkin P.M. 2.11
Grabiner S. 7.21
Gragg W.B. 8.12
Graves-Morris P.P. 8.12
Gribov M.B. 7.7
Grishin A.F. 11.3,11.10
Grothendieck A. 1.13
GrOnbaum B. 10.7
Gubanova A.S. 1.13
Gulisashvili A.B. Ack.,13.6
Gurarii V.P. 6.12,7.17,7.18,7.19, 7.20,10.3
Gurevich A.Yu. 4.2
Gurevich D.I. 7.3
Gyires B. 5.11
Hadamard J. 8.12
Hakim M. S.I0
Halmos P.R. 4.22,4.24,4.34,4.36,4.38
Hamilton D.H. 13.2
Hanin L.G. 8.1
Haplanov M.G. 1.10
Hardy G.H. 6.8
Hartman P. 5.1
Harvey R. 8.10,8.19
Haslinger F. 1.12
Hastings W. 8.3
Hasumi M. 6.18
Havin V.P. 6.2,6.12,6.17,8.1,8.8, 8.9,8.15,8.16,8.19,9.3,10.0, I0.3,S.2
Havinson S.Ya. 8.15,8.16,8.19,11.8
Hayashi E. 3.2
Hayashi M. 2.10,6.18
Hayman W. 8.8,8.16,13.3
Hedberg L.I. Ack.,7.9,8.1,8.10, 8.20,8.21
Hellerstein S. 11.9
Helson H. 4.14,5.8,5.12,6.9,6.10, 6.18
Helton J.W. 2.3,4.34,5.11
Henkin G.M. Ack.,1.10,8.14,10.5, 12.3,S.2,S.I0
Herrero D.A. 4.38
Hilbert D. 5.0
Hilden H.M. 7.7
Hirchowitz A. 13.8
Hochstadt H. 4.3
Hoffman K. 1.6,2.10,2.11,6.18
Holbroock J.A.R. 4.23
H6rmander L. 7.1,7.2
Horowitz C. 7.9,7.14,9.1
Host B. 2.0
Howe R. 4.34,5.10
Hru~Mv S.V. 3.2,3.3,5.2,7.11,7.16, 8.3,8.15,9.1,9.3,9.4,9.6,9.9, 9.12,10.0,10.2,11.6
Hunt R.A. 5.8,6.9,6.10
Hurwitz A. S.8
Huskivadze G.A. 6.2
Hutt H. 9.6
Hvedelidze B.V. 6.2
Ibragimov I.A. 3.2,5.11
Ien E. 5.14
Igari S. 2.7
Ii'in E.M. 4.6
Iohvidov I.S. 4.7
Isaev L.E. 4.10,4.17
Ivanov L.D. 8.15,8.16,8.17,8.18,8.19
Jacewicz Ch.A1. 7.14
Jacobi C.G.J. 5.0
Janson S. 5.0,6.9
Jawerth B. 8.21
Jewell N. 5.1
John F. 8.19
Johnson B.E. 2.5,2.6,2.13,4.37,4.39
Johnson W.B. 1.3
Jones J. 2.13
Jones P.W. 1.8,6.3,6.7,6.9,6.16, 8.2,8.22,13.3
Jonsson A. 10.4,10.7,S.II
JOricke B. Ack.,9.4
Jurkat W.B. 6.4
Kac M. 4.1
Kadampatta S.N. 1. i0
Kadec M.I. II.i
Kahane JrP. 3.0,6.11,7.22,7.23,8.1, 9.1
Kaliman Sh.I. 13.8,13.9
Kamowitz H. 4.39
Kanjin Y. 2.7
Karahanyan M. I. 13.8
Kargaev P.P. Ack. ,3.7,9.9,S.4
Karlovich Yu. I. 5.9
Karlsson J. 8.12
Katsnel ' son V.E. 11.6
Kato T. 4.35
Kau fman R. Ack. ,9.7,13.5,S. 7
Keldysh M.V. 8.0,8.1,8.7,8.8,8.9
Kelleher J.J. 7.4
Kennedy P.B. 8.16
713
Khrushchev S. (see Hru~v S.V) 9.9
Kiselevskii G.E. 4.9,4.39
Kiselman C.O. 1.14
Kisliakov S.V. Ack.,l.2,1.3,1.5, 6.5,S.I,S.6
Kitover A.K. 4.25,4.26,4.39
Klee V. 10.7
Kohn I.I. 5.14
Kokilashvili V.M. 6.2
Kolsrud T. 8.1
Komarchev I.A. 1.3
Kondrat'ev V.A. 8.20
Konyagin S.V. S.11
Koosis P. Ack.,3.1,4.3,8.4,9.5
Korenblum B.I. 7.7,7.8,7.10,7.15, 7.16,7.17
Korevaar J. 9.10
Kostenko N.M. 4.2
Kostyuchenko A.G. 9.8
Kotake T. 9.8
KOthe G. 1.12,1.13
Kottman C. 1.5
Kovalenko V.F. 4.1
Kral J. 8.19,13.4
Krasichkov-Ternovskii I.F. 7.3, 7.4,7.5,7.6,7.7
Krein M.G. Ack.,3.3,4.0,4.8,4.15, 4.16,4.17,4.18,4.21,4.22,5.0, 5.1,5.11,5.15,8.4,11.6,13.1
Krein S.G. 1.6,1.9,1.10,4.21
Kriete T. 8.3
Krol' I.N. 8.20
Kronstadt E. 8.13
Krupnik N.Ya. 4.6,4.29,4.30,5.8,
5.14,6.6
Krushkal S.L. 12.4
Labr~che M. 8.7
Landis E.M. 8.20
Landkof N.S. 1.10
Langer H. 4.7,13.1
Lapin G.P. i0.I
Latushkin Yu.D. 5.15
Lautzenheiser R.G. 4.36
Lavrent'ev M.A. 8.7,8.8
Lax P. 4.3
Lebedev A.V. 4.39
Lehto O. 8.6
Leiterer J. 2.4,4.2
Lelong P. 13.4
Leont'ev A.F. 7.6,10.1,10.6,11.10
Leray J. 1.13
Levi R.N. 4.39
Levin B.Ya. 7.17,7.20,11.0,11.4, ii.5,11.6,11.7,11.9,13.1,S.9
Levinson N. 8.4,9.9
Lin B.L. 1.5
Lin V.Ya. 13.8,13.9,13.10
Lindberg P. 8.10
Lindenstrauss J. 1.3,1.5,1.9,4.24
Linnik I.Yu. 3.12
Lions J.-L. 8.1,9.8
Littlewood J.E. 6.6
Litvinchuk G.S. 5.15
Livshic M.S. 4.0,4.15,4.18
Lodkin A.A. 4.22
Lohwater A.J. 8.15
Lomonosov V.I. 4.29
Lorch L. Ack.
L~w E. S.10
Lozanovskii G.Ya. 3.5
Luecking D. 5.1,8.13
Lukacs E. 13.7
Lundin M. 9.10
Lyons T.J. 8.7
Lyubarskii Yu.I. 11.6
Lyubic Yu.I. 4.40,9.8,S.4
Macias R.A. S.11
~cintyre A.J. 11.5
Magenes E. 8.1,9.8
Magnus W. 4.3
Makai E.Jr. 2.4
714
Makarov B.M. 1.3
Makarov N.G. Ack. ,4.5,4.9,4.33, 9.4
Makarova L.Ya. I. 13
Malamud M.M. Ack. ,9.8
Malgrange B. 7.2,7.3
Malliavin P. 3.1,7.0,9.10,10.1
Mandelbroj t S. 8.4,9.12
Markus A.S. Ack.,4.29,4.30
Markushevich A.I. 5.15
Marshall D.E. 6.13,8.9,8.15,8.16
Marstrand J.M. 8.15
Martineau A. 1.12,1.13,1.14,7.3
Maslov V.P. 4.1
Matsaev V.I. 7.4,7.6,9.8
Matveev V.B. 4.2
Matyska J. 8.19
Maurey B. 1.3
Maz ' ya V.G. Ack. ,8.8,8.9,8.10, 8.20,9.3
McDonald G. 5.7
McIntosh A. 4.32,6.3,S.5
McKean H.P. 3.0,3.1,4.3,8.4
McMillan J.E. 8.8
Meiman N.N. 13.7
Melamud E.Ya. 4.16
Mel ' nikov M.S. 8.8,8.17,8.19
Mergelyan S.N. 8.0,8.3,8.7,8.8, 8.9
Merzlyakov S.G. 7.6
Metzger T.A. 8.8
Meyer M. 1 .12
Meyer Y. 6.1,6.3,S.5
Meyers N.G. 8.2,8.10,8.20
Michael E. 10.7
Michlin S.G. 4.30,5.0,5.14
Mikaelyan L.V. 5.11
Miklyukov V.M. 6.2
Milman V. 1.3
Mirsky L. 4.32
Mitiagin B.S. 1.5,1.9,1.10,1.11, 1.12,S.2
715
van Moerbeke P. 4.3
Mogilevskaya R.L. 4.17
Mogul'skii E.Z. 7.4
de Montessus de Ballore R. 8.12
Moran W. 2.6
Morrel J.H. 5.4
Morrell J.S. 1.3
Moser J. 6.13
Muckenhoupt B. 5.8,6.4,6.9,6.10
Mulla F. 8.21
MUller V. 2.4
Murphy G.J. 2.1,S.3
Naboko S.N. 4.4,4.11,4.24
Napalkov V.V. 7.3,9.13
Narasimhan M.S. 7.0,9.8
Natanson I.P. Ii.i
Nazarov S.A. 8.20
Nehari A. 5.1
Nelson D. 7.15
Nersesyan A.A. 8.7
Nevanlinna R. 11.3,11.4
Neville C. 6.18
Newman D. 4.9
Nguen Thanh Van I. i0
Nikol ' skii N.K. 2.3,3.3,4.9,4.10, 4.13,4.33,5.2,5.13,6.6,7.4,7.5, 7.7,7.8,7.11,7.19,7.21,8.3,11.6, 10.0,10.2
Nilsson P. 8.21
Nirenberg L.I. 5.14,8.19
Norrie D.H. 5.14
Norvidas S.T. 13.7
Novikov R.G. 12.3
Nyman B. 7.17,7.19,7.20
Oberlin D. 1.1,7.14,7.23
Odell E. 1.5
O'Farrell A. 6.7
~ksendal B.K. 4.36
Osadchii N°M. 8.1
Ostrovskii T.V. Ack.,ll.4,11.5,11.6,
II.9,S.8
Oum Ki-Choul 11.4
Ovsepian R.I. 1.2
Paatashvily V.A. 6.2
Painlev~ P. 8.15
Palamodov V.P. Ack.,7.0,7.2,9.8, 10.1
Paraska V.I. 4.30
Parreau M. 2.0,6.18
Pasnicu C. 4.34
Pastur L. 4.1
Paulsen V. 2.4
Pavlov B.S. 4.4,4.10,5.2,9.6, 9.12,10.0,i0.2,11.6
Pearcy C. 4.34
Pearcy K. 13.2
Pedersen G.K. 2.1
Peetre J. 5.0,8.19,8.21,10.8
Pe~czy~ski A. 1.0,1.2,1.3,1.4, 1.5,1.7,4.24,S.I
Peller V.V. Ack.,3.2,3.3,4.21, 4.24,4.25,4.26,5.5,9.3
Pengra P.G. 6.19
Perron O. 8.11,8.12
Petras S.V. 4.4
Petrenko V.P. 11.3
Petunin Yu.I. 1.6,1.9
Pfluger A. 11.4
Phelps R.R. 11.8
Pichorides S.K. 6.6
Pimsner M. 4.22
Pinchuk S.I. 7.3
Pinkus J.D. 4.34
Piranian G. 8.8
Pitt R. 2.0
Plamenevskii B.A. 8.20
PlemelJ J. 5.0
Plesner A.I. 4.0
Plotkin A.I. 1.6
Podoshev L.R. S.9
Polking J.C. 8.10,8.19
716
Polya G. 6.6,8.9,11.9
Pommerenke Ch. 6.11,6.12,6.18,8.19
Poreda S.J. 8.13
Potapov V.P. 4.16,4.17,4.18
Pousson H.R. 5.8
Power S.C. Ack.,5.2
Pranger W. 6.18
Privalov I.I. 5.0,9.6
Pr6ssdorf S. 4.20,4.30,5.14
Pt~k V. 2.2,2.3
Putnam C°R. 4.35,4,36,4,37
Rabindranathan M. 5.8
Rad6 T. 13.4
Raikov D.A. 2.11,7.17
Ransford T. 2.4
Rashevskii P.K. 7.4
Rathsfeld A. 5.14
Read C.J. S.3
Redheffer R.M. 8.4
Reshetihin N.Yu. 4.19
Retherford J.R. 1.3
Rieder D. 7.23
Riesz M. 5.0
Ritt 7.0
Roch S. 5.13
Rochberg R. 2.12,4.24,4.25,5.3,6.8
Rogers C.A. 8.18
Rogers J.T. S.2
Rolewicz S. 1.12
Romanov A.V. 13.7
Romberg B.W. 7.8
Rosenberg J. 4.22
Rose~iblatt M. 3.5
Rosenblum M. 4.12,5.5
Rosenthal H. 1.3
RothA. 8.7
Rovnyak J. 4.9
Rubio de Francia J.L. 6.9
Rubel L.A. 8.13
Rudin W. 2.7,2.10,7.14,7.23,8.13,
I0.8,12.2,S.i0
Rukshin S.E. i0.0
Ruston A.F. 4.28
Ryll J. S.10
Saak ~.M. 8.10
Sabitov I.H. 5.14
Sadosky C. 6.4,6.8
Saff E.B. 8.12
Saginashvili I.M. 5.9
Sahnovich L.A. 4.2,4.11,4.17,4.18
R.-Salinas B. 9.12
Sampson G. 6.4
Sapogov N.A. 3.7
Sarason D. Ack.,2.3,2.11,3.0,3.2, 4.14,4.24,4.33,5.1,5.4,5.5, 5.9,6.0,6.11,6.14,9.4,S.6
Sawyer E. 6.4
Schaeffer A.C. 9.5,11.7,13.7
Scheinberg S. 4.39
Scherer K. 8.22
Schmidt G. 5.14
Schubert C.F. 4.12
Schu~ I. 5.0
Schwartz L. 7.0,7.3,7.4,7.5,9.1, 9.10,I0.I
Sebasti~o-e-Silva J. 1.13,7.5
Sedaev A.A. 1.6
Seeley R.T. 4.30
Segovia C. S.II
Sem~nov E.M. 1.6,1.9
Sem~nov Yu.A. 4.1
Sem~nov-Tian-Shansky M.A. 5.16
Semeguk O.S. i.i0
Semmes S. 3.3,5.0,6.7
Shaginyan A.A. 8.7,8.8
Shamoyan F.A. Ack.,7.7,7.8,7.14
Shapiro H.S. 4.9,6.2,6.11,7.8, 7.9,9.3
Shelepov V.Yu. 6.2
Shields A.L. 4;9,4.24 ,5.1,6.2, 7.8,7.9,8.5,8.13,9.3
Shilov G.E. 1.12,2.11,7.17,7.21,
717
S.3
Shirokov N.A. 8.17,10.3,10.5
Shishkov A.E. 9.8
Shreider Yu.A. 2.6,2.7
Shteinberg A.M. 1.9
Shulman L. 4.8
Shulman V.S. 4.37
Shwartsman P.A. 10.7
Sibony N. 2.12,S.I0
Siddi~i J.A. Ack.,9.10
Silbermann B. 5.13
da Silva Dias C.L. 1.13
Simakova L.A. 4.16
Simon B. 4.1,4.31
Simonenko I.B. 5.0,5.8
Sinanyan S.O. 8.8,8.10,8.17,8.19
Sinclair A.M. 2.5
Singer I.M. 5.10
Sj6gren P. S.ll
Sj61in P. 1,7
Skiba N.I. 1.10
Skoda H. 7.0,S.i0
Skrypnik I.W. 8.20
Smyth M.R.F. 2.1,4.28
Sobolev S.L. 8.20
S6derberg D. 7.21
Sodin M.L. II.3,11.I0,S.9
Solev V.N. 3.2
Solomyak B.M. Ack.,4.9
Solomyak M.Z. 4.21,4.31
Spitkovskii I.M. 5.8,5.9,5.11,5.15
Spencer T. 4.1
Squires W.A. i0.i
Srivastav R.P. 5.14
Stanson C. 6.18
Stein E.M.4.24,6.9,6.10,6.17,8.21,S.11
Stephenson K. 5.4
Stieltjes T.J. 5.0
Stolzenberg G. 12.1
Stone M.H. 4.0
Strassen V. 3.6
Stray A. 6.9, 8.5
Stromberg J.-O. 1.7
Styf B. 7.19
Sucheston L. 4.25
Sudakov V.N. 3.6
Sundberg C. 3.2,5.1,5.7,6.19
Sunder V.S. 4.32
Suris E.L. 7.18
Sylvester J.J. 8.15
Szeg6 G. 5.0,5.8,5.11,5.12,6.9, 6.10
Szeptycki P. 8.21
Sz~efalvi-Nagy B. 3.5,4.10,4.11, 4.12,4.13,4.15,4.22,4.23,4.25, 5.4,5.5
Tairova V.G. S.8
TammM. S.10
Taylor B.A. 1.12, 7.4,7.15,7.16, I0.I
Taylor J.L. 2.0,2.6
Tchebysh~v P.L. 8.0
Teodorescu R. 4.13
Thomas M.P. 7.19,7.21
Tillman H.G. 1.13
Titchmarsh E.S. 11.2
Tkachenko V.A. Ack.,7.5,7.6,9.8
Tolokonnikov V.A. Ack.,4.12,5.0, 7.13
Tomassini G. 7.12
Tomaszewski B. S.10
Tonge A.M. 4.24
Toeplitz O. 5.0
Tortrat A. 3.4
Trant T. 6.19
Treil' S.R. 5.13
Trent T. 8.3
Triebel H. 4.31,8.1
Trubowitz E. 4.3
Truhil'o R. 4.30
Trutnev V.M. 1.13,1.14,7.3,S.2
Trutt D. 2.9,8.3
Tsirel'son B.S. 1.9
718
Tugores F.
Tumanov A.E.
Tumarkin G.C.
Tzafriri L.
10.4
12.3
4.9,6.2
1.9
4.12,6.9,6.16
8.11
8.0
Uchiyama A.
Uchiyama S.
Uryson P.S.
Uy N. 8.15
Vainshtein F.V. 13.9
Valiron G. 7.0,11.4
Val'skii R.E. 4.36,8.16
Varfolomeev A.L. S.2
varga R.S. 8.12
Varopoulos N.Th. 4.24,7.0
Vasilevskii N.L. 4.30
Vasyunin V.I. 4.10,4.12,4.13,4.33, 10.2
Vecua I.N. 5.15
Vecua N.P. 5.15
Verbitskii I.~. Ack.,5.8,5.13,6.6, 6.15,10.3
Vershik A.M. 3.5,4.22
Vinogradov S.A. 7.11,10.0,10.2, 10.3
Virtanen K.I. 8.6
Vitushkin A.G. 4.36,8.0,8.6,8.10, 8.14,8.15,8.16,8.17,8.19,S.2
Vladimirov V.S. 5.12
Vogt D. I.I0,i.ii
Voichick M. 6.18
Voiculesku D. 4.22,4.34,4.38
Vol'berg A.L. Ack.,8.3,9.11,S.ll
Volovich I.V. 5.12
de Vote R. 8.22
Vretblad A. 7.17
Waelbroeck L. 7.1
van der Waerden B.L. S.8
Wagner M.J. 1.10, 1.11
Wallen L.J. 7.7
Wallin H. Ack.,8.12,10.4,10.7,S.11
Walsh J.L. 8.11
Wang D. 5.4
Weis L. 1.2
Weiss G. 4.25,5.3,6.0,6.8,S.ii
Wells J. 7.15
Wells R.O. 8.14
Wendland W.L. 5.14
Wermer J. 2.12,8.14,12.1
West T.T. 2.1,4.28,S.3
Weyl H. 4.32,S.8
Wheeden R.L. 5.8,6.9,6.10
Whitney H. 10.0,10.7
widom H. 1.10,4.21,5.6,5.7,6.18
Wiener N. 2.0,3.0,7.17,8.20
Williams J.P. 4.37,4.38
Williams D.L. 7.15,7.16
Williamson J. 11.9
Wilson R. 11.5
Wojtaszczyk P. 1.4,1.5,1.7,2.10, S.10
Wold H. 4.0
Wolff T. 3.2,3.3,4.12,5.5,6.0, 6.9,6.10,6.11,7.13,8.1,8.10, 8.21,8.22,S.6
Yaglom A.M. 3.2
Yang Ho 4.37
Yosida K. 9.1
Young N.J. 2.2,2.3
Yuen Y. 2.4
Yuzhakov A.P. 1.13
Zaidenberg M.G. 1.6,13.8,13.10
Zafran M. 10.8
Zaharevich M.I. 4.22
Zaharyuta V.P. Ack.,l.10,1.11
Zalcman L. 4.36,8.6,8.15,8.17
Zame W.R. S.2
Zelazko W. 2.8
Zelinskii Yu.B. 1.13
719
Zem~nek J. Ack.,2.4
Zerner 1.11
Ziemer W.P. 8.20
Znamenskii S.V. Ack.,l.13
Zoretti L. 8.16
Zverovich E.I. 5.15
Zygmund A. 1.4,6.11,6.16,9.6,13.7
Zyuzin Yu.V. 13.8
STANDARD NOTATION
Symbols ~,~,~, ~ denote respectively the set of positive integers,
the set of all integers, the real line, and the complex plane.
stands f o r the one-po in t compac t i f i ca t i on of ~ 1~ denotes the
normed Lebesgue measure on T (~(T) ~ 1 )~1 X is the restriction of
a mapping (function) ~ to X. G~(-) is the closure of the set (-) .
V (') is the closed span of the set (°) in a linear topological
space.
I TI denotes the norm of the operator T .
~(.) denotes the sequence of Fourier coefficients of ~ .
~ denotes the Pourier transform of ~ .
~p is a class of operators ~ on a Hilbert space satisfying
, i.e. the space of all holomorphic
trace ( ,A* A ) I~Iz< + ' ~ •
H P is the ~ar~ class in D
functions on ~ with
II*llp ( .[ ,,Ip ~.2 ; ) ) < + = , p; ,O. 0<'I.< 1 T
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