Letter to the EditorComment on (On the Carleman Classes of Vectors ofa Scalar Type Spectral Operator)
Marat V Markin
Department of Mathematics California State University Fresno 5245 N Backer Avenue MS PB 108 Fresno CA 93740-8001 USA
Correspondence should be addressed to Marat V Markin mmarkincsufresnoedu
Received 4 July 2017 Accepted 9 October 2017 Published 1 January 2018
Academic Editor Yuri Latushkin
Copyright copy 2018 Marat V Markin This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The results of three papers in which the author inadvertently overlooks certain deficiencies in the descriptions of the Carlemanclasses of vectors in particular the Gevrey classes of a scalar type spectral operator in a complex Banach space established in ldquoOnthe Carleman Classes of Vectors of a Scalar Type Spectral Operatorrdquo Int J Math Math Sci 2004 (2004) no 60 3219ndash3235 areobserved to remain true due to more recent findings
1 Introduction
Certain deficiencies of the descriptions (established in [1])of the Carleman classes of vectors in particular the Gevreyclasses of a scalar type spectral operator in a complex Banachspace inadvertently overlooked by the author when provingthe results of the three papers [2ndash4] are observed not to affectthe validity of the latter due to more recent findings of [5]
2 Preliminaries
For the readerrsquos convenience we outline in this section certainpreliminaries essential for understanding
Henceforth unless specified otherwise 119860 is supposed tobe a scalar type spectral operator in a complex Banach space(119883 sdot ) and 119864119860(sdot) is supposed to be its strongly 120590-additivespectral measure (the resolution of the identity) assigning toeach Borel set 120575 of the complex plane C a projection operator119864119860(120575) on 119883 and having the operatorrsquos spectrum 120590(119860) as itssupport [6 7]
Observe that in a complex finite-dimensional space thescalar type spectral operators are those linear operators on thespace for which there is an eigenbasis (see eg [6 7]) and ina complexHilbert space the scalar type spectral operators areprecisely those that are similar to the normal ones [8]
Associated with a scalar type spectral operator in a com-plex Banach space is the Borel operational calculus analogous
to that for a normal operator in a complex Hilbert space[6 7 9 10] which assigns to any Borel measurable function119865 120590(119860) rarr C a scalar type spectral operator
119863(sdot) is the domain of an operator 120594120575(sdot) is the characteristicfunction of a set 120575 sube C and N fl 1 2 3 is the set ofnatural numbers and
119865119899 (119860) fl int120590(119860)
119865119899 (120582) 119889119864119860 (120582) 119899 isin N (4)
are bounded scalar type spectral operators on119883 defined in thesame manner as for a normal operator (see eg [9 10])
HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2018 Article ID 2135740 3 pageshttpsdoiorg10115520182135740
2 International Journal of Mathematics and Mathematical Sciences
Z+ fl 0 1 2 is the set of nonnegative integers 1198600 fl 119868and 119868 is the identity operator on119883 and
119890119911119860 fl int120590(119860)
119890119911120582119889119864119860 (120582) 119911 isin C (6)
The properties of the spectral measure and operational calcu-lus are exhaustively delineated in [6 7]
For a densely defined closed linear operator (119860119863(119860)) in a(real or complex) Banach space (119883 sdot) a sequence of positivenumbers 119898119899infin119899=0 and the subspace
le 119888120572119899119898119899 119899 isin Z+ (8)
are called the Carleman classes of ultradifferentiable vectorsof the operator 119860 corresponding to the sequence 119898119899infin119899=0 ofRoumieu and Beurling type respectively
The inclusions119862(119898119899) (119860) sube 119862119898119899 (119860) sube 119862infin (119860) (9)
are obviousIf two sequences of positive numbers 119898119899infin119899=0 and 1198981015840119899infin119899=0
are the well-known Gevrey classes of ultradifferentiablevectors of 119860 of order 120573 of Roumieu and Beurling typerespectively (see eg [11ndash13]) In particular E1(119860) andE(1)(119860) are the classes of analytic and entire vectors of 119860respectively [14 15] andE0(119860) andE(0)(119860) (ie the classes1198621(119860) and 119862(1)(119860) corresponding to the sequence 119898119899 equiv1) are the classes of entire vectors of 119860 of exponential andminimal exponential type respectively (see eg [13 16])
3 Remarks
If the sequence of positive numbers 119898119899infin119899=0 satisfies thecondition
120582119899119898119899 120582 ge 0 (00 fl 1) (17)
first introduced by Mandelbrojt [17] is well defined (cf [13])The function is continuous strictly increasing and 119879(0) = 1
As is shown in [11] (see also [12 13]) the sequence 119898119899infin119899=0satisfying the condition (WGR) for a normal operator 119860 in acomplex Hilbert space119883 the equalities
are understood in the sense of the Borel operational calculus(see eg [9 10]) and the function119879(sdot) is replaceable with anynonnegative continuous and increasing on [0infin) function119865(sdot) satisfying
1198881119865 (1205741120582) le 119879 (120582) le 1198882119865 (1205742120582) 120582 ge 119877 (20)
with some 1205741 1205742 1198881 1198882 gt 0 and 119877 ge 0 in particular with
119878 (120582) fl 1198980 sup119899ge0
120582119899119898119899 120582 ge 0
or 119875 (120582) fl 1198980 [infinsum119899=0
12058221198991198982119899 ]12
120582 ge 0(21)
(cf [13])Notably when 119898119899 fl [119899]120573 (119898119899 fl 119899120573119899) with 120573 gt 0 the
corresponding function 119879(sdot) is replaceable with119865 (120582) = 1198901205821120573 120582 ge 0 (22)
(see [13] for details cf also [1]) and
E120573 (119860) = ⋃
119905gt0
119863(119890119905|119860|1120573)
E(120573) (119860) = ⋂
119905gt0
119863(119890119905|119860|1120573) (23)
International Journal of Mathematics and Mathematical Sciences 3
Observe that equalities (18) can be considered to be operatoranalogues of the classical Paley-Wiener Theorems relatingthe smoothness of the Fourier transform 119891(sdot) of a square-integrable on R function 119891(sdot) to its decay at plusmninfin [18]
In [1 Theorem 31] and [1 Corollary 41] equalities (18)and (23) are generalized to the case of a scalar type spectraloperator 119860 in a reflexive complex Banach space 119883 with thereflexivity requirement dropped the inclusions
are proven onlyIn the recent paper [5] the reflexivity requirement is
shown to be superfluous and the following statements areproven
Theorem 1 (see [5 Theorem 31]) Let 119898119899infin119899=0 be a sequenceof positive numbers satisfying the condition (WGR) given by(15) Then for a scalar type spectral operator 119860 in a complexBanach space (119883 sdot ) equalities (18) are true the scalar typespectral operators 119879(119905|119860|) 119905 gt 0 are understood in the senseof the Borel operational calculus and the function 119879(sdot) definedby (17) is replaceable with any nonnegative continuous andincreasing on [0infin) function 119865(sdot) satisfying condition (20)
Corollary 2 (see [5 Corollary 41]) Let 120573 gt 0 Then for ascalar type spectral operator119860 in a complex Banach space (119883 sdot) equalities (23) are true
In papers [2ndash4] written before [5] the deficiency ofinclusions (24) and (25) for the general case is inadvertentlyoverlooked by the author and wrong conclusions are drawnfrom them in the ldquoonly if rdquo parts of [2 Theorem 51] and [4Theorem 51] and the sufficiency of [3Theorem 31]Thus farthis circumstance leaving the statements effectively provenonly for reflexive spaces when by [1 Theorem 31] and [1Corollary 41] inclusions (24) and (25) turn into equalities(18) and (23) respectively also seems to have escaped theattention of the referees and the authors who have cited [2 3](see eg [19ndash21])
However the good news for all is that due to [5Theorem31] and [5 Corollary 41] inclusionsrsquo (24) and (25) beingactually equalities (18) and (23) respectively without therequirement of reflexivity readily amends the faulty logic inthe proofs of all the foregoing statements making them truefor an arbitrary complex Banach space
Conflicts of Interest
The author declares that there are no conflicts of interest
References
[1] M V Markin ldquoOn the Carleman classes of vectors of a scalartype spectral operatorrdquo International Journal of Mathematicsand Mathematical Sciences no 57-60 pp 3219ndash3235 2004
[2] M V Markin ldquoOn scalar type spectral operators infinitedifferentiable and Gevrey ultradifferentiable C0-semigroupsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2004 no 45 pp 2401ndash2422 2004
[3] M V Markin ldquoA characterization of the generators of analyticC0-semigroups in the class of scalar type spectral operatorsrdquoAbstract and Applied Analysis vol 2004 no 12 pp 1007ndash10182004
[4] M VMarkin ldquoOn scalar-type spectral operators and Carlemanultradifferentiable C0-semigroupsrdquo Ukrainian MathematicalJournal vol 60 no 9 pp 1418ndash1436 2008
[5] M V Markin ldquoOn the Carleman ultradifferentiable vectors ofa scalar type spectral operatorrdquoMethods of Functional Analysisand Topology vol 21 no 4 pp 361ndash369 2015
[6] N Dunford ldquoA survey of the theory of spectral operatorsrdquoBulletin (New Series) of the American Mathematical Society vol64 pp 217ndash274 1958
[7] N Dunford and J T Schwartz Linear Operators Part IIISpectral Operators Interscience Publishers New York NYUSA 1971
[8] J Wermer ldquoCommuting spectral measures on Hilbert spacerdquoPacific Journal of Mathematics vol 4 pp 355ndash361 1954
[9] N Dunford and J T Schwartz Linear Operators Part IISpectral Theory Self Adjoint Operators in Hilbert Space vol 5Interscience Publishers New York NY USA 1986
[10] A I Plesner Spectral Theory of Linear Operators NaukaMoscow Russia 1965 (Russian)
[11] V I Gorbachuk ldquoSpaces of infinitely differentiable vectors ofa nonnegative self-adjoint operatorrdquo Ukrainian MathematicalJournal vol 35 no 5 pp 531ndash534 1983
[12] V I Gorbachuk and M L Gorbachuk Boundary Value Prob-lems for Operator Differential Equations Mathematics and ItsApplications (Soviet Series) vol 48 Kluwer Academic Publish-ers Group Dordrecht Netherlands 1991
[13] V I Gorbachuk and A V Knyazyuk ldquoBoundary values of solu-tions of operator-differential equationsrdquo Russian MathematicalSurveys vol 44 no 3 pp 67ndash111 1989
[14] R W Goodman ldquoAnalytic and entire vectors for representa-tions of Lie groupsrdquo Transactions of the AmericanMathematicalSociety vol 143 pp 55ndash76 1969
[15] E Nelson ldquoAnalytic vectorsrdquo Annals of Mathematics SecondSeries vol 70 pp 572ndash615 1959
[16] Ya V Radyno ldquoThe space of vectors of exponential typerdquoDoklady Akademii Nauk BSSR vol 27 no 9 pp 791ndash793 1983(Russian)
[17] SMandelbrojt Series de Fourier et Classes Quasi-Analytiques deFonctions Gauthier-Villars Paris France 1935
[18] R E A Paley andNWiener Fourier Transforms in the ComplexDomain vol 19 of American Mathematical Society ColloquiumPublications American Mathematical Society New York NYUSA 1934
[19] M Kostic ldquoDifferential and analytical properties of semigroupsof operatorsrdquo Integral Equations and Operator Theory vol 67no 4 pp 499ndash557 2010
[20] M R Opmeer ldquoNuclearity of Hankel operators for ultradiffer-entiable control systemsrdquo Systems amp Control Letters vol 57 no11 pp 913ndash918 2008
[21] B Silvestri ldquoIntegral equalities for functions of unboundedspectral operators in Banach spacesrdquo Dissertationes Mathemat-icae no 464 pp 3ndash60 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
2 International Journal of Mathematics and Mathematical Sciences
Z+ fl 0 1 2 is the set of nonnegative integers 1198600 fl 119868and 119868 is the identity operator on119883 and
119890119911119860 fl int120590(119860)
119890119911120582119889119864119860 (120582) 119911 isin C (6)
The properties of the spectral measure and operational calcu-lus are exhaustively delineated in [6 7]
For a densely defined closed linear operator (119860119863(119860)) in a(real or complex) Banach space (119883 sdot) a sequence of positivenumbers 119898119899infin119899=0 and the subspace
le 119888120572119899119898119899 119899 isin Z+ (8)
are called the Carleman classes of ultradifferentiable vectorsof the operator 119860 corresponding to the sequence 119898119899infin119899=0 ofRoumieu and Beurling type respectively
The inclusions119862(119898119899) (119860) sube 119862119898119899 (119860) sube 119862infin (119860) (9)
are obviousIf two sequences of positive numbers 119898119899infin119899=0 and 1198981015840119899infin119899=0
are the well-known Gevrey classes of ultradifferentiablevectors of 119860 of order 120573 of Roumieu and Beurling typerespectively (see eg [11ndash13]) In particular E1(119860) andE(1)(119860) are the classes of analytic and entire vectors of 119860respectively [14 15] andE0(119860) andE(0)(119860) (ie the classes1198621(119860) and 119862(1)(119860) corresponding to the sequence 119898119899 equiv1) are the classes of entire vectors of 119860 of exponential andminimal exponential type respectively (see eg [13 16])
3 Remarks
If the sequence of positive numbers 119898119899infin119899=0 satisfies thecondition
120582119899119898119899 120582 ge 0 (00 fl 1) (17)
first introduced by Mandelbrojt [17] is well defined (cf [13])The function is continuous strictly increasing and 119879(0) = 1
As is shown in [11] (see also [12 13]) the sequence 119898119899infin119899=0satisfying the condition (WGR) for a normal operator 119860 in acomplex Hilbert space119883 the equalities
are understood in the sense of the Borel operational calculus(see eg [9 10]) and the function119879(sdot) is replaceable with anynonnegative continuous and increasing on [0infin) function119865(sdot) satisfying
1198881119865 (1205741120582) le 119879 (120582) le 1198882119865 (1205742120582) 120582 ge 119877 (20)
with some 1205741 1205742 1198881 1198882 gt 0 and 119877 ge 0 in particular with
119878 (120582) fl 1198980 sup119899ge0
120582119899119898119899 120582 ge 0
or 119875 (120582) fl 1198980 [infinsum119899=0
12058221198991198982119899 ]12
120582 ge 0(21)
(cf [13])Notably when 119898119899 fl [119899]120573 (119898119899 fl 119899120573119899) with 120573 gt 0 the
corresponding function 119879(sdot) is replaceable with119865 (120582) = 1198901205821120573 120582 ge 0 (22)
(see [13] for details cf also [1]) and
E120573 (119860) = ⋃
119905gt0
119863(119890119905|119860|1120573)
E(120573) (119860) = ⋂
119905gt0
119863(119890119905|119860|1120573) (23)
International Journal of Mathematics and Mathematical Sciences 3
Observe that equalities (18) can be considered to be operatoranalogues of the classical Paley-Wiener Theorems relatingthe smoothness of the Fourier transform 119891(sdot) of a square-integrable on R function 119891(sdot) to its decay at plusmninfin [18]
In [1 Theorem 31] and [1 Corollary 41] equalities (18)and (23) are generalized to the case of a scalar type spectraloperator 119860 in a reflexive complex Banach space 119883 with thereflexivity requirement dropped the inclusions
are proven onlyIn the recent paper [5] the reflexivity requirement is
shown to be superfluous and the following statements areproven
Theorem 1 (see [5 Theorem 31]) Let 119898119899infin119899=0 be a sequenceof positive numbers satisfying the condition (WGR) given by(15) Then for a scalar type spectral operator 119860 in a complexBanach space (119883 sdot ) equalities (18) are true the scalar typespectral operators 119879(119905|119860|) 119905 gt 0 are understood in the senseof the Borel operational calculus and the function 119879(sdot) definedby (17) is replaceable with any nonnegative continuous andincreasing on [0infin) function 119865(sdot) satisfying condition (20)
Corollary 2 (see [5 Corollary 41]) Let 120573 gt 0 Then for ascalar type spectral operator119860 in a complex Banach space (119883 sdot) equalities (23) are true
In papers [2ndash4] written before [5] the deficiency ofinclusions (24) and (25) for the general case is inadvertentlyoverlooked by the author and wrong conclusions are drawnfrom them in the ldquoonly if rdquo parts of [2 Theorem 51] and [4Theorem 51] and the sufficiency of [3Theorem 31]Thus farthis circumstance leaving the statements effectively provenonly for reflexive spaces when by [1 Theorem 31] and [1Corollary 41] inclusions (24) and (25) turn into equalities(18) and (23) respectively also seems to have escaped theattention of the referees and the authors who have cited [2 3](see eg [19ndash21])
However the good news for all is that due to [5Theorem31] and [5 Corollary 41] inclusionsrsquo (24) and (25) beingactually equalities (18) and (23) respectively without therequirement of reflexivity readily amends the faulty logic inthe proofs of all the foregoing statements making them truefor an arbitrary complex Banach space
Conflicts of Interest
The author declares that there are no conflicts of interest
References
[1] M V Markin ldquoOn the Carleman classes of vectors of a scalartype spectral operatorrdquo International Journal of Mathematicsand Mathematical Sciences no 57-60 pp 3219ndash3235 2004
[2] M V Markin ldquoOn scalar type spectral operators infinitedifferentiable and Gevrey ultradifferentiable C0-semigroupsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2004 no 45 pp 2401ndash2422 2004
[3] M V Markin ldquoA characterization of the generators of analyticC0-semigroups in the class of scalar type spectral operatorsrdquoAbstract and Applied Analysis vol 2004 no 12 pp 1007ndash10182004
[4] M VMarkin ldquoOn scalar-type spectral operators and Carlemanultradifferentiable C0-semigroupsrdquo Ukrainian MathematicalJournal vol 60 no 9 pp 1418ndash1436 2008
[5] M V Markin ldquoOn the Carleman ultradifferentiable vectors ofa scalar type spectral operatorrdquoMethods of Functional Analysisand Topology vol 21 no 4 pp 361ndash369 2015
[6] N Dunford ldquoA survey of the theory of spectral operatorsrdquoBulletin (New Series) of the American Mathematical Society vol64 pp 217ndash274 1958
[7] N Dunford and J T Schwartz Linear Operators Part IIISpectral Operators Interscience Publishers New York NYUSA 1971
[8] J Wermer ldquoCommuting spectral measures on Hilbert spacerdquoPacific Journal of Mathematics vol 4 pp 355ndash361 1954
[9] N Dunford and J T Schwartz Linear Operators Part IISpectral Theory Self Adjoint Operators in Hilbert Space vol 5Interscience Publishers New York NY USA 1986
[10] A I Plesner Spectral Theory of Linear Operators NaukaMoscow Russia 1965 (Russian)
[11] V I Gorbachuk ldquoSpaces of infinitely differentiable vectors ofa nonnegative self-adjoint operatorrdquo Ukrainian MathematicalJournal vol 35 no 5 pp 531ndash534 1983
[12] V I Gorbachuk and M L Gorbachuk Boundary Value Prob-lems for Operator Differential Equations Mathematics and ItsApplications (Soviet Series) vol 48 Kluwer Academic Publish-ers Group Dordrecht Netherlands 1991
[13] V I Gorbachuk and A V Knyazyuk ldquoBoundary values of solu-tions of operator-differential equationsrdquo Russian MathematicalSurveys vol 44 no 3 pp 67ndash111 1989
[14] R W Goodman ldquoAnalytic and entire vectors for representa-tions of Lie groupsrdquo Transactions of the AmericanMathematicalSociety vol 143 pp 55ndash76 1969
[15] E Nelson ldquoAnalytic vectorsrdquo Annals of Mathematics SecondSeries vol 70 pp 572ndash615 1959
[16] Ya V Radyno ldquoThe space of vectors of exponential typerdquoDoklady Akademii Nauk BSSR vol 27 no 9 pp 791ndash793 1983(Russian)
[17] SMandelbrojt Series de Fourier et Classes Quasi-Analytiques deFonctions Gauthier-Villars Paris France 1935
[18] R E A Paley andNWiener Fourier Transforms in the ComplexDomain vol 19 of American Mathematical Society ColloquiumPublications American Mathematical Society New York NYUSA 1934
[19] M Kostic ldquoDifferential and analytical properties of semigroupsof operatorsrdquo Integral Equations and Operator Theory vol 67no 4 pp 499ndash557 2010
[20] M R Opmeer ldquoNuclearity of Hankel operators for ultradiffer-entiable control systemsrdquo Systems amp Control Letters vol 57 no11 pp 913ndash918 2008
[21] B Silvestri ldquoIntegral equalities for functions of unboundedspectral operators in Banach spacesrdquo Dissertationes Mathemat-icae no 464 pp 3ndash60 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
International Journal of Mathematics and Mathematical Sciences 3
Observe that equalities (18) can be considered to be operatoranalogues of the classical Paley-Wiener Theorems relatingthe smoothness of the Fourier transform 119891(sdot) of a square-integrable on R function 119891(sdot) to its decay at plusmninfin [18]
In [1 Theorem 31] and [1 Corollary 41] equalities (18)and (23) are generalized to the case of a scalar type spectraloperator 119860 in a reflexive complex Banach space 119883 with thereflexivity requirement dropped the inclusions
are proven onlyIn the recent paper [5] the reflexivity requirement is
shown to be superfluous and the following statements areproven
Theorem 1 (see [5 Theorem 31]) Let 119898119899infin119899=0 be a sequenceof positive numbers satisfying the condition (WGR) given by(15) Then for a scalar type spectral operator 119860 in a complexBanach space (119883 sdot ) equalities (18) are true the scalar typespectral operators 119879(119905|119860|) 119905 gt 0 are understood in the senseof the Borel operational calculus and the function 119879(sdot) definedby (17) is replaceable with any nonnegative continuous andincreasing on [0infin) function 119865(sdot) satisfying condition (20)
Corollary 2 (see [5 Corollary 41]) Let 120573 gt 0 Then for ascalar type spectral operator119860 in a complex Banach space (119883 sdot) equalities (23) are true
In papers [2ndash4] written before [5] the deficiency ofinclusions (24) and (25) for the general case is inadvertentlyoverlooked by the author and wrong conclusions are drawnfrom them in the ldquoonly if rdquo parts of [2 Theorem 51] and [4Theorem 51] and the sufficiency of [3Theorem 31]Thus farthis circumstance leaving the statements effectively provenonly for reflexive spaces when by [1 Theorem 31] and [1Corollary 41] inclusions (24) and (25) turn into equalities(18) and (23) respectively also seems to have escaped theattention of the referees and the authors who have cited [2 3](see eg [19ndash21])
However the good news for all is that due to [5Theorem31] and [5 Corollary 41] inclusionsrsquo (24) and (25) beingactually equalities (18) and (23) respectively without therequirement of reflexivity readily amends the faulty logic inthe proofs of all the foregoing statements making them truefor an arbitrary complex Banach space
Conflicts of Interest
The author declares that there are no conflicts of interest
References
[1] M V Markin ldquoOn the Carleman classes of vectors of a scalartype spectral operatorrdquo International Journal of Mathematicsand Mathematical Sciences no 57-60 pp 3219ndash3235 2004
[2] M V Markin ldquoOn scalar type spectral operators infinitedifferentiable and Gevrey ultradifferentiable C0-semigroupsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2004 no 45 pp 2401ndash2422 2004
[3] M V Markin ldquoA characterization of the generators of analyticC0-semigroups in the class of scalar type spectral operatorsrdquoAbstract and Applied Analysis vol 2004 no 12 pp 1007ndash10182004
[4] M VMarkin ldquoOn scalar-type spectral operators and Carlemanultradifferentiable C0-semigroupsrdquo Ukrainian MathematicalJournal vol 60 no 9 pp 1418ndash1436 2008
[5] M V Markin ldquoOn the Carleman ultradifferentiable vectors ofa scalar type spectral operatorrdquoMethods of Functional Analysisand Topology vol 21 no 4 pp 361ndash369 2015
[6] N Dunford ldquoA survey of the theory of spectral operatorsrdquoBulletin (New Series) of the American Mathematical Society vol64 pp 217ndash274 1958
[7] N Dunford and J T Schwartz Linear Operators Part IIISpectral Operators Interscience Publishers New York NYUSA 1971
[8] J Wermer ldquoCommuting spectral measures on Hilbert spacerdquoPacific Journal of Mathematics vol 4 pp 355ndash361 1954
[9] N Dunford and J T Schwartz Linear Operators Part IISpectral Theory Self Adjoint Operators in Hilbert Space vol 5Interscience Publishers New York NY USA 1986
[10] A I Plesner Spectral Theory of Linear Operators NaukaMoscow Russia 1965 (Russian)
[11] V I Gorbachuk ldquoSpaces of infinitely differentiable vectors ofa nonnegative self-adjoint operatorrdquo Ukrainian MathematicalJournal vol 35 no 5 pp 531ndash534 1983
[12] V I Gorbachuk and M L Gorbachuk Boundary Value Prob-lems for Operator Differential Equations Mathematics and ItsApplications (Soviet Series) vol 48 Kluwer Academic Publish-ers Group Dordrecht Netherlands 1991
[13] V I Gorbachuk and A V Knyazyuk ldquoBoundary values of solu-tions of operator-differential equationsrdquo Russian MathematicalSurveys vol 44 no 3 pp 67ndash111 1989
[14] R W Goodman ldquoAnalytic and entire vectors for representa-tions of Lie groupsrdquo Transactions of the AmericanMathematicalSociety vol 143 pp 55ndash76 1969
[15] E Nelson ldquoAnalytic vectorsrdquo Annals of Mathematics SecondSeries vol 70 pp 572ndash615 1959
[16] Ya V Radyno ldquoThe space of vectors of exponential typerdquoDoklady Akademii Nauk BSSR vol 27 no 9 pp 791ndash793 1983(Russian)
[17] SMandelbrojt Series de Fourier et Classes Quasi-Analytiques deFonctions Gauthier-Villars Paris France 1935
[18] R E A Paley andNWiener Fourier Transforms in the ComplexDomain vol 19 of American Mathematical Society ColloquiumPublications American Mathematical Society New York NYUSA 1934
[19] M Kostic ldquoDifferential and analytical properties of semigroupsof operatorsrdquo Integral Equations and Operator Theory vol 67no 4 pp 499ndash557 2010
[20] M R Opmeer ldquoNuclearity of Hankel operators for ultradiffer-entiable control systemsrdquo Systems amp Control Letters vol 57 no11 pp 913ndash918 2008
[21] B Silvestri ldquoIntegral equalities for functions of unboundedspectral operators in Banach spacesrdquo Dissertationes Mathemat-icae no 464 pp 3ndash60 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
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