Introduction to Banach Algebras, Operators, and Harmonic Analysis

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This work has arisen from lecture courses given by the authors on impor-tant topics within functional analysis. The authors, who are all leading re-searchers, give introductions to their subjects at a level ideal for beginninggraduate students, as well as others interested in the subject. The collection hasbeen carefully edited to form a coherent and accessible introduction to currentresearch topics.

The first part of the book, by Professor Dales, introduces the general theoryof Banach algebras, which serves as a background to the remaining material.DrWillis then studies a centrally important Banach algebra, the group algebra ofa locally compact group. The remaining chapters are devoted toBanach algebrasof operators on Banach spaces: Professor Eschmeier gives all the backgroundfor the exciting topic of invariant subspaces of operators, and discusses somekey open problems; Dr Laursen and Professor Aiena discuss local spectraltheory for operators, leading into Fredholm theory.

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3 Local fields, J. W. S. CASSELS4 An introduction to twister theory, second edition, S. A. HUGGET & K. P. TOD5 Introduction to general relativity, L. P. HUGHSTON & K. P. TOD7 The theory of evolution and dynamical systems, J. HOFBAUER & K. SIGMUND8 Summing and nuclear norms in Banach space theory, G. J. O. JAMESON9 Automorphisms of surfaces after Nielson and Thurston, A. CASSON & S. BLEILER11 Spacetime and singularities, G. NABER12 Undergraduate algebraic geometry, M. REID13 An Introduction to Hankel operators, J. R. PARTINGTON15 Presentations of groups, second edition, D. L. JOHNSON17 Aspects of quantum field theory in curved spacetime, S. A. FULLING18 Braids and coverings: Selected topics, V. LUNDSGAARD HANSEN19 Steps in commutative algebra, R. Y. SHARP20 Communication theory, C. M. GOLDIE & R. G. E. PINCH21 Representations of finite groups of Lie type, P. DIGNE & J. MICHEL22 Designs, graphs, codes, and their links, P. J. CAMERON & J. H. VAN LINT23 Complex algebraic curves, F. KIRWAN24 Lectures on elliptic curves, J. W. S. CASSELS25 Hyperbolic geometry, B. IVERSEN26 An introduction to the theory of L-functions and Eisenstein series, H, HIDA27 Hilbert space: Compact operators and the trace theorem, J. R. RETHERFORD28 Potential theory in the complex lane, T. RANSFORD29 Undergraduate cummutative algebra, M. REID31 The Laplacian on a Rlemannian manifold, S. ROSENBERG32 Lectures on Lie groups and Lie algebras, R. CARTER, G. SEGAL & J. MACDONALD33 A primer of algebraic D-modules, S. C. COUTINHO34 Complex algebraic surfaces, A. BEAUVILLE35 Young tableaux, W. FULTON37 A mathematical introduction to wavelets, P. WOJTASZCZYK38 Harmonic maps, loop groups and integrable systems, M. GUEST39 Set theory for the working mathematician, K. CIESIELSKI40 Ergodic theory and dynamical systems, M. POLLICOTT & M. YURI41 The algorithmic resoution of diophantine equations, N. P. SMART42 Equlibrium states in ergodic theory, G. KELLER43 Fourier analysis on finite groups and applications, A. TERRAS44 Classical invariant theory, P. J. OLVER45 Purmutation groups, P. J. CAMERON46 Rlemann surfaces: A primer, A. BEARDON47 Introductory lectures on rings and modules, J. BEACHY48 Set theory, A. HAJNAL & P. HAMBURGER49 An introduction to K-theory for C∗-algebras, M. RØRDAM, F. LARSEN &N. J LAUSTSEN

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INTRODUCTION TO BANACHALGEBRAS, OPERATORS, AND

HARMONIC ANALYSIS

H. GARTH DALESUniversity of Leeds, UK

PIETRO AIENAUniversita degli Studi, Palermo, Italy

JORG ESCHMEIERUniversitat des Saarlandes, Saarbrucken

KJELD LAURSENUniversity of Copenhagen, Denmark

GEORGE A. WILLISUniversity of Newcastle, New South Wales, Australia

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University PressThe Edinburgh Building, Cambridge , United Kingdom

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Contents

Preface page viiPart I Banach algebras H. Garth Dales 1

1 Definitions and examples 32 Ideals and the spectrum 123 Gelfand theory 204 The functional calculus 305 Automatic continuity of homomorphisms 386 Modules and derivatives 487 Cohomology 58

Part II Harmonic analysis and amenability George A. Willis 738 Locally compact groups 759 Group algebras and representations 8610 Convolution operators 9811 Amenable groups 10912 Harmonic analysis and automatic continuity 121

Part III Invariant subspaces Jorg Eschmeier 13513 Compact operators 13714 Unitary dilations and the H∞-functional class 14315 Hyperinvariant subspaces 15416 Invariant subspaces for contractions 16017 Invariant subspaces for subnormal operators 16618 Invariant subspaces for subdecomposable operators 17119 Reflexivity of operator algebras 17820 Invariant subspaces for commuting contractions 186

Appendix to Part III 193Part IV Local spectral theory Kjeld Bagger Laursen 199

21 Basic notions from operator theory 20122 Classes of decomposable operators 212

v

vi Contents

23 Duality theory 22624 Preservation of spectra and index 23025 Multipliers on commutative Banach algebras 241

Appendix to Part IV 254Part V Single-valued extension property and Fredholm

theory Pietro Aiena 26526 Semi-regular operators 26727 The single-valued extension property 28528 SVEP for semi-Fredholm operators 298

Index of symbols 319Subject index 321

Preface

This volume is based on a collection of lectures intended for graduate studentsand others with a basic knowledge of functional analysis. It surveys severalareas of current research interest, and is designed to be suitable preparatoryreading for those embarkingongraduatework.Thevolumeconsists of fiveparts,which are based on separate sets of lectures, each by different authors. Each partprovides an overview of the subject that will also be useful to mathematiciansworking in related areas. The chapters were originally presented as lectures atinstructional conferences for graduate students, and we have maintained thestyles of these lectures.The sets of lectures are an introduction to their subjects, intended to convey

the flavour of certain topics, and to give some basic definitions and motivatingexamples: they are certainly not comprehensive accounts. References are givento sources in the literature where more details can be found.The chapters in Part I are by H. G. Dales. These are an introduction to the

general theory of Banach algebras, and a description of the most importantexamples: B(E), the algebra of all bounded linear operators on a Banach spaceE ; L 1(G), the group algebra of a locally compact group G, taken with theconvolution product; commutative Banach algebras, including Banach algebrasof functions on compact sets in C and radical Banach algebras. Chapters 3–6cover Gelfand theory for commutative Banach algebras, the analytic functionalcalculus, and, in a chapter on ‘automatic continuity’, the lovely results that showthe intimate connection between the algebraic and topological structures of aBanach algebra. Chapters 6 and 7 are an introduction to the cohomology theoryof Banach algebras, at present a very active area of research; we concentrate onthe basic structure, that of derivations into modules.The chapters in Part II, by G. A. Willis, develop the theory of one of the

examples discussed by Dales: these are the group algebras L 1(G). Chapters 8and 9 give a description of locally compact groups G and their structure theory,

vii

viii Preface

and then describe the algebras L 1(G), and the related measure algebra M(G),as a Banach algebra. The Gelfand theory for general commutative Banachalgebras, as described byDales, becomes Fourier transform theory in the specialcase of the algebras L 1(G). In Chapter 10, Willis discusses compact groups,abelian groups, and free groups, and then, in Chapter 11, moves to a veryimportant class, that of amenable groups:many characterizations of amenabilityarise in diverse areas of mathematics. Willis then expands a notion from Part Iby discussing the automatic continuity of linear maps from group algebras.Parts III–V of this book develop the theory of another example mentioned

by Dales: this is the algebra B(E) for a Banach space E . However, they alsoconcentrate on the properties of single operators of various types within B(E).A seminal question in functional analysis is the ‘invariant subspace problem’.

Let E be a Banach space, and let T ∈ B(E). A closed subspace F of E isinvariant for T if T y ∈ F (y ∈ F) ; F is trivial if F = {0} or F = E . Doessuch an operator T always have a non-trivial invariant subspace? A positiveanswer to this question in the case where E is finite-dimensional (of dimensionat least 2) is the first step in the structure theory of matrices. The question forBanach spaces has been the spur for a huge amount of research in operatortheory since the question was first raised in the 1930s. The question is still openin the case where E is a Hilbert space – this is one of the great problems ofour subject – but counter-examples are known when E is an arbitrary Banachspace. Nevertheless, there are many positive results for operators T ∈ B(E)which belong to a special class.The chapters in Part III, by J. Eschmeier, discuss in particular one very impor-

tant technique for establishing positive results: it descends from original workof Scott Brown in 1978. One class of operators considered is that of subdecom-posable operators. Part III concludes with remarks about the extensions, mainlydue to the author, of the positive results to n-tuples of commuting operators.As explained by Dales, every element a of a Banach algebra has a spectrum,

called σ (a); this is a non-empty, compact subset of the complex plane C. Inparticular, each operator T ∈ B(E) has such a spectrum,σ (T ). In the casewhereE is finite-dimensional, σ (T ) is just the set of eigenvalues of T . The notion ofthe spectrum for a general operator T is at the heart of the remaining chapters,by K. B. Laursen and P. Aiena.Laursen discusses the spectral theory of operators in several different classes;

these include in particular the decomposable operators, which were also intro-duced byEschmeier.Weunderstand the nature of an operator T by looking at thedecomposition of σ (T ) into subsets with special properties and also at specialclosed subspaces of E on which T acts ‘in a nice way’. In particular, Laursen

Preface ix

discusses super-decomposable and generalized scalar operators. In Chapter 25,Laursen relates his description to notions introduced by Dales by discussingwhen multiplication operators on commutative Banach algebras have the var-ious properties that he has introduced. A valuable appendix to Part IVsketches the background theory, involving distributions, to the ‘functionalmodel’ of Albrecht and Eschmeier that is the natural setting for many of theduality results that have been obtained.The final chapters, those of P. Aiena, are closely related to those of

Laursen. The basic examples of the decomposable operators of Eschmeier’sand Laursen’s chapters are compact operators on a Banach space and normaloperators on a Hilbert space. It is natural to study the decomposable operatorswhich have similar properties to those of these important specific examples: weare led to the class of ‘Fredholm operators’ and related classes, a main topic ofAiena’s lectures.In these chapters, we see again, from a different perspective, some of the

key ideas – decomposition of the spectrum, invariant subspace, single-valuedextension property, actions of analytic functions, divisible subspaces – thathave featured in earlier chapters. The final chapter by Aiena summarizes recentwork of the author and others.

The lectures on which this book is based were given at two conferences. Thefirst was held in Mussomeli, Sicily, from 22 to 29 September 1999. We are verygrateful to Dr Gianluigi Oliveri, who organized this conference, and to the As-sociazone Culturale Archimede of Sicily who sponsored it. The lectures givenat this conference were those of Dales, Eschmeier, Laursen, and Aiena. Thesecond conference was held at the Sadar Patel University, Vallabh Vidyanagar,Gujarat, India, from 8 to 15 January 2002. We are very grateful to ProfessorSubhashbhai Bhatt and Dr Haresh Dedania for organizing this conference, andto the Indian Board for Higher Mathematics and to the London MathematicalSociety, who supported the conference financially. The lectures given inGujaratwere those of Dales, Willis, and Laursen.As we said, the original lectures were intended for graduate students and

others with a basic knowledge of functional analysis and with a background incomplex analysis and algebra typical of a first degree in mathematics. In bothcases the students were enthusiastic and helpful; their suggestions led to manyimprovements in the exposition, and we are grateful to them for this.In fact, the actual lectures as given did not include all that iswritten downhere:

modest additions have been made subsequently. There is more in a ‘lecture’than can easily be absorbed in one hour. However, we havemaintained the fairlyinformal style of the lecture theatre. At various points, the reader is invited to

x Preface

check statements that are made: these are all routine, and follow in a few linesfrom facts given in the lectures. There are also exercises at the end of eachlecture; the answers to all the exercises are contained in the references thatare specified. We hope that readers will work through the exercises as a steptowards the gaining of familiarity with the subject.There are various cross-references between the sets of lectures; indeed topics

of one set of lectures often reappear, perhaps in a different guise, in otherlectures. All the book depends somewhat on the first six chapters, and Part Vfollows from Part IV. However, otherwise the various parts of the book can beread independently. The references to each set of lectures are contained at theend of the relevant part of this book, and not at the end of the whole book.However, there are two indices for the whole book at the end (pp. 319–326):these are the symbol index and the index of terms.Some attempt has been made to make the notation consistent between the

various sets of lectures, but we have not always achieved this; we give a resumeof some standard notations at the end of this preface.We very much enjoyed giving the original lectures and discussing the theory

and associated examples in the classes that were given in the same week as thelectures.We hope that you enjoy reading them and, especially, working throughthe examples.In rather more detail, we expect the reader to be familiar with the following

topics:

� the definition of a Banach space and a locally convex space, weak topologieson dual spaces;

� standard theorems of functional analysis such as the Hahn–Banach theorem,closed graph theorem, open mapping theorem, and uniform boundednesstheorem;

� the theory of bounded linear operators on a Banach space, duals of suchoperators, compact operators;

� the elementary theory of Hilbert spaces;� undergraduate complex analysis, including Liouville’s theorem;� undergraduate algebra, including the theory of ideals, modules, and homo-morphisms.

Throughout we adopt the following notation:N = {1, 2, 3, . . .};Z = {0,±1,±2, . . .};Z+ = {0, 1, 2, . . .};Q is the field of rational numbers;

Preface xi

R is the field of real numbers;C is the field of complex numbers;I = [0, 1];T = {z ∈ C : |z| = 1};D (z; r ) = {w ∈ C : |w − z| < r};D = D(0; 1);Z is the coordinate functional on C, or on a subset of C;E ′ is the dual space of a topological linear space E ;E[1] is the closed unit ball of a Banach space E ;[x, y] is the inner product of x, y ∈ H , where H is a Hilbert space;〈x, λ〉 is the action of λ ∈ E ′ on x ∈ E , where E is a Banach space.

H. G. D., Leeds

Part I

Banach algebras

H. GARTH DALESUniversity of Leeds, UK

1

Definitions and examples

1.1 Definitions

A Banach algebra is first of all an algebra. We start with an algebra A andput a topology on A to make the algebraic operations continuous – in fact, thetopology is given by a norm.

Definition 1.1.1 Let E be a linear space. A norm on E is a map ‖ · ‖ : E → R

such that:

(i) ‖x‖ ≥ 0 (x ∈ E) ; ‖x‖ = 0 if and only if x = 0 ;(ii) ‖αx‖ = |α| ‖x‖ (α ∈ C, x ∈ E) ;(iii) ‖x + y‖ ≤ ‖x‖ + ‖y‖ (x, y ∈ E) .Then (E, ‖ · ‖) is a normed space. It is aBanach space if every Cauchy sequenceconverges, i.e., if ‖ · ‖ is complete.

Definition 1.1.2 Let A be an algebra. An algebra norm on A is a map ‖ · ‖ :A→ R such that (A, ‖ · ‖) is a normed space, and, further:(iv) ‖ab‖ ≤ ‖a‖ ‖b‖ (a, b ∈ A) .The normed algebra (A, ‖ · ‖) is a Banach algebra if ‖ · ‖ is a complete norm.

In Chapters 1–7, we shall usually suppose that a Banach algebra A is unital:this means that A has an identity eA and that ‖eA‖ = 1. Let A be a Banachalgebra with identity. Then, by moving to an equivalent norm, we may supposethat A is unital. It is easy to check that, for each normed algebra A, the map(a, b) �→ ab, A × A→ A, is continuous.

H. G. Dales, P. Aiena, J. Eschmeier, K. B. Laursen, and G. A. Willis, Introduction to BanachAlgebras, Operators, and Harmonic Analysis. Published by Cambridge University Press.c© Cambridge University Press 2003.

3

4 Part I Banach algebras, H. Garth Dales

1.2 Examples

Let us give some elementary examples.(i) Let S be any non-empty set. Then CS is the set of functions from S into

C. Define pointwise algebraic operations by

(α f + βg)(s) = α f (s)+ βg(s) ,( f g)(s) = f (s)g(s) ,

1(s) = 1 ,

for each s ∈ S, each f, g ∈ CS , and each α, β ∈ C. Then CS is a commutative,unital algebra. We write ∞(S) for the subset of bounded functions on S, anddefine the uniform norm | · |S on S by

| f |S = sup{| f (s)| : s ∈ S} ( f ∈ ∞(S)) .

Check that (∞(S), | · |S) is a unital Banach algebra.(ii) Let X be a topological space (e.g., think of X = R). We write C(X )

for the algebra of all continuous functions on X , and Cb(X ) for the algebraof bounded, continuous functions on X . Check that (Cb(X ), | · |X ) is a unitalBanach algebra.Now take to be a compact space (e.g., = I = [0, 1]). Then we have

Cb() = C(), and so (C(), | · |) is a unital Banach algebra. This is a veryimportant example.(iii) Let D = {z ∈ C : |z| < 1}, the open unit disc. The disc algebra is

A(D) = { f ∈ C(D) : f is analytic on D} .

Check that A(D) is a unital Banach algebra. (You just have to show that A(D)is closed in C(D): why is this?)Each f ∈ A(D) has a Taylor expansion about the origin:

f =∞∑n=0

αn Zn =

∞∑n=0

f (n)(0)Zn

n!.

Here Z is the coordinate functional, so that Z : z �→ z on C. Some functionsin A(D) have the further property that

∞∑n=0|αn| <∞ .

1 Definitions and examples 5

(Are there any functions f in A(D) without this property?) The subset of func-tions with this extra property is called A+(D). Check that A+(D) is a unitalBanach algebra for the norm ‖ · ‖1, where

‖ f ‖1 =∞∑n=0|αn|

(f =

∞∑n=0

αn Zn

).

(iv) Let X be a compact set in the spaceCn . Then P(X ) is the family of func-tions that are the uniform limits on X of the restrictions to X of the polynomials(in n-variables). Check that (P(X ), | · |X ) is a unital Banach algebra. In fact,A(D) = P(D).We shall also be interested in P(T),whereT = {z ∈ C : |z| = 1}is the unit circle.(v) Let X be a compact set in the complex planeC (or inCn). Then A(X ) is the

closed subalgebra of (C(X ), | · |X ) consisting of the functionswhich are analyticon the interior of X, int X . Clearly A(X ) = C(X ) if and only if int X = ∅. AlsoR(X ) is the family of functions on X which are the uniform limits on X of therestrictions to X of the rational functions: these are functions of the form p/q,where p and q are polynomials and 0 /∈ q(X ). Clearly we have

P(X ) ⊂ R(X ) ⊂ A(X ) ⊂ C(X ) .

The question of the equality of various of these algebras encapsulates much ofthe classical theory of approximation.(vi) Let n ∈ N. Then C (n)(I) consists of the functions f on I such that f has

n derivatives on I and f (n) ∈ C(I). Check that C (n)(I) is a Banach algebra forthe pointwise operations and the norm

‖ f ‖n =n∑k=0

1

k!| f (k)|I ( f ∈ C (n)(I)) .

(vii) Let E and F be linear spaces. Then L(E, F) is the collection of alllinear maps from E to F ; it is itself a linear space for the standard operations.Now let E and F beBanach spaces. ThenB(E, F) is the family of all bounded

(i.e., continuous) linear operators from E to F ; it is a subspace of L(E, F) andB(E, F) is itself a Banach space for the operator norm given by

‖T ‖ = sup{‖T x‖ : x ∈ E, ‖x‖ ≤ 1} .

WewriteL(E) andB(E) forL(E, E) andB(E, E), respectively. The productof two operators S and T in L(E) is given by composition:

(ST )(x) = (S ◦ T )(x) = S(T x) (x ∈ E) .


Then trivially ‖ST ‖ ≤ ‖S‖ ‖T ‖ (S, T ∈ B(E)), and (B(E), ‖ · ‖) is a unitalBanach algebra; the identity of B(E) is the identity operator IE . This is our firstnon-commutative example.For example, let E be the finite-dimensional spaceCn (saywith the Euclidean

norm ‖ · ‖2). Then L(E) = B(E) is just the algebraMn = Mn(C) of all n × nmatrices over C (with the usual identifications).(viii) The algebra C[[X ]] of formal power series in one variable consists of

sequences

(αn : n = 0, 1, 2, . . . ) ,

where αn ∈ C, with coordinatewise linear operations and the product

(αr )(βs) = (γn) ,

where γn =∑

r+s=n αrβs . It helps to think of elements of C[[X ]] as formalseries of the form

∞∑n=0

αn Xn ,

with the product suggested by the symbolism. This algebra contains as a sub-algebra the algebra C[X ] of polynomials in one variable – these polynomialscorrespond to the sequences (αn) that are eventually zero.A weight on Z+ is a function ω : Z+ → R+ \ {0} such that ω(0) = 1 and

ω(m + n) ≤ ω(m)ω(n) (m, n ∈ Z+) .

Check that ωn = e−n and ωn = e−n2 define weights on Z+. For such a weightω, define

1(ω) ={(αn) ∈ C[[X ]] : ‖α‖ω =

∞∑n=0|αn|ωn <∞

}.

Check that 1(ω) is a subalgebra of C[[X ]], and that ( 1(ω), ‖ · ‖ω) is a com-mutative, unital Banach algebra.(viii) Let G be a group, and let

1(G) ={f ∈ CG : ‖ f ‖1 =

∑s∈G| f (s)| <∞

}.


Then ( 1(G), ‖ · ‖1) is a Banach space. We can think of an element of 1(G) as∑s∈G

αsδs ,

where∑ |αs | <∞; here δs(s) = 1 and δs(t) = 0 (t �= s).

We define a product on 1(G) that is not the pointwise product; it is denotedby � and is sometimes called convolution multiplication. In this multiplication,

δs � δt = δst (s, t ∈ G) ,

where st is the product in G. (Actually this formula defines the product.) Thus

( f � g)(t) =∑{ f (r )g(s) : rs = t} (t ∈ G) . (1.2.1)

Check that 1(G) is a unital Banach algebra for this product and the norm ‖ · ‖1.It is commutative if and only ifG is an abelian group. Special case: takeG = Z,a group with respect to addition.(ix) (Strictly, this example needs the theory of the Lebesgue integral on R.)

The Banach space L 1(R) has the norm ‖ · ‖1 given by

‖ f ‖1 =∫ ∞−∞| f (t)| dt .

For functions f, g ∈ L 1(R), define their convolution product f � g by

( f � g)(t) =∫ ∞−∞

f (t − s)g(s)ds (t ∈ R) .

Integration theory shows that f � g is defined almost everywhere (a.e.) andthat f � g gives an element of L 1(R); further, ‖ f � g‖1 ≤ ‖ f ‖1 ‖g‖1, and sowe obtain a commutative Banach algebra (which does not have an identity).This example is central to the theory of Fourier transforms.(x) Let U be a non-empty, open set in C (or in Cn). Then H (U ) denotes the

set of analytic (or holomorphic) functions onU . Clearly H (U ) is an algebra forthe pointwise operations. However the algebra H (U ) is not a Banach algebra.For each compact subset K of U , define

pK ( f ) = | f |K ( f ∈ H (U )) .

Then each pK is an algebra seminorm on H (U ). The space H (U ) is a Frechetspace with respect to the family of these seminorms; in this topology, fn → fif and only if ( fn) converges to f uniformly on compact subsets of U . Thealgebra is a Frechet algebra because pK ( f g) ≤ pK ( f )pK (g) in each case.


A related algebra is H∞(U ), the algebra of bounded analytic functions onU .Check that this algebra is a Banach algebra with respect to the uniform norm| · |U .

1.3 Philosophy of why we study Banach algebras

There are several reasons why we study Banach algebras. They:

� cover many examples;� have an abstract approach that leads to clear, quick proofs and new insights;� blend algebra and analysis;� have beautiful results on intrinsic structure.

1.4 Basic properties

We begin our study of general Banach algebras by considering invertible ele-ments in such algebras.

Definition 1.4.1 Let A be a unital algebra. An element a ∈ A is invertible ifthere exists an element b ∈ A with ab = ba = eA. The element b is unique; itis called the inverse of a, and written a−1. The set of invertible elements of Ais denoted by InvA.

Check that a, b ∈ InvA⇒ ab ∈ InvA and (ab)−1 = b−1a−1.Now let (A, ‖ · ‖) be a unital Banach algebra. Check that, for each a ∈ A, we

have

limn→∞‖a

n‖1/n = inf{‖an‖1/n : n ∈ N} ≤ ‖a‖ .

Theorem 1.4.2 Let (A, ‖ · ‖) be a unital Banach algebra.(i) Suppose that a ∈ A and lim ‖an‖1/n < 1. Then eA − a ∈ InvA.(ii) InvA ⊃ {b ∈ A : ‖eA − b‖ < 1}.(iii) InvA is an open subset of A.(iv) The map a �→ a−1, InvA→ InvA, is continuous.

Proof (i) The series eA +∑∞

n=1 an converges to (eA − a)−1.

(ii) This is immediate from (i).(iii) Take a ∈ InvA, and then take b ∈ A with ‖b‖ < ‖a−1‖−1. Note that

a − b = a(eA − a−1b) and ‖a−1b‖ < 1. By (i), eA − a−1b ∈ InvA. Hencea − b ∈ InvA.


(iv) Exercise: use the inequality that

‖b−1 − a−1‖ ≤ 2‖a−1‖2‖b − a‖

whenever a, b ∈ InvA with ‖b − a‖ ≤ 1/2‖a−1‖. �

1.5 Exercises

1. Check the details of the examples.2. Prove Theorem 1.4.2(iv).3. Identify InvA for as many as possible of the examples A given in §1.2. (Easyfor A = C(), A = A(D), A = H (U ), A = B(E); harder for the algebraA = A+(D); not possible in general for 1(G).) Show that InvC[[X ]] ={(αn) : α0 �= 0}.

4. For f ∈ L1(T) (in particular for f ∈ C(T)), the Fourier coefficients are

f (k) = 1

2π

∫ π

−πf (eiθ )e−ikθdθ (k ∈ Z) .

Let sn(θ ) =∑+n−n f (k)e

ikθ and set

σn( f ) = 1

n + 1(s0 + · · · + sn) .

Then Fejer’s theorem says that: for each f ∈ C(T), σn( f )→ f uniformlyon T.Deduce that the following are equivalent for f ∈ C(T) :

(a) f ∈ P(T) ;(b) f = F | T for some F ∈ A(D) ;(c) f (−k) = 0 (k ∈ N) .

We can now identify A(D) with P(T) (why?), and regard A(D) as a closedsubalgebra of C(T) – if we should wish to do this!

5. Let

W (T) = { f ∈ C(T) : ‖ f ‖1 =∞∑

k=−∞| f (k)| <∞} .

Check that (W (T), ‖ · ‖1) is a commutative, unital Banach algebra (for thepointwise operations). Check that the map

∞∑n=−∞

αnδn �→∞∑

n=−∞αn Z

n, 1(Z)→ W (T) ,


is an isometric isomorphism. (W stands for N. Wiener, who was the first tostudy these algebras.)

6. Let L1(I) be theBanach space of (equivalence classes of) integrable functionson I, with the norm

‖ f ‖1 =∫ 10| f (t)| dt ( f ∈ L1(I)) .

For f, g ∈ L1(I), define f � g by

( f � g)(t) =∫ t

0f (t − s)g(s) ds (t ∈ I) .

Show that L1(I) is a Banach algebra for this product. It is called the Volterraalgebra, and is denoted by V .Set u(t) = 1 (t ∈ I), so that

(u � f )(t) =∫ t

0f (s) ds .

Calculate u�n and ‖u�n‖1, where u�n denotes the nth power of u in the algebraV . The map V : f �→ u � f on L1(I) is the Volterra operator, discussed inlater chapters.

1.6 Additional notes

1. By an algebra A, we always mean a linear space over C together witha multiplication such that a(bc) = (ab)c, a(b + c) = ab + ac, (a + b)c =ac + bc, and α(ab) = (αa)b = a(αb) for a, b, c ∈ A and α ∈ C. The alge-bra has an identity eA if eAa = aeA = a (a ∈ A). Suppose that A does nothave an identity. Then A# = C� A is a unital algebra for the product

(α, a)(β, b) = (αβ, αb + βa + ab) (α, β ∈ C, a, b ∈ A) ;

if A is a Banach algebra, then so is A# for the norm ‖(α, a)‖ = |α| + ‖a‖.2. For f ∈ CS , define f (s) = f (s), the complex conjugate of f (s). Then themap f �→ f is an involution on CS and on C(). Check that | f |2 = | f f |in the latter case. The algebra C() with this involution is the canonicalexample of a commutative, unital C∗-algebra; see §3.5.

3. Let be a locally compact space (e.g., R). For a continuous function f on, supp f , the support of f , is the closure of the set {x ∈ : f (x) �= 0}. Wewrite C00() for the algebra of functions of compact support, and C0() for


the algebra of functions f that vanish at infinity, i.e., {x ∈ : | f (x)| ≥ ε}is compact for each ε > 0. Check that (C0(), | · |) is a Banach algebra. Is(C00(), | · |) also a Banach algebra? Is it dense in (C0(), | · |)?

4. A closed, unital subalgebra A of an algebra (C(), | · |) such that, foreach x, y ∈ with x �= y there exists f ∈ A with f (x) �= f (y), is called auniform algebra.

5. In the text, we defined 1(G) for a group G. Check that the construction(with the product being defined in (1.2.1)) also works for a semigroup Sinstead of G – save that 1(S) is unital only if S has an identity.

6. There is a common generalization of L1(R) and 1(G). Each locally compactgroup G has a left Haar measure m, and L1(G), consisting of measurablefunctions f on G with

‖ f ‖1 =∫G| f (t)| dm(t) <∞ ,

becomes a Banach algebra for the product

( f � g)(t) =∫Gf (s)g(s−1t) dm(s) .

This is the group algebra of G. Note that G need not be abelian. See Part II7. There is no norm ‖ · ‖ on H (U ) such that (H (U ), ‖ · ‖) is a Banach algebra:see Dales (2000, 5.2.33(ii)).

8. Most of the above is inRudin (1973, 10.1–10.7) andRudin (1996, 18.1–18.4).For uniform algebras, including the disc algebra A(D), see Gamelin (1969).The disc algebra is utilized in Part III, Theorem 14.12. All the examples aregiven in substantial detail in Dales (2000). See, for example, Dales (2000,§2.1). Uniform algebras and group algebras are discussed in §4.3 and §3.3of Dales (2000), respectively. The group algebras L 1(G) are a main topic ofPart II of this book; for the related measure algebra M(G), see Proposition9.1.2. For the theory of topological algebras, including Frechet algebras, seeDales (2000, §2.2).

2

Ideals and the spectrum

We now establish some basic results about Banach algebras. A key idea is thatof the spectrum of an element. Throughout A is a unital algebra with identity eA.It is pleasing to see, first, that the basic ideas of our subject can be proved so

quickly, and that the proofs are an attractive blend of basic results from complexanalysis and functional analysis.

2.1 The spectrum

Let us first look at the spectrum of an element in a Banach algebra. The conceptgeneralizes that of the eigenvalues of a matrix.

Definition 2.1.1 Let A be a unital algebra, and let a ∈ A. The resolvent setof a is

ρA(a) = {z ∈ C : zeA − a ∈ InvA} ;

the spectrum of a is σA(a), the complement of ρA(a) in C, so that

σA(a) = C \ ρA(a) ;

the resolvent function of a is the function

Ra : z �→ (zeA − a)−1, ρA(a)→ InvA .

Usually we write ρ(a) for ρA(a), etc.We shall use the following, easily checked identity: for each z,w ∈ ρ(a), we

have

Ra(w)− Ra(z) = (z − w)Ra(z)Ra(w) . (2.1.1)

12

2 Ideals and the spectrum 13

Now suppose that A is a Banach algebra, and let a ∈ A. It follows fromTheorem 1.4.2(ii) that

σ (a) ⊂ {z ∈ C : |z| ≤ ‖a‖} .

Definition 2.1.2 Let A be a unital algebra, and let a ∈ A. The spectral radiusof a is

νA(a) = ν(a) = sup{|z| : z ∈ σ (a)} .The element a is quasi-nilpotent if ν(a) = 0 (i.e., σ (a) = {0} or σ (a) = ∅); theset of quasi-nilpotents is denoted by Q(A).

Check that, if a is nilpotent (i.e., an = 0 for some n ∈ N), then a ∈ Q(A);for T ∈ B(Cn) = L(Cn) ∼= Mn , the spectrum of the matrix T is the (finite)

set of eigenvalues of T ;for f ∈ C(), σ ( f ) is equal to f (), the range of f , and ν( f ) = | f |, so

that the only quasi-nilpotent in C() is 0.The following is the key basic theorem of our subject.

Proposition 2.1.3 Let A be a unital Banach algebra, and let a ∈ A.(i) The resolvent set ρ(a) is open in C.(ii) For each λ ∈ A′, the function λ ◦ Ra is analytic on ρ(a).(iii) The spectrum σ (a) is compact and non-empty.(iv) For each n ∈ N and r > ν(a), we have

an = 1

2π i

∫|ζ |=r

ζ n(ζeA − a)−1 dζ .

(v) ν(a) = limn→∞ ‖an‖1/n.

Proof (i) The map

θ : z �→ zeA − a, C → A ,

is continuous, and InvA is open. So the set ρ(a) = θ−1(InvA) is open in C.(ii) Fix z ∈ ρ(a), and let w ∈ ρ(a) \ {z}. Set f = λ ◦ Ra . Then

f (w)− f (z)

w − z = λ

(Ra(w)− Ra(z)

w − z)= λ(−Ra(w)Ra(z))

→−λ(Ra(z)2) as w → z ,

using 1.4.2(iv). Thus f is analytic on ρ(a).


(iii) By (i), the spectrum σ (a) is closed. We know that σ (a) is bounded. Soσ (a) is compact.Assume towards a contradiction that σ (a) = ∅. Take λ ∈ A′. By (ii), the

function λ ◦ Ra is entire. But

Ra(z) = z−1(eA − z−1a)−1→ 0 as |z| → ∞ .

By Liouville’s theorem, λ ◦ Ra = 0. Hence λ(Ra(0)) = 0. This is true for eachλ ∈ A′, and so Ra(0) = 0 by the Hahn–Banach theorem. But this is a contra-diction. Thus σ (a) is non-empty.(iv) For r > ‖a‖, the series ∑∞

k=0 ak/zk+1 is uniformly convergent to

(zeA − a)−1 on {z ∈ C : |z| = r}, and so the equation holds for this value of r .Since Ra is analytic on ρ(a), the equation holds for all r > ν(a) by Cauchy’stheorem.(v) Let z ∈ σ (a) and n ∈ N. We check easily that zn ∈ σ (an). Hence we have

|z|n ≤ ‖an‖, and so ν(a) ≤ inf ‖an‖1/n .Take r > ν(a), and set Mr = sup{‖Ra(z)‖ : |z| = r}. Then, by (iv), we have

‖an‖ ≤ rn+1Mr (n ∈ N) .

This shows that lim sup ‖an‖1/n ≤ r . The result follows. �

Part (iii) of the above result is the fundamental theorem of Banach algebras;part (v) is the spectral radius formula.Consequences: a ∈ Q(A) if and only if ‖an‖1/n → 0;

an → 0 as n→∞ if and only if ν(a) < 1 .

Example Let A = 1(ω) ⊂ C[[X ]] for a weight ω. Then ‖Xn‖ = ωn and

ν(X ) = limn→∞ω

1/nn .

So X is quasi-nilpotent if and only if ω1/nn → 0 as n→∞; this is the casewhen ωn = exp(−n2), for example.Recall that a unital algebra A is a division algebra if InvA = A \ {0}.

Theorem 2.1.4 (Gelfand–Mazur) Let A be a unital normed algebra which isa division algebra. Then A = CeA.

Proof Define θ : z �→ zeA, C → A. Then θ is a monomorphism. Take a ∈ A.By Proposition 2.1.3(iii), σ (a) �= ∅ (see the additional notes for the case where


A is not a Banach algebra), and so there exists z ∈ C with zeA − a /∈ InvA. Byhypothesis, InvA = A \ {0}. Hence θ (z) = a, and θ is a surjection. �

2.2 Ideals and the radical

Let A be an algebra. For subsets S and T of A, we write

S · T = {ab : a ∈ S, b ∈ T } and

ST ={ n∑j=1

α j a j b j : α j ∈ C, a j ∈ S, b j ∈ T},

so that ST = lin S · T , where ‘lin’ denotes the linear span. We write S[2] forS · S and S 2 for lin S[2]. A linear subspace I of A is a left ideal if AI ⊂ A,and an ideal if AI ∪ I A ⊂ I . A left ideal M is maximal if M �= A and if thereare no left ideals I with M ⊆

/I ⊆

/A. Every left ideal is contained in a maximal

left ideal (in the case where A is unital).For an ideal I in A, A/I is the quotient algebra: of course,

(a + I )(b + I ) = ab + I (a, b ∈ A) .

Check that, if A is a normed [Banach] algebra, and I is a closed ideal in A, thenA/I is a normed [Banach] algebra for the quotient norm.

Example Let be a compact space, with a closed subset F . Define:

I (F) = { f ∈ C() : f | F = 0} ;J (F) = { f ∈ C() : f = 0 on a neighbourhood of F} .

Check that I (F) and J (F) are ideals inC(), that I (F) is closed, and that J (F)is dense in I (F). When is I (F) a maximal ideal?

Proposition 2.2.1 Let A be a unital Banach algebra.

(i) Let I be a [left] ideal in A. Then I is also a [left] ideal in A.(ii) Let M be a maximal [left] ideal in A. Then M is closed.

Proof (i) This is immediately checked.(ii) Set I = M . Assume that I = A. It follows from Proposition 1.4.2(iii)

that we have M ∩ InvA �= ∅, a contradiction of the fact that M �= A. HenceI = M because M is maximal. �


The radical of an algebra A is defined to be the intersection of the maximalleft ideals of A#; it is denoted by rad A. A (necessarily non-unital) algebra Ais radical if rad A = A. The algebra A is semisimple if rad A = {0}. It is easilychecked that, in the unital case,

rad A = {a ∈ A : eA − ba ∈ InvA (b ∈ A)} .In fact rad A is an ideal in A, and A/rad A is a semisimple algebra.

Proposition 2.2.2 Let A be a unital Banach algebra. Then rad A is a closedideal and A/rad A is a semisimple Banach algebra. �

Some people think that semisimple Banach algebras are ‘good’, and rad A isthe ‘bad’ bit; we should like to remove the bad bit by writing

A = (A/rad A)⊕ rad Ain some sense (but this is not always possible).


(i) rad A ⊂ Q(A).(ii) Suppose that I is a left ideal of A with I ⊂ Q(A). Then I ⊂ rad A.(iii) In the case where A is commutative, rad A = Q(A).

Proof (i) Let a ∈ rad A. Then eA − a/z ∈ InvA for all z �= 0, and so we haveσ (a) = {0}.(ii) Let a ∈ I . For each b ∈ A, ba ∈ I because I is a left ideal. So ba ∈ Q(A)

by (i) and eA − ba ∈ InvA. Hence a ∈ rad A.(iii) Take a ∈ Q(A) and b ∈ A. We have (ba)n = bnan (n ∈ N) because A

is commutative, and so

‖(ba)n‖1/n = ‖bnan‖1/n ≤ ‖b‖‖an‖1/n → 0 as n→∞ .

Hence ν(ba) = 0 and ba ∈ Q(A). ThusQ(A) is an ideal, so thatQ(A) = rad Aby (i) and (ii). �

For example, the commutative Banach algebra C() is semisimple.Notation: for a Banach space E , and for x0 ∈ E and λ0 ∈ E ′, we define

x0 ⊗ λ0 ∈ B(E) by

x0 ⊗ λ0 : x �→ 〈x, λ0〉x0, E→E .


These maps show that, given x0, y0 ∈ E \ {0}, there exists S ∈ B(E) such thatSx0 = y0. An operator T ∈ B(E) is finite-rank if T (E) is finite-dimensional.Each finite-rank operator is a linear combination of rank-one operators of theform x0 ⊗ λ0 (where x0 ∈ E and λ0 ∈ E ′). The collection of finite-rank opera-tors on E is denoted by F(E); it is clearly an ideal in B(E).

Proposition 2.2.4 Let E be a non-zero Banach space. Then B(E) is semi-simple.

Proof Take T �= 0 in B(E). There exist x0, y0 ∈ E \ {0} with T x0 = y0.Choose S ∈ B(E) with Sy0 = x0. Then clearly (IE − ST )x0 = 0, and soIE − ST /∈ InvB(E). Hence T /∈ radB(E), and so radB(E) = {0}. �

However, the algebra B(E) contains many non-zero nilpotent operators, andhence radB(E) �= Q(B(E)).Let E be a Banach space. An operator T ∈ B(E) is compact if T (E[1])

is compact in E , where E[1] denotes the closed unit ball of E . The collectionK(E) of compact operators is a closed ideal inB(E). ClearlyF(E) ⊂ K(E); theclosureA(E) = F(E) ofF(E) in B(E) is the ideal of approximable operators.For many Banach spaces, we have A(E) = K(E) (technically, this is true atleast when E has the approximation property), but Banach spaces E such thatA(E) ⊆

/K(E) are known.

In many ways, the ideal K(E) is the natural ‘infinite-dimensional’ gen-eralization of the algebras Mn of all n × n matrices over C. (Of course,K(E) = B(E) = Mn when the Banach space E has finite dimension n.) It wasan early achievement of functional analysis to fully analyse the spectrum of acompact operator.

Theorem 2.2.5 Let E be an infinite-dimensional Banach space, and supposethat T ∈ K(E). Then:

(i) σ (T ) is a countable compact set of the form {zn : n ∈ N} ∪ {0}, where{zn : n ∈ N} is either finite or a sequence which converges to 0;

(ii) each z ∈ σ (T ) \ {0} is an eigenvalue of T , and the correspondingeigenspace {x ∈ E : T x = zx} is finite-dimensional;

(iii) for each z ∈ C \ {0}, the subspace (z IE − T )(E) is closed in E and

dim ker(z IE − T ) = dim(E/(z IE − T )(E)) ,

so that, in particular, z IE − T is surjective if and only if it is injective. �


2.3 Exercises

Throughout, A is a unital Banach algebra.

1. Let B be a closed subalgebra of A with eA ∈ B. Let a ∈ A. Show thatσB(a) ⊃ σA(a) and ∂σB(a) ⊂ ∂σA(a). (Here ∂ denotes the topological fron-tierwith respect toC.)What does thismean for thewayσB(a) can be obtainedfrom σA(a)? Calculate σA(Z ) and σB(Z ) when A = C(T) and B = A(D).

2. Let a ∈ A. Show that there is a maximal (with respect to inclusion) com-mutative subalgebra B of A containing a. Show that B is unital, that B isclosed, and that σB(a) = σA(a).

3. Let a ∈ A, and letU be an open neighbourhood of σ (a) inC. Show that thereexists δ > 0 such that σ (b) ⊂ U whenever b ∈ A with ‖a − b‖ < δ. (Thissays that the map σ from A is upper-semicontinuous in a certain sense.)

4. Let

C((X )) ={ ∞∑n=n0

αn Xn : αn ∈ C, n0 ∈ Z

}.

Show that C((X )) is a field (use the identification of InvC[[X ]]); it is thequotient field of C[[X ]], called the Laurent field (why?). Show that it is nota normed algebra for any norm.It can be shown that C[[X ]] is not a Banach algebra for any norm. (This

is quite hard.) Is it obvious whether or not it is a normed algebra for somenorm?

5. Let a, b ∈ A. Show that σ (ab) and σ (ba) differ by at most the point 0. (Hint:if c inverts eA − ab, consider eA + bca.) Calculate σ (L), σ (R), σ (LR), andσ (RL) when R and L are, respectively, the right and left shifts on the usualBanach space 1(N).

6. Let M be the operator (αn) �→ (αn/(n + 1)) on 2, and let R be the rightshift operator. Set T = MR. Prove that T is compact and that σ (T ) = {0}.Does T have any eigenvalues? Calculate ‖T n‖ for n ∈ N, and check thatlimn→∞ ‖T n‖1/n = 0. (For these operators, and many others, see Part IV.)

7. Take A = 1(ω) for the weight ω = (ωn), where ωn = e−n2 (n ∈ Z+).Show that

rad A = {(αn) : α0 = 0} and InvA = {(αn) : α0 �= 0} .

In particular, the commutative, unital Banach algebra A is not semisimple.8. LetV be the Volterra algebra, and let u ∈ V be as in Exercise 1.5.6. Calculatethe spectral radius ν(u). Deduce that V is a radical Banach algebra.



1. The definition of spectrum, etc., makes sense for an arbitrary unital algebra.For example, calculate σ (X ) for the algebra A = C[[X ]]. Let H (C) be thealgebra of entire functions on C. What are the possibilities for σ ( f ) whenf ∈ H (C)?

2. Let A be a normed algebra. Then A has a completion, say B (Bonsall andDuncan 1973, 1.12; Dales 2000, §2.1). Clearly σB(a) ⊂ σA(a) for a ∈ A. SoσA(a) �= ∅ for each a ∈ A even for a normed algebra A.

3. Suppose that A is not unital. Then σA(a) is defined to be the spectrum of(0, a) in C� A = A#. The spectrum can also be defined directly by usingquasi-inverses (Bonsall and Duncan 2000, §3; Dales 2000, 1.3.4): this isactually quite important.

4. When A is non-unital, you must deal with maximal modular ideals, ratherthan maximal ideals. A left ideal I in A ismodular if there exists u ∈ A suchthat a − au ∈ I for each a ∈ A.

5. Our radical is strictly the Jacobson radical. There are many other radicals ofan algebra, but the Jacobson radical is certainly the most important one forBanach algebra theory. For the algebraic theory of radA, see Dales (2000,§1.5).

6. For §2.1, see Rudin (1973, Chapter 10) and Dales (2000, §2.3). For §2.2,see Bonsall and Duncan (1973, Chapter 3) and Dales (2000, §2.3). For theproperties of B(E) and K(E) as Banach algebras, see Dales (2000, §2.5);for individual operators, see the later chapters. Compact operators are ex-tensively discussed in Part III.

3

Gelfand theory

Thewonderful ‘Gelfand theory’ of commutativeBanach algebraswas laid downby Gelfand in (1941), more than 60 years ago. We still essentially follow theroute of the master.

3.1 CharactersDefinition 3.1.1 Let A be an algebra. A character on A is an epimorphismϕ : A→ C.

Thus a character ϕ is a non-zero linear functional on A such that

ϕ(ab) = ϕ(a)ϕ(b) (a, b ∈ A) .

(Characters are sometimes called multiplicative linear functionals.) We write�A for the set of all characters on A; �A is called the character space of A. IfA has an identity eA, then clearly ϕ(eA) = 1 (ϕ ∈ �A).

Example Let A = C() and εx0 ( f ) = f (x0) ( f ∈ A) for some x0 ∈ . Thenεx0 ∈ �A.

Let A be an algebra, and let ϕ ∈ �A. We write

Mϕ = kerϕ = {a ∈ A : ϕ(a) = 0} .

Proposition 3.1.2 Let A be a unital algebra. Then the map ϕ �→ Mϕ is abijection from �A onto the set of maximal ideals of codimension 1 in A.

Proof Clearly each Mϕ is a maximal ideal of codimension 1.Suppose that Mϕ = Mψ . Then ϕ = ψ because ϕ(eA) = ψ(eA).

20

3 Gelfand theory 21

Let M be a maximal ideal of codimension 1. Then the quotient map

ϕ : A→ A/M = C

is a character with Mϕ = M . �

Now let A be a unital Banach algebra. The next (easy) result is the key tomuch of Banach algebra theory.

Theorem 3.1.3 Let ϕ ∈ �A. Then ϕ is continuous and ‖ϕ‖ = 1.

Proof First proof. The maximal ideal Mϕ is closed by Proposition 2.2.1(ii),and so ϕ is continuous.Second proof. Assume towards a contradiction that there exists a ∈ A such

that ‖a‖ < 1 and |ϕ(a)| = 1. By replacing a by a/ϕ(a), we can suppose thatϕ(a) = 1. By Theorem 1.4.2(i), there exists b ∈ A with b(eA − a) = eA. Wehave

ϕ(b)− ϕ(b)ϕ(a) = ϕ(eA)

because ϕ is a homomorphism. But ϕ(a) = ϕ(eA) = 1, and so we have acontradiction. Thus ϕ is continuous and ‖ϕ‖ ≤ 1. Since ϕ(eA) = 1, we have‖ϕ‖ = 1. �

We thus see that �A is a subset of the space

{λ ∈ A′ : λ(eA) = 1 = ‖λ‖} ,

which is called the state space of A.The weak-∗ topology on A′ is denoted by σ = σ (A′, A), so that λν → λ in

(A′, σ ) if and only if λν(a)→ λ(a) for each a ∈ A. Recall that the unit ball ofA′, written A′[1], is weak-∗ compact. The space�A is taken to have the relativeweak-∗ topology from A′: we shall call it the Gelfand topology.Take a ∈ A. We define a on A′ by

a(λ) = λ(a) (λ ∈ A′) .

The definition of the weak-∗ topology shows immediately that a is continuouson (A′, σ ).

Theorem 3.1.4 The space �A is compact (in the weak-∗ topology).


Proof For each a, b ∈ A, define

Ka,b ={λ ∈ A′ : λ(ab) = λ(a)λ(b)

};

also define KeA ={λ ∈ A′ : λ(eA) = 1

}. Then each Ka,b and the set KeA are

all compact in A′ [why?], and clearly

�A =⋂{Ka,b : a, b ∈ A} ∩ KeA .

Thus �A is compact. �

Wenow regard a as being defined on�A; for each a ∈ A, we have a ∈ C(�A)and |a|�A

≤ ‖a‖.For a general Banach algebra, the character space�A may be empty; it does

not have a significant role. But we shall now see that it is highly significant forcommutative Banach algebras.

3.2 Gelfand theory

Throughout this subsection A is a commutative, unital Banach algebra.

Proposition 3.2.1 Every maximal ideal of A has the form Mϕ for someϕ ∈ �A.

Proof By Proposition 3.1.2, we must show each maximal ideal M of A has co-dimension 1.Weknow thatM is closed, and so A/M is aBanach algebra. But, ina commutative algebra, A/M is a field (this is a commutative division algebra).By the Gelfand–Mazur theorem, 2.1.4, A/M = C, and so indeed we havedim(A/M) = 1. �

Thus �A corresponds to the set of maximal ideals of A; for this reason �A

is sometimes called the maximal ideal space of A.Recall that C(�A) is a Banach algebra for the uniform norm on �A.

Theorem 3.2.2 (Gelfand representation theorem) Let A be a commutative,unital Banach algebra. Then:

(i) the map G : a �→ a, A→ C(�A), is a norm-decreasing homomorphism;(ii) σ (a) = a(�A) = {a(ϕ) : ϕ ∈ �A};(iii) νA(a) = |a|�A

;(iv) a ∈ InvA if and only if a ∈ InvC(�A);(v) rad A = Q(A) = kerG.

3 Gelfand theory 23

Proof We show that (ii) holds. Indeed,

z ∈ σ (a)⇔ zeA − a /∈ InvA⇔ zeA − a belongs to some maximal ideal⇔ zeA − a ∈ Mϕ for some ϕ ∈ �A

⇔ z = ϕ(a) for some ϕ ∈ �A

⇔ z ∈ a(�A) .

The rest is immediately checked. �

Corollary 3.2.3 The following are equivalent:

(a) A is semisimple;(b) G is a monomorphism;(c) ν is a norm on A. �

Thus a semisimple, commutative, unital Banach algebra is identified as asubalgebra of C() for the compact space = �A. Such algebras are calledBanach function algebras. Recall that there are commutative, unital Banachalgebras which are not semisimple.

3.3 Examples

Let A be a unital Banach function algebra contained in C() for some compact. We suppose that, if x �= y in , then there exists f ∈ A with f (x) �= f (y).For x ∈ , define

εx ( f ) = f (x) ( f ∈ A) .

The functionals εx are the evaluation maps on A.

Lemma 3.3.1 The map η : x �→ εx , → �A, is an embedding.

Proof Certainly εx ∈ �A. If εx = εy , then x = y. Thus the map η is aninjection.Let xν → x0 in . Then f (xν)→ f (x0) for each f ∈ A, and so εxν → εx0

in �A (for the weak-∗ topology). Thus the map η is continuous, and so it is ahomeomorphism by an elementary result from topology. �


Now regard as a closed subset of�A. We say that A is natural if = �A,so that the only characters on A are the evaluation maps.Notation: for f ∈ C(), we define

Z( f ) = {x ∈ : f (x) = 0} ,

the zero set of f .

Proposition 3.3.2 Let A be a unital Banach function algebra on. Then A isnatural on if and only if, given f1, . . . , fn ∈ A with

⋂nj=1 Z( f j ) = ∅, there

exist g1, . . . , gn ∈ A with

f1g1 + · · · + fngn = 1 . (3.3.1)

Proof Take f1, . . . , fn ∈ A, and set

I ={

n∑j=1

fi g j : g1, . . . , gn ∈ A},

an ideal in A.Suppose that A is natural and

⋂nj=1 Z( f j ) = ∅. Then, for each ϕ ∈ �A, there

exists f j with f j (ϕ) �= 0. Hence I �⊂ Mϕ for any ϕ. So I = A and there existg1, . . . , gn satisfying (3.3.1).Conversely, suppose that A is not natural, so that there exists ϕ ∈ �A \.

For each x ∈ , take fx ∈ Mϕ with fx (x) = 1; we have fx (y) �= 0 for y ina neighbourhood of x . By an elementary compactness argument, there existf1, . . . , fn ∈Mϕ with

⋂nj=1 Z( f j )=∅, and so there cannot exist g1, . . . , gn ∈ A

to satisfy (3.3.1). �

Corollary 3.3.3 For each compact space , the Banach function algebraC() is natural.

Proof Take f1, . . . , fn ∈ C() with⋂n

j=1 Z( f j ) = ∅, and set

g =n∑j=1

f j f j =n∑j=1| f j |2 ,

so that g(x) > 0 (x ∈ ) and g ∈ InvC(). Set g j = f j g−1 ( j = 1, . . . , n).Then (3.3.1) is satisfied. �

3 Gelfand theory 25

3.4 Commutative group algebras

We show here that, for commutative group algebras, the Gelfand transformcoincides with the well-known Fourier transform.Let G be a locally compact abelian (or LCA) group. (Think of the cases

where G = Z or G = T or G = R.) A character on G is a group morphismfrom G onto T. The set � of all continuous characters is itself an abelian groupwith respect to pointwise multiplication

〈s, γ1 + γ2〉 = 〈s, γ1〉〈s, γ2〉 (s ∈ G, γ1, γ2 ∈ �) ,wherewewrite 〈s, γ 〉 for γ (s). Let� have the topology of uniform convergenceon compact subsets of G: then � is also an LCA group, called the dual group ofG. The Pontryagin duality theorem asserts that the dual of � is G, in a naturalway.For example: the dual of Z is T, in the sense that every continuous character

on Z has the form n �→ zn for some z ∈ T; the dual of T is Z; and the dual ofR is R, but the dual group is best thought of as a ‘different copy of R’.Let G be an LCA group. For f ∈ L 1(G), the Fourier transform of f is f ,

defined on � by the formula

f (γ ) =∫Gf (s)〈−s, γ 〉 dm(s) (γ ∈ �) .

For example, in the case where G = T, we have already defined f on Z inExercise 1.5.4. Now suppose that G = R. Then

f (y) =∫ ∞−∞

f (t) e−iyt dt (y ∈ R) .

In the general case, set

A(�) = { f : f ∈ L 1(G)} ,the range of the Fourier transform. The Riemann–Lebesgue Lemma shows that

A(�) ⊂ C0(�) .

In fact, A(�) is a self-adjoint, translation-invariant, dense subalgebra of C0(�):it is a Banach function algebra on the locally compact space �.

Theorem 3.4.1 Let G be an LCA group. Then the Fourier transform

F : f �→ f , L 1(G)→ A(�) ⊂ C0(�) ,

is a continuous homomorphism with ‖F‖ = 1. �


We can now identify the character space of L 1(G) with the dual group � (sothat ‘characters on G’ correspond to ‘characters on L 1(G)’, and this is why thetwo meanings of the word ‘character’ become somewhat blurred).

Theorem 3.4.2 Let G be an LCA group. Then each character on L 1(G) hasthe form f �→ f (γ ) for some γ ∈ �. Thus A(�) is a natural Banach functionalgebra on �, the algebra L 1(G) is semisimple, and the Gelfand transformcoincides with the Fourier transform in this case. �

3.5 C∗-algebras

Definition 3.5.1 An involution on an algebra A is a map ∗ : A→ A suchthat:

(i) (αa + βb)∗ = αa� + βb� (a, b ∈ A, α, β ∈ C) ;(ii) (a∗)∗ = a (a ∈ A) ;(iii) (ab)∗ = b∗a∗ (a, b ∈ A) .

Definition 3.5.2 A C∗-algebra is a Banach algebra A with an involution suchthat

‖a∗a‖ = ‖a‖2 (a ∈ A) .

Examples (i) The algebras C() are commutative, unital C∗-algebras for theinvolution f �→ f .(ii) Let H be a Hilbert space with inner product denoted by [ · , · ]. For

T ∈ B(H ), take T ∗ to be the adjoint of T , defined by the formula:

[T x, y] = [x, T ∗y] (x, y ∈ H ) .Then B(H ) is a unital C∗-algebra for the involution T �→ T ∗.(iii) Let A be a closed subalgebra ofB(H ) such that T ∗ ∈ Awhenever T ∈ A

(so that A is ∗-closed). Then A is a C∗-algebra.Let A be a commutative, unital, semisimple Banach algebra. We know that

the Gelfand transform embeds A in C(�A). When is this map a surjection? Theanswer is given by the (commutative) Gelfand–Naimark theorem.

Theorem 3.5.3 (Gelfand–Naimark) Let A be a commutative, unital C∗-algebra. Then the Gelfand transform

a �→ a, A→ C(�A) ,

is an isometric ∗-isomorphism. �

3 Gelfand theory 27

3.6 Exercises

1. Let A be a natural Banach function algebra on, and let I be an ideal in A.The hull of I is

h(I ) =⋂{Z( f ) : f ∈ I } .

Show that h(I ) is a non-empty, closed subset of . For a closed subset F of, define

k(F) = { f ∈ A : f | F = 0} ,

the kernel of F . Show that I ⊂ k(h(I )).Prove that the map F �→ k(h(F)) is a closure operation on the family of

Gelfand-closed subsets of �A. The topology it defines is the hull-kerneltopology. By considering the disc algebra, show that this topology need notbe Hausdorff. Prove that the hull-kernel topology coincides with the Gelfandtopology if and only if A is regular, in the sense that, for each closed subsetF of �A and each ϕ ∈ �A \ F , there exists f ∈ A with f (F) = {0} andf (ϕ) = 1. Prove that the algebras C() and L 1(G) are regular, but thedisc algebra. A(D) is not regular. The hull-kernel topology is important inPart IV of this book.

2. Show that every closed ideal in C() has the form k(F) for some closedsubset F of .

3. Let1 and2 be compact spaces. Show that C(1) and C(2) are isomor-phic (as algebras) if and only if1 and2 are homeomorphic (as topologicalspaces).

4. Show that C (n)(I) is natural for each n ∈ N.5. Let A be a commutative, unital Banach algebra, and let a ∈ A. Then

C[a] ={

n∑j=0

α j aj : α1, . . . , αn ∈ C, n ∈ N

}

in the smallest unital subalgebra of A containing a. We say that a is apolynomial generator of A if C[a] is dense in A. Consider the map

ϕ �→ ϕ(a), �A → σ (a) .

This map is always a continuous surjection. Show that, in the case where ais a polynomial generator of A, the map is a homeomorphism, and concludethat we can then identify �A with σ (a).


6. Now suppose that A is a (possibly non-commutative) unital C∗-algebra, andthat a ∈ A. The algebra

C[a, a∗] ={

n∑i, j=1

αi j ai (a∗) j : αi j ∈ C, n ∈ N

}is the smallest unital subalgebra of A containinga anda∗. TheC∗-subalgebragenerated by a is defined to be C∗(a) = C[a, a∗].Let B = C∗(a). In the casewherea isnormal (i.e.,aa∗ = a∗a), B is a com-

mutative, unital C∗-algebra. Show that �B is then homeomorphic to σ (a).Thus theGelfand–Naimark theoremgives an isometric unital∗-isomorphismC∗(a)→ C(σ (a)). Its inverse is an isometric, unital ∗-isomorphism

�a : C(σ (a))→ C∗(a) ⊂ A

with �a(Z ) = a. We setf (a) = �a( f ) ( f ∈ C(σ (a))) ,

so that f (a) is ‘a continuous function of a’. The map �a is a continuousfunctional calculus for a.

7. Let X be a completely regular topology space. Show that (Cb(X ), | · |X ) is aC∗-algebra for the involution f �→ f . Its character space is called βX , theStone–Cech compactification of X . Show that the embedding

x �→ εx , X → βX,

is continuous and has dense range. Every bounded, continuous function onX has a continuous extension to βX .


1. In the case where A is a non-unital Banach algebra, we can only say that thecharacter space �A is locally compact. The Gelfand representation is thena homomorphism into C0(�A). We have seen that this happens in the casewhere A is L 1(R).

2. The proof of the commutative Gelfand–Naimark theorem 3.5.3 is now aneasy exercise. The (harder) non-commutative Gelfand–Naimark theoremshows that everyC∗-algebra is ∗-isomorphic to a closed, ∗-closed subalgebraof B(H ) for some Hilbert space H .

3. We proved very easily that every character on a Banach algebra is automat-ically continuous. It is a remarkable fact that it has been an open questionfor more than 50 years whether or not every character on a Frechet algebrais continuous. This is called Michael’s problem. For the best partial results,

3 Gelfand theory 29

see Dixon and Esterle (1986). Beware of fallacious ‘proofs’ that appear inthe literature.

4. Let A be a Banach algebra. A bounded approxmiate identity in A is a net(eα) in A such that supα ‖eα‖ <∞ and

limα(‖a − eαa‖ + ‖a − aeα‖) = 0 (a ∈ A) .

An important theorem of Cohen says that a Banach algebra Awith a boundedapproximate identity factors, in the sense that A = A[2].It is important to note that many Banach algebras which do not have an

identity do have a bounded approxmiate identity, and hence this factorizationproperty. For example, every group algebra L 1(G) and every C*-algebrahas a bounded approximate identity. For more on bounded approximateidentities, see Part II, Chapter 9.

5. Let A be a natural Banach function algebra. Then A is weakly regular if, foreach proper closed subspace F of �A, there exists f ∈ A with f �= 0 suchthat F ⊂ Z( f ), and A has the unique uniform norm property (UUNP) if Aadmits exactly one uniform norm, namely the spectral radius νA( · ). For therelationships between these properties and regularity, see Bhatt and Dedania(2002).

6. Gelfand theory is covered in Rudin (1973, Chapter 11), Bonsall and Duncan(1973, §17) and Dales (2000, §2.3), for example. There have been enormousstudies of the algebras A(�). For a sample, see Dales (2000, §4.5); varioussubstantial texts on this topic are listed in the bibliography. See §9.3 of Part IIfor more on the Fourier transform of L 1(G).The general (non-commutative)Gelfand–Naimark theorem (involving the

GNS construction) is in all the books listed on C∗-algebras. For example,look at Kadison and Ringrose (1983, §4.5). See also Dales (2000, 3.2.29).A book discussing the Stone–Cech compactification is Gillman and Jerison(1960); see also Dales (2000, §4.2). Cohen’s factorization theorem for aBanach algebra with a bounded approximate identity, and several importantgeneralizations, are given inDales (2000, §2.9) andmany other sources listedin the references.

4

The functional calculus

In Exercise 3.6.7 we described a ‘continuous functional calculus’ �a for anormal element a of a C*-algebra A. We now replace A by an arbitrary Banachalgebra and obtain a weaker, but very important, ‘analytic functional calculus’.But first we need some more examples of character spaces.

4.1 More character spaces

Definition 4.1.1 Let X be a compact set in Cn. Then the polynomial convexhull of X is

X = {z ∈ Cn : |p(z)| ≤ |p|X for all polynomials p} .

Proposition 4.1.2 Let X be a compact set in Cn. Then �P(X ) = X .

Proof Let ϕ be a character on C[X1, . . . , Xn] (the algebra of polynomials in nvariables). Then ϕ(p) = p(z0) for some z0 = (ϕ(X1), . . . , ϕ(Xn)) ∈ Cn . Thischaracter extends to a (continuous) character on P(X ) if and only if z0 ∈ X .The result follows. �

For a general compact set X inCn , it can be difficult to identify X . However,it is quite easy to show that, for X ⊂ C, the polynomially convex hull X is theunion of X and the bounded components of C \ X . Indeed, let V be a boundedcomponent of C \ X , and let z ∈ V . Then, for each polynomial p ∈ C[X ], wehave |p(z)| ≤ |p|∂V by the maximum modulus principle. But ∂V ⊂ X , and so|p(z)| ≤ |p|X , whence z ∈ X . The converse is also fairly easy.For example �P(T) = D, so P(T) really sits on D, not on T. Indeed P(T) is

isometrically isomorphic to A(D).

30

4 The functional calculus 31

Proposition 4.1.3 The Banach function algebra A+(D) is natural.

Proof Let A = A+(D). Then A is polynomially generated by Z . It is easy tocheck that σ (Z ) = D. So A+(D) is natural by Exercise 3.6.5. �

This shows that, if f ∈ A+(D) and f (z) �= 0 (z ∈ D), then 1/ f also belongsto A+(D); this is not obvious just from the definition of A+(D).Recall that 1(Z) (with convolution product) andW (T) (with pointwise prod-

uct) are isomorphic; they are the ‘same’ algebra, so the following is a specialcase of 3.4.2.

Proposition 4.1.4 The Banach function algebra W (T) is natural.

Proof Let ϕ ∈ �W (T), and set ζ = ϕ(Z ). Then |ζ | ≤ ‖ϕ‖ ‖Z‖ = 1. We haveZ = Z−1 and so ζ �= 0 and ϕ(Z ) = ζ−1. Also |ζ−1| ≤ ‖ϕ‖ ‖Z‖ = 1. Thusζ ∈ T.It follows that, for each trignometric polynomial p of the form

∑nk=−n ck Z

k

(so that p(eiθ ) =∑nk=−n cke

ikθ ), we have ϕ(p) = p(ζ ). But these polynomialsare dense in W (T), and so ϕ = εζ and W (T) is natural. �

Corollary 4.1.5 (Wiener) Let f ∈ C(T) have absolutely convergent Fourierseries, and suppose that 1/ f ∈ C(T). Then 1/ f has absolutely convergentFourier series. �

Wiener’s original proof of this was rather long; the Banach algebra proof ismuch nicer.

4.2 Analytic functional calculus

The idea of this section is to show how to define a function f (a) of an elementa of a Banach algebra and a certain analytic function f .Let A be a unital algebra (not necessarily commutative), and let a ∈ A. Then

we can certainly define p(a) for a polynomial p ∈ C[X ]: indeed, in the casewhere p =∑n

j=0 α j Xj , we set

p(a) =n∑j=0

α j aj

(where a0 = eA). We see that the map�a : p �→ p(a), C[X ]→ A ,

is a unital homomorphism with �a(X ) = a.


Now suppose that r is a rational function, say r has the form p/q, where bothp, q ∈ C[X ] and q �= 0 (and r is defined on {z ∈ C : q(z) �= 0}). Provided thatq(a) ∈ Inv A, we can sensibly define

r (a) = p(a)q(a)−1

in A. Check that r (a) is well-defined, and that the map a �→ r (a) is a homo-morphism.The next step requires A to be a unital Banach algebra. Let f be an analytic

function on the disc U = D(0; r ), centre 0, radius r > ν(a), say

f (z) =∞∑j=0

α j zj (|z| < r ) .

Then the series∑∞

j=0 α j aj converges in A because

∑∞j=0 |α j |‖a j‖ <∞ – and

we define f (a) to be its sum. Clearly f (a) is well-defined and the map

�a : f �→ f (a), H (U )→ A ,

is a unital homomorphism with �a(Z ) = a.The common generalization of these ideas is the single-variable analytic

functional calculus.Thus, fix a unital Banach algebra A, an element a ∈ A, and an open neigh-

bourhood U of σ (a) in C.

Fact There is an open set V in C with σ (a) ⊂ V ⊂ V ⊂ U such that V hasonly finitely many components, such that the closures of these componentsare pairwise disjoint, and such that ∂V consists of a finite number of closed,rectifiable curves (V is called a Cauchy domain). Set � = ∂V , the frontier ofV . Then the A-valued integral

�Ua ( f ) =1

2π i

∫�

f (ζ ) dζ

ζeA − a =1

2π i

∫�

f (ζ )(ζeA − a)−1 dζ

exists for each f ∈ H (U ) and is independent of the choice of V .

Definition 4.2.1 A functional calculus map for a (onU ) is a unital homomor-phism

θ : H (U )→ A

such that θ (Z ) = a. The map is continuous if θ ( fn)→ θ ( f ) whenever fn → funiformly on compact subsets of U.


Theorem 4.2.2 The map f �→ f (a) = �Ua ( f ) is a continuous functional cal-culus map for a. Further:

(i) if f = p/q ∈ H (U ), then q(a) ∈ InvA and f (a) = p(a)q(a)−1;(ii) if f =∑∞

j=0 α j Zj has radius of convergence r , where r > ν(a), then

f (a) =∞∑j=0

α j aj ;

(iii) if ϕ ∈ �A, then ϕ( f (a)) = f (ϕ(a)); in the case where A is commutative,

f (a) = f ( a ) ∈ C(�A) ;

(iv) if B is a maximal commutative subalgebra of A containing a, then alsof (a) ∈ B.

Proof Most of this is immediate.Clearly �a is linear; we check that it is a homomorphism.Let f, g ∈ H (U ), and choose a contour �1 surrounding σ (a) inU , as above,

to specify �Ua ( f ). Let V be the open set bounded by �1, and choose a contour�2 surrounding σ (a) in V to specify �Ua (g). We have

�Ua ( f )�Ua (g) =

(1

2π i

)2 ∫�2

∫�1

f (ζ )g(η)(ζeA − a)−1(ηeA − a)−1 dζdη .(4.2.1)

The integrand contains the term (ζeA − a)−1(ηeA − a)−1 = Ra(ζ )Ra(η), and,by (2.1.1), this is (Ra(ζ )− Ra(η))/(η − ζ ) whenever η �= ζ . Thus the right-hand side of (4.2.1) can be written as the sum of two double integrals. ByFubini’s theorem and the fact that

∫�2f (ζ )(η − ζ )−1dζ = 0 for each η ∈ �1,

one of these two integrals is 0. By Cauchy’s integral formula, we have∫�1g(η)(η − ζ )−1dη = 2π ig(ζ ) for ζ ∈ �2, and so

�Ua ( f )�Ua (g) =

(1

2π i

)2 ∫�2

f (ζ )(ζeA − a)−1{∫

�1

g(η)(η − ζ )−1dη}dζ

= 1

2π i

∫�2

f (ζ )g(ζ )(ζeA − a)−1dζ = �Ua ( f g) .

Thus �Ua : H (U )→ A is a homomorphism.


Take U to be a disc around 0 with σ (a) ⊂ U . Then

�a(Z ) = 1

2π i

∫�

ζdζ

ζeA − a =1

2π i

∫�

(eA + a

ζ+ a2ζ 2 + · · ·

)dζ = a

because (1/2π i)∫�ζ−1dζ = 1 if n = 1 and = 0 if n �= 1. It follows that

�Ua ( f ) = f (a)whenever f ∈ R(U ), the algebra of rational functionswith polesoff U .The remainder is left as an easy exercise. �

Corollary 4.2.3 (Spectral mapping theorem) Let a ∈ A and f ∈ H (U ) forU ⊃ σ (a). Then σ ( f (a)) = f (σ (a)). �

Corollary 4.2.4 (Wiener–Levy) Let f ∈ C(T) have absolutely convergentFourier series, and let F be analytic on a neighbourhood of f (T) = σ ( f ).Then F ◦ f has an absolutely convergent Fourier series. �

Proof Apply the theorem toW (T), and note thatwe have f (T) = σ ( f ) becauseW (T) is natural. �

The functional calculus can be used to define many specific, important ele-ments in A. For example, we have obvious definitions of

exp a, sin a, cos a

(for each a ∈ A). Now suppose thatU = C \ R−, and define log and Z1/2 onUas analytic functions (with log 1 = 0 and 11/2 = 1). Then we can define log aand a1/2 whenever σ (a) ⊂ U . The usual rules apply – for example, it followsimmediately that exp(log a) = a and (a1/2)2 = a.

4.3 The idempotent theorem

Let A be an algebra. An idempotent in A is an element p such that p2 = p.Two idempotents p and q are orthogonal, written p ⊥ q, if pq = qp = 0.The following result follows easily from the functional calculus.

Theorem 4.3.1 Let E be a Banach space, and let T ∈ B(E). Suppose that

σ (T ) = σ1 ∪ σ2,

where σ1 and σ2 are disjoint, non-empty, closed subsets of σ (T ). Then thereexist idempotents P, Q ∈ B(E) with T P = PT and P + Q = IE such that

σ (T | P(E)) = σ1 and σ (T | Q(E)) = σ2.


Proof There exists a function f analytic on a neighbourhood of σ (T ) such thatf | σ1 = 1 and f | σ2 = 0. Define P = �T ( f ) and Q = IE − P . Then P andQ are idempotents in B(E), T P = PT , and P + Q = IE . Further, P(E) andQ(E) are closed linear subspaces of E .For ζ ∈ C \ σ1, there exists a function g analytic on a neighbourhood of σ (T )

with (ζ1− Z )g = f . Set S = �T (g). Then

(ζ IE − T )S = S(ζ IE − T ) = P ,

and so ζ /∈ σ (T | P(E)). Hence we have σ (T | P(E)) ⊂ σ1. Similarly we seethat σ (T | Q(E)) ⊂ σ2.Now take ζ ∈ C \ (σ (T | P(E)) ∪ σ (T | Q(E))). Then

(ζ IP(E) − T | P(E))−1P + (ζ IQ(E) − T | Q(E))−1Q

is the inverse of ζ IE − T in B(E), and so ζ ∈ ρ(T ). Thus we have shown thatσ (T | P(E)) = σ1 and σ (T | Q(E)) = σ2. �

Let T ∈ B(E), and let z be an isolated point of σ (T ). Then the correspondingidempotent with spectrum {z} is denoted by Pz . The resolvent function RT isan analytic function near z; if RT has a pole of order k at z, then k is exactlythe index of z – this is the minimum n ∈ Z+ such that

ker(z IE − T )n = ker(z IE − T )n+1;

k is also the minimum n ∈ Z+ such that (z IE − T )n Pz = 0. The func-tional calculus in the case where σ (T ) is totally disconnected is called theRiesz functional calculus. See the discussion of Riesz operators in Part IV,Chapter 22.In the case where T ∈ K(E), each z ∈ σ (T ) \ {0} is a pole of RT (and has

finite index); the projection Pz has non-zero, finite-dimensional range, and thisis Pz(E) = ker(z I − T )k , where k is the order of the pole.There is a very well-developed theory of compact operators on a Hilbert

space. For example, let H be a Hilbert space, and take T ∈ K(H ) with T = T ∗(so that T is self-adjoint), say σ (T ) = {zn : n ∈ Z+}. Then z0 = 0 and (zn)is a null sequence in R, and there is an orthonormal basis (en) in H suchthat

T x =∞∑n=1

zn[x, en]en (x ∈ H ) ,

where the series for T converges in the operator norm.


4.4 Exercises

1. Letω be a weight function onZ+, and set A = 1(ω) and ρ = limn→∞ ω1/nn .Show that, in the casewhereρ > 0, the character space�A is homeomorphicto the closed disc D(0; ρ), and that the Gelfand transform is the map

a =∞∑n=0

αn Xn �−→ a =

∞∑n=0

αn Zn on D(0; ρ) ,

Note that a is continuous on D(0; ρ), and analytic on the open disc D(0; ρ).Deduce that A is semisimple whenever ρ > 0. What happens if ρ = 0?

2. Let A be a unital Banach algebra.

(i) Define exp a for a ∈ A. Show that, if a, b ∈ A with ab = ba, then

exp(a + b) = (exp a)(exp b) .

Is this necessarily true without the condition that ab = ba?(ii) Let a ∈ A be such that σ (a) ⊂ U = C \ R−. Note that there exists afunction log ∈ H (U )with log 1 = 0 and exp(log z) = z (z ∈ U ). Showthat a = exp(log a).

(iii) Denote by Inv0A the component of InvA containing eA. Prove that

Inv0A = {(exp a1) · · · (exp an) : a1, . . . , an ∈ A} .


1. We have defined the functional calculus for a fixed open neighbourhood Uof σ (a). In fact, the full theory deals with Oσ (a), the algebra of germs ofanalytic functions on σ (a) formed by varying the neighbourhoodsU , whereOσ (a) is given the locally convex inductive-limit topology from the locallyconvex spaces H (U ), and we finally obtain a map �a : Oσ (a)→ A.

2. In the case where A is a unital C∗-algebra and a ∈ A is normal, we havedefined (in Exercise 3.6.6) a unital homomorphism

f �→ f (a), C(σ (a))→ A .

This homomorphism extends the analytic functional calculus for a, in thesense that the two definitions of f (a) agree when f is the restriction toσ (a) of a function analytic on a neighbourhood of σ (a). For a related func-tional calculus, with domain H ∞(D), see Part III, Corollary 14.1.14; similarseveral-variable versions of this functional calculus arementioned in Part III,Chapter 20.


3. Let A be a commutative, unital Banach algebra, and let a = (a1, . . . , an)belong to A(n), the n-fold Cartesian product of A with itself. Then the jointspectrum σ (a) of a is

σ (a) = {(ϕ(a1), . . . , ϕ(an)) : ϕ ∈ �A} .

There is a several-variable analytic functional calculus

�a : f �→ f (a), Oσ (a)→ A ,

a continuous, unital homomorphism with �a(Z j ) = a j for j = 1, . . . , n.Again the spectral mapping theorem holds: σ ( f (a)) = f (σ (a)) ⊂ Cn for

a ∈ A(n) and f ∈ Oσ (a). This result depends on deep results in the theory ofanalytic functions of several complex variables.For another joint spectrum, see Part III, Chapter 20.

4. Theorem 4.3.1 features in Parts III and IV. The several-variable analyticfunctional calculus can be used to prove an extension of Theorem4.3.1 calledSilov’s idempotent theorem. (It is not known how to prove this without usingthe powerful several-variable calculus.) The theorem is as follows.

Theorem Let A be a commutative Banach algebra, and suppose that K isa compact and open subset of �A. Then there is a unique idempotent p inA such that p is the characteristic function of K . �

5. For polynomial convexity, see Gamelin (1969) and Stout (1971). The func-tional calculus is in all the references, including Dales (2000, §2.4). For thespectral theory of compact operators, see Meise and Vogt (1997), for ex-ample; the Riesz functional calculus is discussed in Part IV of this book.There are two different approaches to the several-variable functional cal-culus for commutative Banach algebras: for these, see Bourbaki (1960),Gamelin (1969, §III. 4), Stout (1971, Chapter 1, §8), and Palmer (1994,§3.5). A powerful extension is given in Zame (1979). For various relatedfunctional calculus maps, see Part III. More results on joint spectra and theseveral-variable functional calculus can be found in Eschmeier and Putinar(1996).

5

Automatic continuity of homomorphisms

There is a deep and surprising connection between the algebraic and topologicalproperties of a Banach algebra: sometimes the algebraic properties determinethe topological properties. Let A be a unital Banach algebra. Then we havealready seen that a character on A (defined algebraically as an epimorphismfrom A ontoC) is automatically continuous. In factmanyother homomorphismsbetween Banach algebras A and B are automatically continuous. We exploresome of these ideas in the present chapter.

5.1 Automatic continuity

The first result is a very old theorem of Gelfand.

Proposition 5.1.1 Let A be a Banach algebra, and let B be a commutative,semisimple Banach algebra. Then each homomorphism θ : A→ B is automat-ically continuous.

Proof Let an → 0 in A and θ (an)→ b in B. By the closed graph theorem, itsuffices to prove that b = 0.Take ϕ ∈ �B . Then ϕ ◦ θ ∈ �A ∪ {0}, and so both ϕ and ϕ ◦ θ are contin-

uous. We have

(ϕ ◦ θ )(an)→ (ϕ ◦ θ )(0) = 0

and

(ϕ ◦ θ )(an) = ϕ(θ (an))→ ϕ(b)

as n→∞, and so ϕ(b) = 0. Thus b ∈⋂{kerϕ : ϕ ∈ �B}.

38

5 Automatic continuity of homomorphisms 39

Since B is commutative,⋂{kerϕ : ϕ ∈ �B} = rad B (by 3.2.2(v)); since B

is semisimple, rad B = 0. Hence b = 0 and θ is continuous. �

Definition 5.1.2 Let (A, ‖ · ‖) be a Banach algebra. Then A has a uniquecomplete norm if each norm with respect to which A is a Banach algebra isequivalent to the given norm ‖ · ‖.

Corollary 5.1.3 Let (A, ‖ · ‖) be a commutative, semisimple Banach algebra.Then A has a unique complete norm.

Proof Let ||| · ||| be another complete algebra norm on A. Then the identitymap (A, ‖ · ‖)→ (A, ||| · |||) is continuous by Proposition 5.1.1. �

Even if we deviate slightly from semisimplicity by considering commutativeBanach algebras with one-dimensional radicals, the above result may fail – seeExercise 5.3.1.It was a major open question for many years whether every (perhaps non-

commutative) semisimple Banach algebra has a unique complete norm. Thiswas eventually proved in 1967 by B. E. Johnson. We shall present a proof dueto B. Aupetit (1982), as simplified by T. J. Ransford (1989).One standard tool in automatic continuity theory is the separating space of a

linear map; it measures ‘how far a linear map is from being continuous’.

Definition 5.1.4 Let E and F be Banach spaces, and let T : E → F be alinear map. Then the separating space of T is

S(T ) = {y ∈ F : there exists xn → 0 in E with T xn → y} .

It is easily checked that S(T ) is a closed linear subspace of F , and itfollows from the closed graph theorem that T is continuous if and only ifS(T ) = {0}.Let A and B beBanach algebras, and let θ : A→ B be a homomorphismwith

θ (A) = B. Then it is also easy to check that S(θ ) is an ideal in B. Proposition5.1.1 really shows that S(θ ) ⊂ rad B in the case where B is commutative; weshall establish this even for non-commutative B.

Lemma 5.1.5 Let (A, ‖ · ‖) be a unital Banach algebra, let a ∈ A, and letε > 0. Then there is a norm ||| · ||| on A such that ||| · ||| is equivalent to ‖ · ‖,|||eA||| = 1, and |||a||| ≤ ν(a)+ ε.


Proof Set b = a/(ν(a)+ ε). Then S = {bn : n ∈ Z+} is a bounded semigroupin (A, ·). For c ∈ A, set

p(c) = sup{‖sc‖ : s ∈ S}, |||c||| = sup{p(cd) : d ∈ A, p(d) ≤ 1} .

Check that this works. �

Let A be an algebra. Then A[X ] denotes the algebra of all polynomials withcoefficients in A.

Lemma 5.1.6 Let A be a Banach algebra, let p ∈ A[X ], and take R > 1.Then

(νA(p(1)))2 ≤ sup

|z|=RνA(p(z)) · sup

|z|=1/RνA(p(z)) .

Proof Let q ∈ A[X ], and take λ ∈ A′ with ‖λ‖ = 1 and λ(q(1)) = ‖q(1)‖.Set F = λ ◦ q. By the maximum modulus theorem applied to the functionz �→ F(z)F(1/z) on the annulus {z ∈ C : 1/R ≤ |z| ≤ R}, we obtain

|F(1)|2 ≤ sup|z|=R

|F(z)| · sup|z|=1/R

|F(z)| .

It follows that

‖q(1)‖2 ≤ sup|z|=R

‖q(z)‖ · sup|z|=1/R

‖q(z)‖ . (5.1.1)

Apply (5.1.1) with q = p2n, where n ∈ N. By the spectral radius formula,

2.1.3 (v), we have

‖p2n (z)‖1/2n → νA(p(z)) as n→∞

for each z ∈ C. The sequence (‖p2n (z)‖1/2n ) is monotone decreasing and thefunction z �→ ‖p2n (z)‖1/2n is continuous, and so it follows from Dini’s theoremthat, for each r ∈ R+, we have

sup|z|=r

‖p2n (z)‖1/2n → sup|z|=r

νA(p(z)) as n→∞ .

Thus the result follows from (5.1.1). �

Theorem 5.1.7 (Aupetit) Let A and B be Banach algebras, and suppose thatT : A→ B is a linear map such that νB(Ta) ≤ νA(a) (a ∈ A).


(i) Suppose that b ∈ S(T ). Then

(νB(Ta))2 ≤ νA(a)νB(Ta − b) (a ∈ A) .

(ii) T (A) ∩S(T ) ⊂ Q(B).

Proof (Ransford) (i) Choose (an) in A with an → 0 and Tan → b as n→∞,and take a ∈ A.For each ε > 0, we may, by Lemma 5.1.5, choose norms on A and B which

are equivalent to the given norms and which are such that

‖a‖ ≤ νA(a)+ ε, ‖Ta − b‖ ≤ νB(Ta − b)+ ε (a ∈ A, b ∈ B) .

We apply Lemma 5.1.6 in the case where p = (Ta − Tan)+ (Tan)X be-longs to B[X ], where n ∈ N. We have p(1) = Ta, and so, for each R > 1,

(νB(Ta))2 ≤ sup

|z|=RνB(p(z)) · sup

|z|=1/RνB(p(z)) . (5.1.2)

Now νB(p(z)) ≤ ‖Ta − Tan‖ + |z| ‖Tan‖. Also p(z) = T (a − an + zan),and so, by hypothesis,

νB(p(z)) ≤ νA(a − an + zan) ≤ ‖a − an‖ + |z| ‖an‖ .

Thus, from (5.1.2), we have

(νB(Ta))2 ≤ (‖a − an‖ + R ‖an‖)(‖Ta − Tan‖ + ‖Tan‖ /R) .

This holds for each n ∈ N, and so, letting n→∞, we see that

(νB(Ta))2 ≤ ‖a‖ (‖Ta − b‖ + ‖b‖ /R) .

But this holds for each R > 1, and so, letting R→∞, we obtain

(νB(Ta))2 ≤ ‖a‖ ‖Ta − b‖ ≤ (νA(a)+ ε)(νB(Ta − b)+ ε) .

Finally this holds for each ε > 0, and so the result follows.(ii) This is immediate from (i). �

Theorem 5.1.8 (Johnson) Let A and B be Banach algebras, and suppose thatθ : A→ B is an epimorphism. Then S(θ ) ⊂ rad B. If B is semisimple, then θis automatically continuous.


Proof It is easy to see that Theorem5.1.7 applieswith θ for T . Since θ (A) = B,Theorem 5.1.7(ii) shows that S(θ ) ⊂ Q(B). Since S(θ ) is an ideal in B, itfollows from Proposition 2.2.3(ii) that S(θ ) ⊂ rad B. �

Corollary 5.1.9 A semisimple Banach algebra has a unique completenorm. �

The sharp-eyed will notice that we have not quite generalized Proposition5.1.1 to the casewhere B is not necessarily commutative. To obtain an analogousresult, we certainly need to suppose that θ (A) = B, but we actually assumedthat θ (A) = B. Thus we have the following question, which has been open forat least 30 years.

Question 5.1.10 Let A and B be Banach algebras, and suppose that B issemisimple. Let θ : A→ B be a homomorphism with θ (A) = B. Is θ automat-ically continuous?

If you can solve this, please let me know quickly.

5.2 Homomorphisms from Banach algebras

Let θ : A→ B be a homomorphism between Banach algebras. We have givensome conditions on the range B that ensure that θ is automatically continuous.What about conditions on the domain algebra A? The key tool in this case isthe continuity ideal I(θ ) of the homomorphism θ .

Definition 5.2.1 Let θ : A→ B be a homomorphism. Then

I(θ ) = {a ∈ a : θ (a)b = bθ (a) = 0 (b ∈ S(θ ))} .

Thus I(θ ) is the (two-sided) annihilator of the separating space S(θ ).It is an easy exercise to check that I(θ ) is an ideal in A. (However, there is

no reason for I(θ ) to be closed in A unless θ really is continuous – which iswhat we are trying to show.) It is also easy to check that I(θ ) is just the set ofelements a ∈ A such that both of the maps x �→ θ (ax) and x �→ θ (xa) fromA into B are continuous. In the case where A is unital our aim is to show thateA ∈ I(θ ), for then I(θ ) = A and θ is continuous on the whole of A.The following main boundedness theorem (MBT) is due to Bade and Curtis

(1960).

Theorem 5.2.2 Let A and B be Banach algebras, and let θ : A→ B bea homomorphism. Suppose that (an) and (bn) are sequences in A such that


ambn = 0 whenever m �= n. Then there is a constant C > 0 such that‖θ (anbn)‖ ≤ C ‖an‖ ‖bn‖ (n ∈ N) . (5.2.1)

Proof We may suppose that ‖an‖ = ‖bn‖ = 1 (n ∈ N).Assume towards a contradiction that there is no C such that (5.2.1) holds.

Then there is an injective map (i, j) �→ n(i, j), N× N → N, such that

‖θ (ui, j vi, j‖ ≥ 4i+ j (i, j ∈ N) , (5.2.2)

where ui, j = an(i, j) and vi, j = bn(i, j). Set

vi =∞∑=1

vi,/2 (i ∈ N) ;

for each i , the series for vi converges in A, and ‖vi‖ ≤ 1. For each i ∈ N, choosej(i) ∈ N such that ‖θ (vi )‖ ≤ 2 j(i), and set

a =∞∑k=1

uk, j(k)/2k

(again the series converges in A); we have

avi =∞∑k=1

∞∑=1

uk, j(k)vi,/2k+ ,

and soavi = ui, j(i)vi, j(i)/2i+ j(i) becauseambn = 0wheneverm �= n. By (5.2.2),we have ‖θ (avi )‖ ≥ 2i+ j(i). However,

‖θ (avi )‖ ≤ ‖θ (a)‖ ‖θ (vi )‖ ≤ 2 j(i) ‖θ (a)‖ .

We conclude that ‖θ (a)‖ ≥ 2i for each i ∈ N, clearly a contradiction. �

Definition 5.2.3 Let A be a unital algebra. A continued bisection of the iden-tity for A is a pair {(pn), (qn)} of sequences of idempotents such that eA =p1 + q1 and such that, for each n ∈ N, we have pn ⊥ qn, pn = pn+1 + qn+1,and Apn A = Aqn A.

It is easy to check in this case that qm ⊥ qn (m �= n).

Theorem 5.2.4 Let A be a unital Banach algebra with a continued bisectionof the identity. Then every homomorphism from A into a Banach algebra isautomatically continuous.


Proof Let {(pn), (qn)} be as in Definition 5.2.3.Let θ : A→ B be a homomorphism into a Banach algebra B. Assume to-

wards a contradiction that qn /∈ I(θ ) for infinitely many n ∈ N. Then we maysuppose that the map

x �→ θ (qnx), A→ B ,

is discontinuous for infinitely many n ∈ N, and so there exists (xn) ⊂ A suchthat ‖xn‖ = 1 and ‖θ (qnxn)‖ ≥ n ‖qn‖2 for infinitely many n ∈ N. Apply theMBTwith an = qn and bn = qnxn (so that ambn = 0 whenm �= n): there existsC > 0 such that

‖θ (qnxn)‖ ≤ C ‖qn‖2 (n ∈ N) .

Thus C ≥ n for infinitely many n ∈ N, a contradiction.We have shown that there exists k ∈ Nwith qk ∈ I(θ ). But now, successively,

we see that pk, pk−1, qk−1, . . . , p1, q1, eA belong to I(θ ), and so I(θ ) = A andθ is continuous. �

Corollary 5.2.5 Let E be a Banach space such that E ∼= E ⊕ E. Then allhomomorphisms from B(E) into a Banach algebra are continuous.

Proof We have

E & E ⊕ E & (E ⊕ E)⊕ E & · · · ;

at the nth stage, E is linearly homeomorphic to the direct sum of n copies ofitself. Let Pn and Qn be the projections of E onto the first and second of thesen components, respectively. Clearly IE = P1 + Q1 and,

Pn ⊥ Qn, Pn = Pn+1 + Qn+1 (n ∈ N) .

LetUn be the operator on E which exchanges the first two of the n componentsat the nth stage of the decomposition. Then Pn = UnQnUn and Qn = UnPnUnfor each n ∈ N. Thus {(Pn), (Qn)} is a continued bisection of the identity. �

5.3 Exercises

1. Write out the details of the following example.Let A be the sequence space ( 2, ‖ · ‖2) with coordinatewise product.

Check that A is a commutative, semisimple Banach algebra. Set

A = A ⊕ C


as a linear space, with multiplication given by (a, z)(b,w) = (ab, 0). ThenA is an algebra and radA = {0} ⊕ C, which has dimension 1. Check that Ais a Banach algebra for the norm

‖(a, z)‖ = ‖a‖2 + |z| ((a, z) ∈ A) .

Let λ be a linear functional on A such that λ | 1 is the functional

(αn) �→∞∑n=1

αn,

and define

|||(a, z)||| = max{‖a‖2 , |λ(a)− z|} ((a, z) ∈ A) .

Check that (A, ||| · |||) is a Banach algebra (even though λ is necessarilydiscontinuous on A), but that ||| · ||| is not equivalent to ‖ · ‖ on A.

2. We have talked about unique complete norms. Since B(E) is a semisimpleBanach algebra, it has a unique complete norm. In fact, a beautiful result ofEidelheit (1940) shows an even stronger result. Write out the details of thefollowing theorem.Let E be a (non-zero) Banach space, and letA = B(E), a Banach algebra

for the operator norm, ‖ · ‖. Let ||| · ||| be any algebra norm onA. Then thereis a constant C such that ‖T ‖ ≤ C |||T ||| (T ∈ A).To obtain a contradiction, assume that there is no suchC . Take a sequence

(Sn) in A such that |||Sn||| = 1 (n ∈ N) and such that ‖Sn‖ → ∞. Thenthere exists x0 ∈ F with the sequence (‖Snx0‖) unbounded, and then thereexists λ ∈ E ′ such that the sequence (|λ(Snx0)|) is unbounded. Now definezn = λ(Snx0) (n ∈ N).Define T x = λ(x)x0 (x ∈ E), so that T ∈ B(E). Check that we have

T SnT = znT (n ∈ N), and obtain a contradiction.3. Let be a compact space, and let ‖ · ‖ be an algebra norm on C(). Showthat | f | ≤ ‖ f ‖ ( f ∈ C()).

4. Let E and F be Banach spaces, let T : E → F be a linear map, and then letQ : F → F/S(T ) be the quotient map. Show that QT is continuous.Suppose thatG is another Banach space, and that S ∈ B(F,G). Show that

SS(T ) = S(ST ) ,

so that ST is continuous if and only if SS(T ) = {0}.5. Show that to resolve Question 5.1.10 is equivalent to solving the followingquestion.Let A and B be Banach algebras, let θ : A→ B be a homomorphism,

and let b ∈ S(θ ). Is it necessarily true that σ (b) = {0}?


(It can be shown quite easily that σ (b) is always a connected set containing0, but nothing further seems to be known.)


1. Let A be a commutative Banach algebra. We know that, in the case whereA is semisimple, A has a unique complete norm. However, this does notcharacterize semisimpleBanach algebras. For example, automatic continuitytheory shows that the Banach algebras 1(ω) all have a unique completenorm, but they are not necessarily semisimple. Many other examples aregiven in Dales (2000).

2. Despite an immense amount of work on the ‘uniqueness-of-norm’ problem,basic questions remain open.For example, let (A, ‖ · ‖) be a commutative Banach algebra which is an

integral domain. Does A necessarily have a unique complete norm?A Banach algebra A is topologically simple if the only closed ideals in

A are the trivial ones {0} and A. It is not known whether or not there is acommutative, topologically simple Banach algebra other than C (this couldbe the hardest question in Banach algebra theory). It was shown by Cusack(1977) that, if there is a commutative Banach algebra which is an integraldomain and which does not have a unique complete norm, then there is acommutative, topologically simple, radical Banach algebra.

3. Let H be a Hilbert space. We have shown that all homomorphisms fromB(H ) are continuous. What about other C∗-algebras?First note that it is a standard triviality that all ∗-homomorphisms between

C∗-algebras are automatically continuous, (see Dales (2000, 3.2.4) and allbooks on C∗-algebras). But we are interested in homomorphisms which arenot ∗-homomorphisms.The following result is proved in Sinclair (1976, 12.4) by a different

argument from the one we used in Theorem 5.2.4. Let A be a unital C∗-algebra such that A has no proper closed ideals of finite codimension. Thenevery homomorphism from A into a Banach algebra is continuous.

4. The above result does not cover the commutative C∗-algebras of the formC() for a compact, infinite space. It was a question of Kaplansky (1949)whether or not every homomorphism from C() is automatically contin-uous; this is equivalent to the question whether every algebra norm ‖ · ‖on C() is equivalent to the uniform norm | · |. (Kaplansky proved thatnecessarily | f | ≤ ‖ f ‖ ( f ∈ C()); see Exercise 5.3.3.)A major advance was due to Bade and Curtis (1960). Let θ : C()→ B

be a discontinuous homomorphism into a Banach algebra B. Then there


is a finite subset F = {x1, . . . , xn} of (the singularity set) such that therestriction of θ to the ideal J (F) is continuous. Thus θ is continuous on a‘big subalgebra’ of C(). Further developments, due to Esterle (1978a) andSinclair (1975) showed that there is a maximal ideal M of C(), a primeideal P ⊆

/M , and an embedding of the algebra M/P into a radical Banach

algebra. However it was left open whether or not such embeddings exist.Finally it was shown independently by Dales (1979) and Esterle (1978b)

that such embeddings do exist – at least if the continuum hypothesis (CH)be assumed – for every infinite compact space. It is a striking fact that thisresult cannot be proved in the theory ZFC – see Dales and Woodin (1987)for an exposition, and Dales and Woodin (1996) for further developments.

5. There is a non-commutative analogue of the Bade–Curtis theorem: seeSinclair (1974). It is a conjecture of Albrecht and Dales (1983) that thefollowing are equivalent for a C∗-algebra A: (a) there is a discontinuoushomomorphism from A; (b) there exists k ∈ N and infinitely many maximalideals Mn in A such that A/Mn

∼= Mk .For a short proof that (b) implies (a) (with CH), see Dales and Runde

(1997). For proofs that (a) implies (b) for various classes of C∗-algebras seeAlbrecht and Dales (1983) and Ermert (1996).

6. Is every epimorphism from a C∗-algebra onto a Banach algebra automat-ically continuous? There is an elegant proof of this for commutative C∗-algebras in Esterle (1980) (see also Dales 2000, 5.4.27), but the general caseis open.

7. Let A be a Banach algebra with finite-dimensional radical. We would like toknow necessary and sufficient conditions for A to have a unique completenorm. For an attack on this and a plausible conjecture, see Dales and Loy(1997).

8. Johnson’s original proof of the uniqueness-of-norm theorem is in Johnson(1967). There are several proofs in Dales (2000) – uniqueness-of-norm the-orems and counter–examples are in §5.1.All the results which are mentioned in the Additional notes are discussed

in detail in Dales (2000, Chapter 5). In particular, the structure of all(discontinuous) homomorphisms from C() is given in full detail.

6

Modules and derivations

After homomorphisms between algebras, the next most elementary maps froman algebra A are derivations into an A-bimodule E . We discuss the algebraictheory, and then the Banach version.

6.1 Modules

We first recall the elementary theory of modules over an algebra.

Definition 6.1.1 Let A be an algebra. A left A-module is a linear space Eover C and a map (a, x) �→ a · x, A × E → E, such that:

(i) a · (αx + βy) = αa · x + βa · y (α, β ∈ C, a ∈ A, x, y ∈ E);(ii) (αa + βb) · x = αa · x + βb · x (α, β ∈ C, a, b ∈ A, x ∈ E);(iii) a · (b · x) = ab · x (a, b ∈ A, x ∈ E).A right A-module is a linear space E over C and a map (a, x) �→ x · a,A × E → E, such that:

(i) (αx + βy) · a = αx · a + βy · a (α, β ∈ C, a ∈ A, x, y ∈ E);(ii) x · (αa + βb) = αx · a + βx · b (α, β ∈ C, a, b ∈ A, x ∈ E);(iii) (x · a) · b = x · ab (a, b ∈ A, x ∈ E) .An A-bimodule is a space E which is a left A-module and a right A-moduleand which is such that

a · (x · b) = (a · x) · b (a, b ∈ A, x ∈ E) .Suppose that A is commutative and that E is an A-bimodule such that

a · x = x · a (a ∈ A, x ∈ E) .Then E is an A-module.

48

6 Modules and derivatives 49

For example, let I be a left ideal in A. Then I and A/I are left A-modules; theyare A-bimodules in the case where I is an ideal. Again, let A be commutative,and let ϕ ∈ �A. Then the linear space C is an A-module for the map

(a, z) �→ ϕ(a)z, A × C → C ;

it is denoted by Cϕ . Finally, let A be a Banach function algebra on . ThenC() is a Banach A-module (for the module product equal to the pointwiseproduct).Let E be a left A-module. Then the map ρ : A→ L(E) defined by

ρ(a)(x) = a · x (a ∈ A, x ∈ E)is a homomorphism, and every such homomorphism defines a left A-module.The map ρ is called a representation of the algebra A on the linear space E .Throughout, one could replace the language of modules by that of representa-tions. This is the approach taken in Palmer (1994, 2001).Suppose that A is unital. Then a left A-module E is unital if

eA · x = x (x ∈ E) .Let E be a left A-module. Then we write

A · E = {a · x : a ∈ E, x ∈ E}, AE = lin A · E .We write aE for {a}E , etc. The module E is simple if A · E �= {0} and if {0}and E are the only submodules of E . It is easy to check that, for a simplemoduleE , we have A · x = E for each x ∈ E \ {0}.

Proposition 6.1.2 Let A be a unital algebra. An ideal I in A is primitive if

I = {a ∈ A : aE = {0}}for some simple left A-module E.

It is easy to check that I is a primitive if and only if

I = {a ∈ A : aA ⊂ M}for some maximal left ideal M of A, and in this case the simple module E thatarises in the definition is isomorphic to A/M as a left A-module. Further, eachprimitive ideal is the intersection of the maximal left ideals which contain it.In some ways, primitive ideals play the same role for general algebras that thekernels of characters play for commutative algebras.


Let E and F be left A-modules. A left A-module homomorphism is a linearmap T : E → F such that

T (a · x) = a · T x (a ∈ A, x ∈ E) .

Similarly we define A-bimodule homomorphisms.

Definition 6.1.3 Let A be a Banach algebra, and let E be a Banach spacewhich is a left A-module. Then E is a weak Banach left A-module if the map

ρ(a) : x �→ a · x, E → E ,

is continuous for each a ∈ A, and E is a Banach left A-module if the map

(a, x) �→ a · x, A × E → E ,

is continuous. Similarly, for right A-modules and A-bimodules.

Thus E is weak Banach if, for each a ∈ A, there exists a constant Ca > 0such that

‖a · x‖ ≤ Ca ‖x‖ (x ∈ E) ,

and E is Banach if there exists a constant C > 0 such that

‖a · x‖ ≤ C ‖a‖ ‖x‖ (a ∈ A, x ∈ E) ;

in this latter case we may suppose that C = 1 by moving to an equivalent normon E .Let A be a Banach algebra, and let I be a closed left ideal in A. Then I and

A/I are Banach left A-modules. Let A be commutative, and take ϕ ∈ �A. ThenCϕ is a Banach A-module. Let A and B be Banach algebras, and let θ : A→ Bbe a homomorphism. Then B is a weak Banach A-bimodule for the maps

a · b = θ (a)b, b · a = bθ (a) (a ∈ A, b ∈ B) ,

but we only know that B is a Banach A-bimodule in the case where θ iscontinuous.


(i) Each primitive ideal in A is closed.


(ii) Let E be a simple left A-module. Then there is a norm ‖ · ‖ on E such that(E, ‖ · ‖) is a Banach left A-module.

Proof (i) By Proposition 2.2.1(ii), each maximal left ideal is closed, and weknow that a primitive ideal is an intersection of maximal left ideals.(ii) We may suppose that E = A/M as a left A-module for a maximal

left ideal M , and then E is a Banach left A-module for the quotientnorm ‖ · ‖. �

We now come to the important concept of a dual module.Let E be a Banach space. Then the action of λ ∈ E ′ on x ∈ E is denoted by

〈x, λ〉.

Definition 6.1.5 Let A be a Banach algebra, and let E be a Banach A-bimodule. For a ∈ A and λ ∈ E ′, define a · λ and λ · a by

〈x, a · λ〉 = 〈x · a, λ〉, 〈x, λ · a〉 = 〈a · x, λ〉 (x ∈ E) .Then a · λ, λ · a ∈ E ′, and E ′ is a Banach A-bimodule, called the dual moduleto E.

Check that E ′ has the stated properties.For example, take E = A. Then A′ is a Banach A-bimodule for the operations

〈b, a · λ〉 = 〈ba, λ〉, 〈b, λ · a〉 = 〈ab, λ〉 (a, b ∈ A, λ ∈ A′) .This module is called the dual module of A.

6.2 DerivationsDefinition 6.2.1 Let A be an algebra, and let E be an A-bimodule. A linearmap D : A→ E is a derivation if

D(ab) = a · Db + Da · b (a, b ∈ A) . (6.2.1)

Here is an obvious example: let A = C (1)(I), E = C(I), and D : f �→ f ′,A→ E . Then E is a Banach A-module for the pointwise product, and D is acontinuous derivation.Equation (6.2.1) is the derivation identity. The set of derivations from A into

E is denoted by Z1(A, E); it is a linear subspace ofL(A, E). For example, takex ∈ E , and set

δx (a) = a · x − x · a (a ∈ A) .


Then, for a, b ∈ A, we have

δx (ab) = a · (b · x − x · b)+ (a · x − x · a) · b = δx (a) · b + a · δx (b) ,

and so δx is a derivation. Derivations of this form are termed inner derivations,and an inner derivation δx is implemented by x ; derivations which are not innerderivations are called outer derivations. In particular, the map

δb : a �→ ab − ba, A→ A ,

is an inner derivation on the algebra A. The set of inner derivations from A toE is a linear subspace N 1(A, E) of Z1(A, E).Note that Z1(A,Cϕ) consists of linear functionals d : A→ C such that

d(ab) = ϕ(a)d(b)+ d(a)ϕ(b) (a, b ∈ A) .

These maps are point derivations at ϕ.For example, let A = A(D), the disc algebra. Then the map f �→ f ′(0) is a

continuous point derivation at the character ε0 on A.Let A be a Banach algebra, and let E be a Banach A-bimodule. The space

of continuous derivations from A to E is denoted by

Z1(A, E) ,

and the space of (necessarily continuous) inner derivations is now N 1(A, E).A particular case to consider is that of derivations D : A→ A for a Banach

algebra A; in this case, we can also define the maps Dn : A→ A for eachn ∈ N. It seems that the range of such a derivation must be ‘small’ in somesense.

Theorem6.2.2 (Singer andWermer) Let A bea commutativeBanachalgebra,and let D : A→ A be a continuous derivation. Then D(A) ⊂ radA.

Proof Let z ∈ C. Then zD ∈ Z1(A, A).It is easy to check (using Leibniz’s identity, Exercise 6.3.4 (iii)) that exp(zD)

is an automorphism in B(A). For ϕ ∈ �A and a ∈ A, define

ϕz(a) = ϕ((exp(zD))(a)) = ϕ(a)+∞∑n=1

ϕ(Dna)

n!zn (z ∈ C) .


Then the map

F : z �→ ϕz(a), C → C ,

is an entire function. Also, for each z ∈ C, we have ϕz ∈ �A, and so it followsthat |ϕz(a)| ≤ ‖a‖. Thus F is bounded. By Liouville’s theorem, F is constant.In particular, ϕ(Da) = 0.This is true for each ϕ ∈ �A, and so Da ∈ rad A. �

The following result is the non-commutative generalization of the Singer–Wermer theorem.

Theorem 6.2.3 (Sinclair) Let A be a Banach algebra, and let D : A→ A bea continuous derivation. Then D(P) ⊂ P for each primitive ideal P of A.

Proof Let P be a primitive ideal, say P = {a ∈ A : aE = {0}} for a simpleleft A-module E . Then E is a Banach left A-module.Assume towards a contradiction that D(P) �⊂ P . Then there exists an

element a0 ∈ P with Da0 · x �= 0 for some x ∈ E . There exists b ∈ A withb · Da0 · x = x . But now D(ba0) · x = x because a0 · x = 0. It follows that(D(ba0))n · x = x (n ∈ N).By a result in the additional notes, below, we have

Dn((ba0)n)− n!(D(ba0))n ∈ P (n ∈ N) .

Hence n!x = Dn((ba0)n) · x , and so

‖x‖1/n ≤(1

n!

)1/n‖D‖ ‖ba0‖ ‖x‖1/n → 0 as n→∞ .

Thus x = 0, a contradiction. The result follows. �

6.3 Exercises

1. Let E and F be linear spaces. Then the tensor product E ⊗ F is definedby the following universal property: for each bilinear map S : E × F → G,there is a linear map T : E ⊗ F → G such that

T (x ⊗ y) = S(x, y) (x ∈ E, y ∈ F) .


Show that, if E is a left A-module and F is a right A-module, then E ⊗ Fis an A-bimodule for products satisfying

a · (x ⊗ y) = a · x ⊗ y, (x ⊗ y) · a= x ⊗ y · a (a ∈ A, x ∈ E, y ∈ F) .(6.2.2)

2. Let E and F be Banach spaces. Then the projective norm on E ⊗ F isdefined by

‖z‖π = inf{

n∑j=1‖x j‖‖y j‖ : z =

n∑j=1x j ⊗ y j ∈ E ⊗ F

}.

The completion of (E ⊗ F, ‖ · ‖π ) is the projective tensor product of Eand F , denoted by E ⊗ F . This tensor product has the following universalproperty: for each continuous bilinear map S : E × F → G into a Banachspace, there is a continuous linear map T : E ⊗ F → G such that

T (x ⊗ y) = S(x, y) (x ∈ E, y ∈ F) .

Suppose that E and F are Banach left and right A-modules, respectively.Show that E⊗F is a Banach A-bimodule for maps satisfying (6.2.2).In particular, we have the following important example that will return

in Chapter 7: the space A⊗A is a Banach A-bimodule for operations thatsatisfy the equations

a · (b ⊗ c) = ab ⊗ c, (b ⊗ c) · a = b ⊗ ca (a, b, c ∈ A) .

3. Let A be a Banach algebra.

(i) Let E and F be Banach left A-modules. For a ∈ A and T ∈ B(E, F),define

(a · T )(x) = a · T x, (T × a)(x) = T (a · x) (x ∈ E) .

Show that B(E, F) is a Banach A-bimodule for the maps

(a, T ) �→ a · T and (a, T ) �→ T × a .

(ii) Let E and F be Banach left and right A-modules, respectively. Showthat the map τ �→ Tτ is an isometric A-bimodule isomorphism fromthe dual A-bimodule (E⊗F)′ onto B(E, F ′). Here we are defining(Tτ x)(y) = 〈x ⊗ y, τ 〉 (x ∈ E, y ∈ F).


4. Let A be an algebra, let E be an A-module, and let D ∈ Z1(A, E).(i) Show that p · Dp · p = 0 for each idempotent p of A.(ii) If a ∈ A and a · Da = Da · a, then

D(an) = nan−1 · Da (n ≥ 2) .

(iii) In the case where E = A, we have Leibniz’s identity:

Dn(ab) =n∑k=0

(n

k

)Dka · Dn−kb (a, b ∈ A, n ∈ N) .

5. Let A be a unital algebra. Show that Z1(A,Cϕ) can be identified with thespace of linear functionals λ on Mϕ such that λ | M2ϕ = 0 and λ(eA) = 0.What is the analogous result when A is a unital Banach algebra, and we areconsidering continuous point derivations?

6. (i) Show that all point derivations on the disc algebra A = A(D) are con-tinuous. (Use the fact that, if z ∈ T and f ∈ Awith f (z) = 0, then thereexist g, h ∈ A with g(z) = h(z) = 0 and f = gh: this follows fromCohen’s factorization theorem, described in the notes to Chapter 3.)

(ii) Show that there are many discontinuous point derivations on the Banachalgebra (C (n)(I), ‖ · ‖n). (See Example 1.2(v).)


1. The theory of derivations fromaBanach algebra A into aBanach A-bimoduleE is concerned with the following questions.(I) When is every derivation from A into a specific Banach A-bimoduleautomatically continuous? For which Banach algebras A is it truethat every derivation from A into an arbitrary Banach A-bimoduleis automatically continuous? For an account of this topic, see Part II,Chapter 12.

(II) When is every continuous derivation from A into a class of BanachA-bimodules necessarily an inner derivation? For which commutativeBanach algebras A is it true that every continuous derivation from Ainto a Banach A-module is necessarily zero? Is there a canonical formfor an arbitrary continuous derivation from a Banach algebra A into aBanach A-bimodule?

(III) Can an arbitrary derivation be decomposed into the sumof a continuousderivation and a discontinuous derivation of a special type?


Many attractive results are known about these questions, but there re-main many challenging open questions. Perhaps the most important isProblem 9.1.13 of Part II; in the notation of Chapter 7, this asks ifH1(L 1(G), M(G)) = {0} for every locally compact group G.

2. Theorem 6.2.2 was proved in Singer andWermer (1955); the authors conjec-tured that the result should hold without the assumption that the derivationD be continuous. This was finally proved in the powerful paper (1988) ofThomas, building on an earlier result of Johnson (1969).

3. We used the following algebraic calculation in Theorem 6.2.3.Let D be a derivation on an algebra A, and let I be an ideal in A. Then,

for each a1, . . . , an ∈ I , we have

Dn(a1 · · · an)− n!(Da1) · · · (Dan) ∈ I .

The proof is as follows.We first make the claim that, for each k ≥ 2, we have D j (b1 · · · bk) ∈ I

whenever j ∈ {0, . . . , k − 1} andb1, . . . , bk ∈ I . The claim is true for k = 2,for certainly

D(b1b2) = b1 · Db2 + Db1 · b2 ∈ I

whenever b1, b2 ∈ I . Assume that the above claim holds for k, and now takeelements b1, . . . , bk+1 ∈ I . Then, by Leibniz’s identity,

D j (b1, . . . , bk+1) =j∑

i=0

(n + 1i

)Di (b1 · · · bk) · D j−1(bk+1) ( j ∈ N) ,

and so D j (b1 · · · bk+1) ∈ I ( j = 0, . . . , k). By induction on k, the claimholds.We now prove the main result by induction on n. The result is certainly

true if n = 1. Assume that the result is true for n, and take a1, . . . , an+1 ∈ I .By Leibniz’s identity,

Dn+1(a1 · · · an+1) =n+1∑j=0

(n + 1j

)D j (a1 · · · an)Dn+1− j (an+1) .

Each term on the right-hand side of this equality belongs to I , save perhapsfor the term (n + 1)Dn(a1 · · · an)D(an+1). By the inductive hypothesis, wehave

Dn(a1 · · · an)− n!(Da1) · · · (Dan) ∈ I ,


and so

Dn+1(a1 · · · an+1)− (n + 1)!(Da1) · · · (Dan+1)= Dn+1(a1 · · · an+1)− (n + 1)Dn(a1 · · · an)(Dan+1)+ (n + 1)(Dn(a1 · · · an)− n!(Da1) · · · (Dan))(Dan+1)

belongs to I . The induction continues.4. The natural non-commutative analogue of Thomas’s general version of theSinger–Wermer theorem would be the following statement.Let D be a derivation on a Banach algebra A. Then D(P) ⊂ P for each

primitive ideal P of A.This result has not been proved so far. However Thomas (1993) has proved

that D(P) ⊂ P for all but finitely many primitive ideals P , and that each ofthese exceptional primitive ideals has finite codimension in A.A closely related open question is the following. Let D be a derivation on

a Banach algebra A. Suppose that a ∈ A and that a · Da = Da · a. Doesit follow that Da is a quasi-nilpotent element? The Kleinecke–Shirokovtheorem states that this is true in the casewhere D is continuous.An attractivesurvey of these questions is given in Mathieu (1994).

5. FormanyBanach algebras A, all derivations from A into an arbitrary BanachA-bimodule E are continuous. For example, this is true for all C�-algebrasA. However there are discontinuous derivations from the disc algebra A(D)into certain Banach A(D)-modules (despite the fact that all point derivationson A(D) are automatically continuous).

6. The algebraic theory of modules and primitive ideals is contained in Dales(2000, §1.4) and Palmer (1994). For Banach modules, see Dales (2000,§2.6). The ‘attractive results’ about derivations that I know are all containedin Dales (2000, §5.6). Theorem 6.2.3 is from Sinclair (1969).

7

Cohomology

There is a substantial interest nowadays in the cohomology of Banach algebras;it is a development of an earlier, purely algebraic theory. This algebraic theory isnow usually presented in the language of homological algebra (exact sequences,functors, Ext, Tor, injective and projective resolutions, . . . ), and this approachhas been adopted by Banach algebraists – see Helemskii (1989). We shall avoidthis terminology here, and give an approach that originates with Hochschild.

7.1 Hochschild cohomology

Notation: let A be an algebra, and let E be an A-bimodule. We write Ln(A, E)for the linear space of all n-linearmaps from A × · · · × A (n copies) into E . (SetL0(A, E) = E .) The formal definition of the Hochschild cohomology groupsHn(A, E) is the following.

Definition 7.1.1 Let A be an algebra, and let E be an A-bimodule. For eachx ∈ E, define

δ0(x) : a �→ a · x − x · a, A→ E ,

and, for n ∈ N and T ∈ Ln(A, E), define δnT ∈ Ln+1(A, E) by

δnT (a1, . . . , an+1) = a1 · T (a2, . . . , an+1)+ (−1)n+1T (a1, . . . , an) · an+1

+n∑j=1(−1) j T (a1, . . . , a j−1, a ja j+1, a j+2, . . . , an+1) .

Let n ∈ Z+. It is clear that δn is a linear map fromLn(A, E) intoLn+1(A, E);these maps are the connecting maps. A direct but tedious calculation shows that

58

7 Cohomology 59

δn+1 ◦ δn = 0, and so im δn ⊂ ker δn+1. Indeed we have a complex:

L•(A, E) : 0 −→ Eδ0−→ L(A, E) δ1−→ L2(A, E) −→ · · · −→

Ln(A, E) δn−→ Ln+1(A, E) δn+1−→ Ln+2(A, E) −→ · · ·

(7.1.1)

of linear spaces and linear maps. For n ∈ N, the elements of ker δn and im δn−1

are the n-cocyles and the n-coboundaries, respectively; we set

Zn(A, E) = ker δn and Nn(A, E) = im δn−1 .

Definition 7.1.2 Let A be an algebra, and let E be an A-bimodule. For n ∈ N,the nth cohomology group of A with coefficients in E is

Hn(A, E) = Zn(A, E)/Nn(A, E) ;

also, H 0(A, E) = ker δ0 = {x ∈ E : a · x = x · a (a ∈ A)} .

In fact, the cohomology groups Hn(A, E) are linear spaces. Despite theimpressive generality of our definition of Hn(A, E) for each n ∈ N, we shallreally only consider H 1(A, E) in this chapter (but see the notes); let us see whatH 1(A, E) measures.For T ∈ L(A, E), we have T ∈ im δ0 if and only if there exists x ∈ E with

T (a) = a · x − x · a (a, b ∈ A) .

Also

(δ1T )(a, b) = a · Tb − T (ab)+ Ta · b (a, b ∈ A) . (7.1.2)

Thus N 1(A, E) and Z1(A, E) coincide with our previous definitions of thisnotation in §6.2, and H 1(A, E) is the quotient of the space of all derivationsby the space of inner derivations; clearly, H 1(A, E) = {0} if and only if everyderivation from A into E is inner.Now let A be a Banach algebra, and let E be a Banach A-bimodule. Then

we write Bn(A, E) for the Banach space of bounded maps in Ln(A, E), andthen define the complex B•(A, E) as in (7.1.1), above, with B replacing L. Theelements of ker δn and im δn−1 are now the continuous n-cocyles and continuousn-coboundaries, respectively, and the spaces thereof are

Zn(A, E) and N n(A, E) .


We then define the nth continuous cohomology group of A with coefficients inE as

Hn(A, E) = Zn(A, E)/N n(A, E) .

This is a seminormed space. (In general,N n(A, E) is not closed in Zn(A, E).)Thus H1(A, E) is the quotient of the space of continuous derivations from Ato E by the space of inner derivations.Although we shall not discuss H 2(A, E) in these lectures, we do note that,

for each T ∈ Z2(A, E), we have the 2-cocycle identity

a · T (b, c)− T (ab, c)+ T (a, bc)− T (a, b) · c = 0 (a, b, c ∈ A) ,

and thus H 2(A, E) = {0} if and only if each such T has the form δ1S for someelement S ∈ L(A, E). Even at the level of n = 2, direct calculations are not soeasy.

7.2 Amenable Banach algebras

Algebraists were interested in characterizing those algebras A such thatH 1(A, E) = {0} for each A-bimodule E .Let A be a unital algebra, and let π : A ⊗ A→ A be the linear map such

that π (a ⊗ b) = ab (a, b ∈ A). A diagonal for A is an element u ∈ A ⊗ Asuch that π (u) = eA and a · u = u · a (a ∈ A).

Theorem 7.2.1 Let A be an algebra. Then the following are equivalent:(a) H 1(A, E) = {0} for every A-bimodule E;(b) A is unital and has a diagonal in A ⊗ A;(c) A is semisimple and finite-dimensional. �

The analogue for Banach algebras is given in the notes.However, the class of Banach algebras that has proved to be important is that

of amenable algebras.

Definition 7.2.2 (Johnson) Let A be a Banach algebra. Then A is amenable ifH1(A, E ′) = {0} for every Banach A-bimodule E.

Thus A is amenable if every continuous derivation into a dual module isinner. We shall explore this notion a little. Throughout A is a unital Banach

7 Cohomology 61

algebra. It can be checked that, to show A is amenable, it is sufficient to showthatH1(A, E ′) = {0} for each unital Banach A-bimodule E .Let E and F be Banach spaces, and let T ∈ B(E, F). The dual of T is the

element T ′ ∈ B(F ′, E ′) specified by

〈x, T ′λ〉 = 〈T x, λ〉 (x ∈ F, λ ∈ F ′) ;

this map is continuous when E ′ and F ′ have their weak-∗ topologies. Thesecond dual of T is T ′′ = (T ′)′ in B(E ′′, F ′′).The following generalization of the notion of a diagonal was introduced in

Johnson (1972b).

Definition 7.2.3 Let A be a unital Banach algebra. A virtual diagonal for A isan elementM in (A⊗A)′′ such that π ′′(M) = eA and a · M = M · a (a ∈ A).

Theorem 7.2.4 Let A be a unital Banach algebra. Then A is amenable if andonly if A has a virtual diagonal.

Proof Let A have a virtual diagonal M, let E be a unital Banach A-bimodule,and let D ∈ Z 1(A, E ′).For each x ∈ E , define �x ∈ (A⊗A)′ by

〈a ⊗ b, �x 〉 = 〈x, a · Db〉 = 〈x · a, Db〉 (a, b ∈ A) ,

and then define λ ∈ E ′ by

〈x, λ〉 = 〈M, �x 〉 (x ∈ E) .

For a, b, c ∈ A and x ∈ E , we have

〈b ⊗ c, �a · x−x · a〉 = 〈a · x − x · a, b · Dc〉 = 〈x, b · Dc · a − ab · Dc〉

and

〈b ⊗ c, a · �x −�x · a〉 = 〈b ⊗ ca − ab ⊗ c, �x 〉= 〈x, bc · Da + b · Dc · a − ab · Dc〉 ,

and so

〈b ⊗ c, �a · x−x · a〉 = 〈b ⊗ c, a · �x −�x · a〉 − 〈x, bc · Da〉 .


Hence, for each v ∈ A⊗A, it follows that

〈v,�a · x−x · a〉 = 〈v, a · �x −�x · a〉 − 〈x · π (v), Da〉 (a ∈ A, x ∈ E) .(7.2.1)

Let v run through a bounded net (va) ⊂ A⊗A that converges in the weak-∗topology to M. Then π (va)→ eA in the weak topology on A; a standardaveraging procedure (Mazur’s theorem) allows us to suppose that π (va)→ eAin the norm-topology of A, and then x · π (va)→ x in E . Thus, from (7.2.1),we have

〈a · x − x · a, λ〉 = 〈a · M−M · a, �x 〉 − 〈x, Da〉= −〈x, Da〉 (a ∈ A, x ∈ E) .

Hence Da = a · λ− λ · a (a ∈ A) in E ′, and D is inner. This shows that Ais amenable.Now suppose that A is amenable. Define v = eA ⊗ eA in A⊗A. Then

π ′′(a · v − v · a) = π (a ⊗ eA − eA ⊗ a) = 0 ,

and so δv ∈ Z1(A, kerπ ′′). Set

X = (A⊗A)′/π ′(A′) .Then X ′ is identifiedwith kerπ ′′ as a Banach A-bimodule. Since A is amenable,there existsw ∈ kerπ ′′ with δw = δv . Define u = v − w . Then, for each a ∈ A,we have

a · u − u · a = (δu − δw )(a) = 0 and π ′′(u) = π ′′(v) = eA ,

and so u is a virtual diagonal. �

Let G be a group, and let A = 1(G). Given f, g ∈ A, we define f ⊗ g onG ⊗ G by

( f ⊗ g)(s, t) = f (s)g(t) (s, t ∈ G) .

By an earlier remark, there is an isometric isomorphism A⊗A→ 1(G × G)that identifies f ⊗ g in A ⊗ A with f ⊗ g as defined above; we thus identifyA⊗A as 1(G × G).

7 Cohomology 63

The dual space of 1(G) is ∞(G), and the actions of δs ∈ 1(G) on f ∈ ∞(G)are given by

(δs · f )(t) = f (ts), ( f · δs)(t) = f (st) (t ∈ G) .

The dual of 1(G × G) is (A⊗A)′ = ∞(G × G), and this is a Banach A-bimodule: the module operations · satisfy the equations

(δs · F)(u, v) = F(u, vs), (F · δs)(u, v) = F(su, v) (u, v ∈ G) ,

where s ∈ G and F ∈ ∞(G × G). Next we see that the dual of the inducedproduct map π : A⊗A→ A is π ′ : ∞(G)→ ∞(G × G), where

π ′( f )(u, v) = f (uv) (u, v ∈ G, f ∈ ∞(G)) .

Thus a virtual diagonal is a continuous linear functional M on ∞(G × G) suchthat

〈F · δs, M〉 = 〈δs · F, M〉 (s ∈ G, F ∈ ∞(G × G)) (7.2.2)

and

〈π ′( f ), M〉 = f (eG) ( f ∈ ∞(G)) . (7.2.3)

Definition 7.2.5 Let G be a group. A mean on ∞(G) is a continuous linearfunctional � on (∞(G), | · |G) such that �(1) = ‖�‖ = 1. The mean � isleft-invariant if

〈 f, �〉 = 〈 f · δx , �〉 (s ∈ G, f ∈ ∞(G)) .

The group G is amenable if there is a left-invariant mean on G.

Thus a mean on G is an element of the state space of the C∗-algebra ∞(G).It is easy to see that every abelian group and every compact group is amenable,but the free group on two generators F2 is not amenable.Here is the theorem that suggested the name ‘amenable’ for the class of

Banach algebras that we are considering.

Theorem 7.2.6 (Johnson) Let G be a group. Then the Banach algebra 1(G)is amenable if and only if the group G is amenable.


Proof Suppose that G is amenable, and let � be a left-invariant mean on∞(G). For F ∈ ∞(G × G), set

F(t) = F(t, t−1) (t ∈ G) ,

so that F ∈ ∞(G) with |F |G ≤ |F |G×G , and then define

〈F,M〉 = 〈F,�〉 . (7.2.4)

Then M ∈ ∞(G × G)′ with ‖M‖ = 1.We claim that M is a virtual diagonal for ∞(G). For F ∈ ∞(G × G) and

s, t ∈ G, we have

F · δs(t) = (F · δs)(t, t−1) = F(st, t−1)

and

δs · F(t) = (δs · F)(t, t−1) = F(t, t−1s) ,

and so δs · F(st) = F · δs(t). Since � is left-invariant, (7.2.2) follows from(7.2.4). Now take f ∈ ∞(G), and set F = π ′( f ) ∈ ∞(G × G). Then

F(t) = f (t t−1) = f (eG) (t ∈ G) ,

and so 〈π ′( f ),M〉 = f (eG), giving (7.2.3).Thus M is a virtual diagonal, as claimed, and so ∞(G) is amenable by

Theorem 7.2.4.For the converse, let M be a virtual diagonal for ∞(G). For f ∈ ∞(G), set

f (u, v) = f (v) (u, v ∈ G) ,

so that f ∈ ∞(G × G) and ∣∣ f ∣∣G×G = | f |G , and then define

〈 f,�〉 = 〈 f ,M〉 . (7.2.5)

Then�(1) = 1.Also, for each s ∈ G and f ∈ ∞(G),we have f · δs = δs · fand f · δs = f , and so

〈 f · δs,�〉 = 〈 f · δs,M〉 = 〈δs · f ,M〉 = 〈 f · δs,M〉 = 〈 f ,M〉 = 〈 f,�〉 .

Thus� is left-invariant. It may be that ‖�‖ �= 1, but this can be fixed by a littletrick, and so G is amenable. (See also 11.1.2.) �

7 Cohomology 65

Proposition 7.2.7 Let A and B be Banach algebras, and let θ : A→ B be acontinuous homomorphism with θ (A) = B. Suppose that A is amenable. ThenB is amenable.

Proof This is an easy exercise. �

Theorem 7.2.8 For each compact space , the Banach algebra C() isamenable.

Proof SetG = C(,R), regarded as an abelian groupwith respect to addition.The map

θ :∑h∈G

αhδh �→∑h∈G

αhexp(ih), 1(G)→ C() ,

is a continuous homomorphism, where δh is the point mass at h. By theStone–Weierstrass theorem, θ has dense range. Since 1(G) is amenable, so isC(). �


1. The algebraic cohomology groups Hn(A, E) were first defined byHochschild (1945, 1946).

2. We can regard L(A, E) itself as an A-bimodule for the products �, where

(a � T )(b) = a · Tb, (T � a)(b)

= T (ab)− Ta · b (a, b ∈ A, T ∈ L(A, E)) .

The formal identification of Ln+1(A, E) with Ln(A,L(A, E)) then leadsto the representation of each Ln(A, E) as an A-bimodule. We then havea useful reduction-of-dimension formula: for k, p ∈ N, we have a linearisomorphism Hk+p(A, E) & Hk(A,Lp(A, E)).

3. Let Ln(A, E) be the space A ⊗ · · · ⊗ A ⊗ E , where there are n copies ofA. Then L1(A, E) is an A-bimodule for the products given by

(a ⊗ x) � b = a ⊗ x · b, b � (a ⊗ x)= ba ⊗ x − b ⊗ a · x (a, b ∈ A, x ∈ E) .

The spacesLn(A, E) form a complex of A-bimodules for certain connectingmaps dn , and the homology groups Hn(A, E) are defined to be

ker dn−1/im dn .


We have Hn(A, E) = {0} if and only if Hn(A, E×) = {0}, where E× is the(algebraic) dual space of E .

4. The Banach algebra analogue of Theorem 7.2.1 is the following. Now π isa (continuous) map from A⊗A onto A.

Theorem 7.3.1 Let A be a Banach algebra. Then the following are equiv-alent:

(a) H1(A, E) = {0} for every Banach A-bimodule E;(b) A is unital and has a diagonal in A⊗A. �

The conditions probably imply that A is semisimple and finite-dimensional,but this has only been proved for various classes of Banach algebras. For astudy of this question, see Runde (1998) and Runde (2002).

5. There is a reduction-of-dimension theorem for the continuous cohomologygroups: for k, p ∈ N, we haveHk+p(A, E) & Hk(A,B p(A, E)). It followsthat, for an amenable Banach algebra A, Hn(A, E ′) = {0} for each n ∈ N

and each Banach A-bimodule E .6. A somewhat different proof of Theorem 7.2.6 is given in Bonsall and Dun-can (1973, §43).

7. Let G be a locally compact group. Then there is a definition of ‘G isamenable’ which is essentially the same as Definition 7.2.5, and then it isagain true (Johnson 1972a) that the group G is amenable if and only ifthe Banach algebra L 1(G) is amenable; see Dales (2000, §5.6). For a proofrelated to that of Theorem 7.2.6, see Stokke (2003). For a further discussionof amenable groups, see Part II.

8. There is an industry which seeks to calculate Hn(A, E) for many specialBanach algebras A and Banach A-bimodules E . For example, let A be avonNeumann subalgebra ofB(H ). ThenH1(A,B(H )) is calculated in verymany cases in Sinclair and Smith (1996). It is a deep theorem of Haagerup(1983) and others that a C∗-algebra A is amenable if and only if A is whatis called nuclear. For a new version of this proof, see Runde (2002).

9. Let A be a uniform algebra on . Then A is amenable if and only ifA = C(). This is a theorem of Scheinberg, given in Helemskii (1989).

10. A Banach algebra A is said to be weakly amenable if H1(A, A′) = {0}.Every C*-algebra and every group algebra of the form L 1(G) is weaklyamenable. A paper by Dales, Ghahramani, and Helemskii (2002) deter-mines when the measure algebra M(G) is amenable and when it is weaklyamenable. It is an interesting open question whether every natural, weaklyamenable uniform algebra on a compact space is necessarily equal toC().

7 Cohomology 67

11. LetG be an amenable group, so that there is a left-invariantmean on ∞(G).Then there is anothermean on ∞(G)which is both left- and right-invariant;see Dales (2000, 3.3.49).

12. Let E be an infinite-dimensional Banach space. Then we suspect that B(E)is never amenable. This has been proved in the case where E is a Hilbertspace by surprisingly deep methods from C*-algebra theory. It has alsobeen proved for the case where E is 1 by Read (2001). We should like tohave a proof that establishes the result at least for each of the spaces p

(for 1 ≤ p ≤ ∞).13. Let A be a Banach algebra with radical R, and set A = A/R. Then∑

: 0→ R → Aq→ A→ 0

is a short exact sequence of Banach algebras which is an extension of A.The sequence

∑is admissible in the case where R is complemented as

a Banach space, and singular if R2 = 0; in the latter case R is a BanachA-bimodule.The algebra A has a strong Wedderburn decomposition (SWD) if there

is a closed subalgebra B of A such that B ∼= A amd A = B⊕ R. Thenwe have a theorem that shows the role of the second cohomology groupsH2(A, · ) of A: in the case where∑ is admissable and singular, A has aSWD if and only ifH2(A,R) = {0}.

14. Algebraic cohomolgy theory is described inDales (2000, §1.9), andBanachcohomology in Dales (2000, §2.8). A substantial account, from a differ-ent perspective, is given in Helemskii (1989). The seminal definition of‘amenable Banach algebra’ is from Johnson (1972a), where we also findTheorem 7.2.6. There is an enormous literature on the cohomology of C*-algebras: for a clear introductory account, see Sinclair and Smith (1995).An attractive new book on amenable Banach algebras is Runde (2002). Aclassic text on amenability is Paterson (1988); see also Part II, Chapter 11,of the present book. For an essay onWedderburn decompositions, includingreferences to earlier work, see Bade, Dales, and Lykova (1999).

The literature on Banach algebras

There is an enormous literature on Banach algebras. A good place to start isthe fine pair of books by Rudin (1973, 1996). From there, one can progress tomore specialized accounts. See also Meise and Vogt (1997).For the general theory, consult Bonsall and Duncan (1973) and Bourbaki

(1960). A more recent account at a fairly elementary level is Helemskii (1993).For a work with an emphasis on the cohomology of Banach algebras, seeHelemskii (1989). A comprehensive account, with an emphasis on the algebraicside, is in Palmer (1994, 2001). Everything mentioned in this part of the book,and a huge amount more, is contained in Dales (2000).The classic books onuniformalgebras areGamelin (1969) andStout (1971).

It seems that this subject is not so fashionable in recent years.There aremany texts on harmonic analysis, which is the study of the Banach

algebras L1(G). For example, see Graham and McGehee (1979), Hewitt andRoss (1963, 1970),Kahane (1970),Katznelson (1976), andReiter andStegeman(2000).For the theory of amenable groups and amenable Banach algebras, see

Paterson (1988) and Runde (2002).The theory of C∗-algebras was barely mentioned in the text; this is a world-

wide industry. See the following texts, for example: Doran and Belfi (1986),Kadison and Ringrose (1983, 1986), Murphy (1990), Pedersen (1979), andTakesaki (1979, 2002).For the connection with abstract algebra and set theory (including the con-

struction of discontinuous homomorphisms fromC()), see Dales andWoodin(1987, 1996).For more advanced work on spectral theory and the (several-variable) func-

tional calculus, see Eschmeier and Putinar (1996).For a collection of articles on Banach algebra theory, including one on the

history of 13 conferences on the subject, see Albrecht and Mathieu (1998).

68

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Aupetit, B. (1982). The uniqueness of the complete norm topology inBanach algebras and Banach Jordan algebras, J. Functional Analysis, 47,1–6.

Bade, W. G. and Curtis, P. C., Jr (1960). Homomorphisms of commutativeBanach algebras, American J. Math., 82, 589–608.

Bade, W. G., Dales, H. G., and Lykova, Z. A. (1999). Algebraic and strongsplittings of extensions of Banach algebras, Memoirs American Math.Soc., 656.

Bhatt, S. J. and Dedania, H. V. (2001). Banach algebras with unique uniformnorm II, Studia Math., 147, 211–35.

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Bourbaki, N. (1960). Elements de mathematiques. Theories spectrales, Paris,Hermann.

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Dales, H. G. (2000). Banach algebras and automatic continuity, LondonMath-ematical Society Monographs, 24, Oxford, Clarendon Press.

Dales, H. G., Ghahramani, F. and Helemskii, A. Ya. (2002). The amenabilityof the measure algebra, J. London Math. Soc., (2), 66, 213–26.

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Dales, H. G. and Loy, R. J. (1997). Uniqueness of the norm topology for Banachalgebras with finite-dimensional radical, Proc. LondonMath. Soc., (3), 26,69–81.

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Dales, H. G. andWoodin, W. H. (1996). Super-real fields: totally ordered fieldswith additional structure, LondonMathematical Society Monographs, 14,Oxford, Clarendon Press.

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Part II

Harmonic analysis and amenability

GEORGE A. WILLISUniversity of Newcastle, New South Wales, Australia

8

Locally compact groups

The set of real numbers, R, plays several fundamental roles in mathematics. Itappears as:

� an algebraic object – either an abelian group or a field;� a topological space – the concepts of continuous function on R and contin-uous curve are the most basic in topology;

� a differential manifold – differentiability of functions is defined first forfunctions on R and differentiability on higher dimensional manifolds isdefined in terms of real parameters; and

� a measure space – the measure of interval length is the basis for theories ofintegration.

These structures are related to each other and all are important in harmonic, orFourier, analysis.Locally compact groups also possess all of these structures. The algebraic

and topological structures are hypothesized but differential and measure spaceproperties then follow automatically. The problems and applications of har-monic analysis extend to the more general setting of locally compact groupsand the algebraic, topological, differential and integral structures continue tobe fundamentally important.

8.1 Definition and examples

The definition of the class of locally compact groups is very simple.


75

76 Part II Analysis and amenability, George A. Willis

Definition 8.1.1 A topological group is a group G which is also a Hausdorfftopological space so that the maps

(x, y) �→ xy : G × G → G and (8.1.1)

x �→ x−1 : G → G (x, y ∈ G) , (8.1.2)

are continuous. The topological group G is a locally compact group if thetopology on G is locally compact.

In these notes group multiplication will be denoted by juxtaposition andthe group identity will be denoted by e. The class of locally compact groupsincludes many important examples.

Examples (a) Any group is locally compact when equipped with the discretetopology. This includes in particular all finite groups. Here are some discretegroups which will be mentioned.The group of integers is denoted byZ. The cyclic group of order n is denoted

by Cn .The free group on two generators, denoted by F2, is a frequently used ex-

ample. Let the generators be a and b. Then a word in a, b, a−1, and b−1 is afinite sequence w = l1l2 · · · ln , where l j = a, b, a−1, or b−1 for each j . A wordw is said to be reduced if no cancellation is possible, which means that wedo not have a and a−1 as adjacent letters in w , and similarly for b and b−1.Thus abba−1ba is reduced and abbb−1a−1ba is not reduced. If w = l1l2 · · · lnis reduced, then n is called the length of w and denoted by |w |.The free group on two generators a and b consists of all reduced words in a,

b, a−1, and b−1, including the emptywordwhich is denoted by e. Twowords aremultiplied by concatenating them and then performing any cancellations until areducedword is achieved. Thus, for example, ifw1 = aba−1 andw2 = ab−1ab,then

w1w2 = ab(a−1a)b−1ab = a(bb−1)ab = aab .

The empty word is the identity element in F2 and, if w = l1l2 · · · ln , then wesee that w−1 = l−1n · · · l−12 l−11 .Free groups on any number of generators are defined similarly. The free

group on a countably infinite number of generators will be denoted by F∞.(b) The groups (R,+) and (C,+) with their usual topology are both locally

compact and abelian, as are (R \ {0},×) and (C \ {0},×).Similarly, (Rn,+) and (Cn,+) are locally compact groups, and so are all

matrix groups overR andC. Thus for instance SL(n,R), GL(n,C) and SO(n)

8 Locally compact groups 77

are locally compact groups with the subspace topology they inherit as subsetsof Rn2 or Cn2 .These matrix groups are all examples of Lie groups. A Lie group is a group

which is also a differential manifold such that the group operations (8.1.1) and(8.1.2) are not just continuous but analytic.(c) The circle group T := {z ∈ C : |z| = 1} with the usual multiplication of

complex numbers is a compact Lie group. It is isomorphic to the rotation groupSO(2).Any product of compact groups with the product topology is a compact

group, by Tychonov’s theorem. Thus T∞ is a compact group but is not a Liegroup. The infinite product (C2)∞ of copies of the cyclic group of order 2 isalso a compact group.(d) The fieldQp of p-adic numbers is a locally compact field. Hence (Qp,+)

and (Qp \ {0},×) are locally compact abelian groups. The open subgroup(Zp,+) of p-adic integers is compact.In the same way as for the fields R and C, matrix groups overQp are locally

compact groups with the subspace topology as subsets of Q n2p . The notion of

an analytic map over Qp may be defined as well and so may the notion of Liegroup over Qp.The field k((X )) of Laurent series over the finite field k is locally compact.

The topology on this field is defined by letting the subgroups

k[[X ]]n :={f =

∞∑j=nk j X

j : k j ∈ k

}

be a base of neighbourhoods of 0. The subring k[[X ]] = k[[X ]]0 of Taylorseries over k is then compact and open, and the topology on k[[X ]] is just theproduct topology on k∞. Matrix groups over k((X )) are also locally compactgroups.(e) Let G be a locally finite graph. Then the automorphism group Aut(G)equipped with the topology of uniform convergence on compact sets is a locallycompact group.In particular, let Tq be the homogeneous tree where every vertex has de-

gree (or valency) q + 1. Then Aut(Tq ) is a locally compact group. A base ofneighbourhoods for the identity e consists of the stabilizer subgroups

V (F) := {x ∈ Aut(Tq ) : x(v) = v (v ∈ F)},

where F ranges over all finite subsets of the vertex set V . Then V (F) is acompact open subgroup for each F .


8.2 Structure theory

Here are a couple of often used facts about topological groups whose proofsare left as exercises.

Lemma 8.2.1 Let G be a locally compact group and V be a neighbourhoodof e. Then there is a symmetric neighbourhood W of e with compact closuresuch that W 2 ⊂ V . �

Lemma 8.2.2 Let G be a topological group. Let K and U subsets of G withK compact, U open and K ⊂ U. Then there is an open neighbourhood V of esuch that

V K ∪ KV ⊂ U . �

The following result is the starting point of the structure theory of locallycompact groups.

Theorem 8.2.3 Let G be a locally compact group. Then the connected com-ponent of the identity, called Ge, is a closed normal subgroup, and the quotientgroup G/Ge is a totally disconnected locally compact group.

Proof Connected components are always closed. We show that Ge is a normalsubgroup.Let x and y be in Ge. Then e and x are in the same connected component and

so ey = y and xy also belong to the same component (because right translationby y is a homeomorphism, see Exercise 8.4.1). Since y ∈ Ge it follows thatxy ∈ Ge. The map x �→ x−1 is also a homeomorphism and it fixes e. HenceG−1e = Ge and so Ge is a subgroup.Next, let x ∈ G. Then the conjugationmap y �→ xyx−1 is a homeomorphism

and so xGex−1 is a connected component of G. Since conjugation fixes e, wehave xGex−1 = Ge.Since Ge is closed, G/Ge is Hausdorff. The image of a compact neighbour-

hood of e inG will be a compact neighbourhood of the identity inG/Ge. HenceG/Ge is locally compact. The connected components inG/Ge consist of singlepoints, that is, G/Ge is totally disconnected. �

The detailed study of the structure of locally compact groups thus dividesinto the study of connected and of totally disconnected groups separately andthen of extensions.Among the examples described above, the groups Rn , SL(n,R), T, and

C \ {0} are connected. Discrete groups are totally disconnected, but some


more interesting examples include matrix groups over Qp and k((X )), infi-nite products (Cn)∞ and automorphism groups of locally finite graphs, such asAut(Tq ).Connected groups are relatively well understood through the following the-

orem. It was proved in the early 1950s by Gleason (1952), Montgomery andZippin (1952), and Yamabe (1953), and is the solution of Hilbert’s 5th problem.Unified accounts are given in Montgomery and Zippin (1955) and Kaplansky(1971).

Theorem 8.2.4 (Gleason, Montgomery and Zippin, Yamabe) Let G be a con-nected locally compact group, and let O be a neighbourhood of e. Then thereis a compact normal subgroup K ⊂ O such that G/K is a Lie group. �

The theorem shows that analytic structure emerges naturally in connectedgroups, even though only topological structure is assumed. Many results aboutconnected locally compact groups are established by factoring out a small com-pact normal subgroup and then applying Lie group techniques to the quotient.This approach is known as ‘approximation by Lie groups’. The theorem alsosays that each connected group is a projective limit of Lie groups or, in otherwords, every connected group is pro-Lie.The following result due to van Dantzig (1936) is the starting point for the

structure theory of totally disconnected locally compact groups.

Theorem8.2.5 (vanDantzig) Let G be a totally disconnected locally compactgroup, and let O be a neighbourhood of e. Then there is a compact opensubgroup U ⊂ O. �

Note that each totally disconnected locally compact space has a base ofneigbourhoods consisting of compact open subsets and therefore has inductivedimension equal to zero. For this reason totally disconnected locally compactgroups are also known as 0-dimensional groups. Van Dantzig’s theorem saysthat there is in fact a base of neighbourhoods of e consisting of compact opensubgroups.In the case where G is compact more can be said.

Corollary 8.2.6 Let G be a totally disconnected compact group. Then G hasa base of neighbourhoods of the identity consisting of compact open normalsubgroups. Hence G is profinite and is isomorphic to a closed subgroup of aproduct of finite groups. �


The theorem does not assert that, for a general totally disconnected locallycompact group, there is a base of neighbourhoods of compact open normalsubgroups however, andgroups such as SL(n,Qp) andAut(Tq ) haveno compactopen normal subgroup. Although particular examples of these types are wellunderstood, the general theory of totally disconnected groups did not progressfar beyond van Dantzig’s theorem for a long time. Relatively recent progresson the structure theory of totally disconnected groups (see Willis 1994, 2001,2002) determines just how such groups may fail to have compact open normalsubgroups. It introduces methods which begin to parallel the Lie techniquesused to analyze connected groups.When G is abelian, the structure of G can be described more fully. The

following theoremmay be found in Hewitt and Ross (1979), Morris (1997) andRudin (1962).

Theorem 8.2.7 Let G be a compactly generated locally compact abeliangroup. Then G is isomorphic toRm × Zn × K, where m and n are non-negativeintegers and K is a compact group. �

A compactly generated group is one which is generated by a compact sub-set. In a locally compact group, any compact neighbourhood of the identitygenerates an open subgroup. Hence every locally compact abelian group hasan open compactly generated subgroup to which Theorem 8.2.7 applies. Therequirement that G be compactly generated is essential: the group (Qp,+) isnot compactly generated and does not satisfy the conclusion of the theorem.Many classes of locally compact groups have been studied and some also

have more-or-less complete structure theories. These classes are given namessuch as the [SIN]-groups (those which have arbitrarily small neighbourhoodsof e invariant under conjugation) and the [Z]-groups (those for which thequotient by the centre is compact). An excellent survey of the current stateof knowledge of classes of locally compact groups can be found in Palmer(2001).

8.3 Haar measure

The familiar Lebesgue measure on R is related to the additive group structurebecause it is invariant under translation, that is, the measure of a set is equalto the measure of its translate by any real number. Equivalently, the integral ofany function on R is equal to the integral of its translate by any real number.All locally compact groups support a translation-invariant Borel measure. Theσ -algebra of Borel subsets of the locally compact group G will be denoted byB(G).


Theorem 8.3.1 (Haar) Let G be a locally compact group. Then there is apositive Borel measure, mG, on G satisfying the following properties for everyB ∈ B(G):

1. mG(x B) = mG(B) for each x ∈ G;2. mG(B) = inf{mG(U ) : B ⊂ U, U is open} and mG(U ) > 0 for every non-empty, open set U;

3. mG(B) = sup{mG(K ) : K ⊂ B, K is compact}.Any measure satisfying these properties is equal to a positive multiple of mG.

�

The measure mG is called the Haar measure on G. Property 1 says that mGis translation-invariant, while Properties 2 and 3 say that mG is a regular Borelmeasure. When there is no ambiguity the Haar measure will be denoted simplyby m.The translation-invariant measure was constructed for compact groups by

von Neumann, and he also proved the uniqueness up to a positive multiple.The translation-invariant measure on locally compact groups was constructedby Haar; see Hewitt and Ross (1979).The Haar measure need not be right-invariant. However, if we fix a Haar

measurem then, for each x ∈ G, the measuremx : B �→ m(Bx), B(G))→ C,is left-invariant and so is equal to a positive multiple of m.

Definition 8.3.2 Let m be a left-invariant Haar measure on G, and, for eachx ∈ G, let �(x) be the positive scalar such that mx = �(x)m. Then the map� : G → (R+,×) is a continuous group homomorphism;� is called themod-ular function on G.If � ≡ 1, then G is a unimodular group.

If G is discrete, then the counting measure is translation-invariant and it isconventional to work with this measure.

Proposition 8.3.3 The locally compact group G is compact if and only ifmG(G) <∞. �

For compact groups it is conventional to normalize the Haar measure so thatmG(G) equals 1. Finite groups are both discrete and compact and these twoconventions are not consistent in this case.In the case of compact groups there is an alternative way to produce the Haar

measure as a limit of convolution powers of a probability measure. The idea ofusing convolution powers of a probability measure, or equivalently a random


walk on a group G, to obtain information about G will be used in some of theresults mentioned in later chapters.

Theorem 8.3.4 (Kawada–Ito) Let G be a compact group, and let µ be aprobability measure on G such that the support of µ generates G. Then thesequence of probability measures(

1

n

n∑j=1

µ j

)∞n=1

converges to the Haar measure on G in the weak-∗ topology on M(G). �

A. Weil shows how a translation-invariant measure m on a group G may beused to define a topology on G which essentially turns it into a locally compactgroup and turns m into Haar measure (Halmos 1950; Weil 1951). Thus locallycompact groups are the only topological groups which support a translation-invariant measure. This is an important observation because, as will be seenlater, the Haar measure is used to produce unitary representations of locallycompact groups.This section concludes with the definitions of some spaces of functions on

groups and of the idea of the translate of a function.

Definition 8.3.5 Let f be a function on the group G. The left translate of fby a ∈ G is the function a f defined by

a f (y) = f (a−1y) (y ∈ G) .

The right translate of f by a ∈ G is the function fa defined by

fa(y) = f (ya) (y ∈ G) .

Definition 8.3.6 Let G be a locally compact group.

(i) The space of bounded continuous functions on G will be denoted by C(G)and the subspace of functions which converge to zero at infinity by C0(G).The norm of a function f in C(G) is

‖ f ‖∞ := sup{| f (x)| : x ∈ G} .

(ii) Let 1 ≤ p <∞. The space of measurable functions (with functions identi-fied if they are equal almost everywhere) on G satisfying

∫G | f |p dm <∞


will be denoted by L p(G). The norm of a function f in L p(G) is

‖ f ‖p =(∫

G| f |pdm

)1/p.

(iii) The space of essentially bounded measurable functions (with functionsidentified if they are equal locally almost everywhere) on G will be denotedby L∞(G). The norm of a function f in L∞(G) is

‖ f ‖∞ = ess sup {| f (x)| : x ∈ G} .

(iv) The space of bounded measures on G will be denoted by M(G). The normof a measure µ in M(G) is

‖µ‖ = sup{

n∑j=1|µ(E j )| : G =

n⋃j=1E j is a Borel partition of G

}.

Throughout Part II of this book:

� the dual space of C0(G) will be identified with M(G) via the pairing

〈 f, µ〉 =∫Gf dµ ( f ∈ C0(G), µ ∈ M(G)) ;

� the dual space of L p(G) for 1 ≤ p <∞will be identified with Lq (G), whereq is the conjugate index to p with 1/p + 1/q = 1, via the pairing

〈 f, g〉 =∫Gf g dm ( f ∈ L p(G), g ∈ Lq (G)) .

Left-invariance of the Haar measure is equivalent to the left-invariance of theintegral.

Proposition 8.3.7 Let f be a measurable function on the locally compactgroup G. Then∫

Gx f dm =

∫Gf (x−1y) dm(y) =

∫Gf (y)dm(y) =

∫Gf dm (x ∈ G) .

�

This identity, when applied to the dual pairing of L p(G) and Lq (G), impliesthat the dual of left translation by a is the same as left translation by a−1.


Proposition 8.3.8 Let 1 ≤ p <∞, and let f ∈ L p(G) and g ∈ Lq (G).Then

〈a f, g〉 = 〈 f, a−1g〉 (a ∈ G) . �

8.4 Exercises

1. The left translation by a ∈ G of the group G is the map x �→ ax, G → G.Show that left translation is a homeomorphism.

2. Prove Lemmas 8.2.1 and 8.2.2.3. LetK be a locally compact field. Show that the connected component of 0 isan ideal inK, and deduce thatK is either connected or totally disconnected.(This is the first step in the classifcation of locally compact fields discussedbelow in the additional notes.)

4. Let M be an n × n matrix with positive entries such that every row andcolumn sum is equal to 1. Thus M = (Mi j

), where Mi j > 0,

∑ni=1 Mi j = 1,

and∑n

j=1 Mi j = 1 for i, j = 1, 2, . . . , n.(i) Show thatm := (1, 1, . . . , 1) is an eigenvector forM with eigenvalue 1.(ii) Show that E := {x ∈ Cn :

∑nj=1 x j = 0} is invariant under M and that

Cn = Cm ⊕ E .(iii) Show that every eigenvector in E forM has an eigenvaluewithmodulus

less than 1.(iv) Deduce that the sequence ofmatrices (Mn)∞n=1 converges to a projection

with range Cm and kernel E .(v) Let G be a finite group, and let µ be a probability measure on G withsupport equal to G. Use the result of part (iv) to show that (µn)∞n=1converges to the Haar measure on G.


1. The structure theory of locally compact fields is more complete than thatof locally compact groups. It was shown by D. van Dantzig in the 1930sthat there are only two connected locally compact fields, namely R andC. Shortly afterwards L. Pontryagin completely classified the non-discretetotally disconnected locally compact fields. They are Qp and their finiteextensions (these are all the characteristic 0 fields) and k((X )) for k somefinite field (these are all the positive characteristic fields).The locally compact skew fields were completely classified by

N. Jacobson, also in the 1930s. There is one extra connected skew field,


namely the quaternions, and the totally disconnected skew fields are alldivision algebras over Qp or k((X )). See Weil (1995) and Jacobson (1980).Pontryagin (1946) calls this the ‘logical necessity’ of these fields.

2. Discrete, compact, and abelian groups are all unimodular. An example ofa group which is not unimodular is the so-called ax + b-group. This is thegroup of affine motions of the line; it is isomorphic to the matrix group{(

a b0 1

): a ∈ R \ {0}, b ∈ R

}.

3. The rate of convergence of the sequence of probability measures in Theorem8.3.4 is of some interest; see work of Diaconis and Saloff-Coste (1995, 1996)on finite groups. They compute the rate of convergence to the Haar measureof powers of specific probability measures.

9

Group algebras and representations

Several Banach algebras may be associated with each locally compact group.All of them are generalizations to locally compact groups of the algebraic groupalgebra CG of the discrete group G.Abstract harmonic analysis is essentially the study of the structure of these

Banach algebras. The development of Banach algebra theory was driven, inpart, by work on harmonic analysis.Throughout this chapter, G is a locally compact group.

9.1 Convolution algebrasDefinition 9.1.1 Let µ and ν belong to M(G). Their convolution is thebounded measure µ � ν defined by

µ � ν(B) = (µ× ν) ({(x, y) ∈ G × G : xy ∈ B}) (B ∈ B(G)) .

For each µ ∈ M(G), the adjoint of µ is the measure µ∗ defined by

µ∗(B) = µ(B−1) (B ∈ B(G)) .

Proposition 9.1.2 Convolution (µ, ν) �→ µ � ν is an associative algebraproduct on M(G), and satisfies ‖µ � ν‖ ≤ ‖µ‖‖ν‖. The adjoint mapµ �→ µ∗ isa conjugate linear involution on M(G), and (µ � ν)∗ = ν∗ � µ∗. Thus (M(G), �)is a Banach ∗-algebra, called the measure algebra of the locally compactgroup G. �

The subspace of M(G) consisting of the discrete measures on G will bedenoted by Md (G), the subspace of continuous measures by Mc(G), and thesubspace of measures absolutely continuous with respect to Haar measure byMa(G).

86

9 Group algebras and representations 87

Proposition 9.1.3

(i) Md (G) is a subalgebra of M(G), and Mc(G) and Ma(G) are closed two-sided ideals in M(G).

(ii) M(G) = Mc(G)⊕ Md (G), and the quotient algebra M(G)/Mc(G) is iso-morphic to Md (G). �

Proposition 9.1.4 (Hewitt and Ross 1979, Theorem 20.10) The space Ma(G)is isometrically isomorphic to the Lebesgue space L1(G) := L1(G,m) via

f ∈ L1(G) �→ f · m ∈ Ma(G) .

This isomorphism induces the convolution product � on L1(G) specified by

f � g(x) =∫Gf (y)g(y−1x) dm(y) ( f, g ∈ L1(G), a.e.(m)x ∈ G) ,

and the isometric involution ∗ specified by

f ∗(x) = f (x−1)�(x)−1 ( f ∈ L1(G), x ∈ G) . �

The convolution formulæ in the proposition agree with those defined in Part Ion Banach algebras for the special cases where G is discrete or equal to R.

Definition 9.1.5 The algebra (L1(G), � , ∗) is the group algebra of the locallycompact group G. When G is discrete, L1(G) may be denoted by 1(G).

The identification of Ma(G) with L1(G) induces a convolution,µ � f , for allµ ∈ M(G) and f ∈ L1(G). Since Ma(G) is an ideal in M(G), µ � f belongsto L1(G).

Proposition 9.1.6 For each f ∈ L1(G), the map x �→ δx � f =x f is contin-uous with respect to the given topology on G and the norm topology on L1(G).

�

WhenG is discreteM(G) = L1(G) = 1(G) and this algebra has the identityδe. When G is not discrete, L1(G) has a bounded approximate identity.

Definition 9.1.7 Let A be a Banach algebra. A left bounded approximateidentity for A is a net {uλ} in A such that supλ ‖uλ‖ <∞ and

limλ‖a − uλa‖ = 0 (a ∈ A) .

88 Part II Analysis and amenability, George Willis

The notions of right bounded approximate identity and two-sided boundedapproximate identity are defined similarly. Let X be a Banach left A-module(see Part I, Chapter 5). Then a left bounded approximate identity {uλ} for A isan approximate identity for X if uλ · x → x (x ∈ X ).

Proposition 9.1.8 Let {Eλ} be a base of compact neighbourhoods of the iden-tity in G. This forms a net directed by inclusion. For each λ, let uλ = 1Eλ/m(Eλ).Then ‖uλ‖1 = 1 for every λ and for every f ∈ L1(G), and

‖ f − f � uλ‖1 λ→ 0 and ‖ f − uλ � f ‖1 λ→ 0 . �

Thus L1(G) has a two-sided bounded approximate identity. The proof is anexercise. One reason for the significance of bounded approximate identitiesis the following factorization theorem. The basic version of the theorem isdue to P. J. Cohen, but it has many applications and has been strengthened bymany authors, such as (Kisynski 2000). See also Bonsall and Duncan (1973,Theorem 11.10) and Dales (2000, Theorem 2.9.24).

Theorem 9.1.9 (Cohen factorization) Let A be a Banach algebra with a leftbounded approximate identity {uλ}, and let X be a Banach left A-module suchthat {uλ} is an approximate identity for X. Then, for every element x ∈ X, thereare a ∈ A and y ∈ X such that x = a · y. In particular, if I is a closed leftideal in A and (bn) is a sequence in I with ‖bn‖ n→ 0, then there are a ∈ A anda sequence (cn) in I with ‖cn‖ n→ 0 in I such that bn = acn (n ∈ N). �

Corollary 9.1.10 Each f in L1(G) is a product f = g � h,wherewe also haveg, h ∈ L1(G). Moreover, if ( fn) is a sequence in L1(G) such that ‖ fn‖ n→ 0,then there are g, hn in L1(G) such that ‖hn‖ n→ 0 and fn = g � hn (n ∈ N).�

The group algebra L1(G) retains all information about G, as was shown byWendel (1952).

Theorem 9.1.11 (Wendel) Let T : L1(G1)→ L1(G2) be an isometric alge-bra isomorphism. Then there is a continuous group isomorphism ϑ : G1→ G2and there is a group character χ on G2 such that

T f (x) = χ (x)( f ◦ ϑ−1)(x) ( f ∈ L1(G1), x ∈ G2) .

Proof Suppose first that G1 is discrete, so that L1(G1) = 1(G). Then the unitball in 1(G1) is the convex hull of its extreme points and these are preciselythe points of the form tδx , where |t | = 1 and x ∈ G1. Since T is an isometry,


it follows that the unit ball in L1(G2) is the convex hull of its extreme points.Hence the Haar measure on G2 is the counting measure and it follows that G2is discrete.For each x ∈ G1, T (δx ) is an extreme point of the unit ball in 1(G2) and so

equals tδy , where |t | = 1 and y ∈ G2. Define ϑ(x) = y and χ (h) = t . That ϑis a group isomorphism and χ is a character follows from the fact that T is analgebra isomorphism.The proof when G1 is not discrete is sketched after the next result, which it

uses. �

Since L1(G) is an ideal in M(G), each bounded measure µ on G defines amultiplication operator Mµ : f �→ µ � f on L1(G). Clearly, Mµ satisfies theidentity,

Mµ( f � g) = Mµ( f ) � g ( f, g ∈ L1(G)) .

A linear operator satisfying this identity is called a left multiplier. As was firstshown by Wendel (1952), every left multiplier arises in this way.

Theorem 9.1.12 (Wendel) Let M be a left multiplier on L1(G). Then M iscontinuous, and there is a unique measure µ ∈ M(G) such that M = Mµ.

Proof Proof Suppose that ( fn) is a sequence in L1(G) such that ‖ fn‖ n→ 0.Then, by Corollary 9.1.10, there is g ∈ L1(G) and a sequence (hn) in L1(G)such that ‖hn‖ n→ 0 and fn = g � hn for each n. Hence

M( fn) = M(g � hn) = M(g) � hnn→ 0 .

Therefore M is continuous. (In fact, the continuity of M was hypothesized byWendel and this hypothesis was removed later by B. E. Johnson.)Choose a bounded approximate identity {uλ} for L1(G). Then {M(uλ)} is

a bounded net in L1(G). Recalling that L1(G) is a subspace of M(G) andthat M(G) is the dual space of C0(G), it follows that {M(uλ)} has a weak-∗accumulation point, µ say, in M(G). It may be verified that M = Mµ. �

The proof of the non-discrete case of Theorem 9.1.11 may be completed bylifting the isometric isomorphism T to be an isomorphism ofmultiplier algebrasand hence, following Wendel’s Theorem, an isomorphism M(G1)→ M(G2).Since the extreme points of the unit ball in M(G) are precisely those measuresof the form tδg , where |t | = 1 and g ∈ G, the argument used in the discretecase applies.


Recall that a derivation on a Banach algebra A is a linear map D satisfyingthe identity D(ab) = aD(b)+ D(a)b (a, b ∈ A). See Chapter 6 for more onderivations. Each measure µ ∈ M(G) determines a derivation, Dµ, on L1(G)defined by

Dµ( f ) = f � µ− µ � f ( f ∈ L1(G)) .At first sight it might seem that it would be as straightforward as the proof ofWendel’s Theorem to show that every derivation on L1(G) arises in this way.This seems to be difficult to prove however.

9.1.13 Problem Let D be a derivation on L1(G). Does there exist a measureµ ∈ M(G) such that D = Dµ?

This question has been answered affirmatively by B. E. Johnson for [SIN]-groups (and hence for all discrete groups), connected groups, and amenablegroups (Johnson 1972, 2001). It has not been solved for all totally disconnectedgroups, and nor has it been solved for extensions of connected groups by discreteones. (In fact, a positive solution to this problem has recently been announcedby V. Losert.)

9.2 Representations on Hilbert space

There is a correspondence between ∗-representations of L1(G) on Hilbert spaceand continuous (with respect to the strong operator topology) unitary represen-tations of G. (A unitary representation of G is a homomorphism from G intothe group of unitary operators on some Hilbert space.)

Proposition 9.2.1 Let V : G → B(H) be a unitary representation of G on theHilbert space H such that the map x �→ Vxv, G → H, is continuous for everyv ∈ H. For f ∈ L1(G), define

RV ( f ) =∫Gf (x)Vx dm(x) . (9.2.1)

ThenRV : L1(G)→ B(H) is a continuous ∗-representation of L1(G).Conversely, every non-degenerate ∗-representation of L1(G) on Hilbert

space is equal toRV for some unitary representation V of G.

Proof The proof of the first part is an exercise. For the proof of the second part,note that, if R is a non-degenerate ∗-representation of L1(G), then Theorem9.1.9 implies that, for every v ∈ H, there are f ∈ L1(G) and v ′ ∈ H such that v =R( f )v ′. A unitary representation V of G on H is then well-defined by setting

Vgv = R(δg � f )v ′ (g ∈ G, v ∈ H) .

ThenR may be recovered from V via the definition (9.2.1). �


Locally compact groups and their group algebras have natural representationson the Hilbert space L2(G) := L2(G,m). The existence of Haar measure is thuscrucial for the existence of unitary representations of such groups.

Definition 9.2.2 For each x ∈ G, let Ux be the unitary operator on L2(G)defined by

Ux f (y) = f (x−1y) ( f ∈ L2(G), y ∈ G) .Then U : x �→ Ux is a representation of G on L2(G) and is continuous with re-spect to the given topology on G and the strong operator topology onB(L2(G)).The map U is the left regular representation of G on L2(G).The strong operator closed subalgebra of B(L2(G)) generated by

{Ux : x ∈ G}is the von Neumann algebra of G; it is denoted by V N (G).

As shown in Proposition 9.2.1, there is a representation of L1(G) on L2(G)corresponding to the left regular representation of G.

Definition 9.2.3 For each g ∈ L1(G), let R(g) be the operator on L2(G)defined by

R(g) f (y) =∫Gg(x) f (x−1y) dm(x) ( f ∈ L2(G), y ∈ G) .

Then R : L1(G)→ B(L2(G)) is an injective ∗-homomorphism and ‖R‖ = 1.The mapR is the left regular representation of L1(G) on L2(G).The norm-closed subalgebra of B(L2(G)) generated by R(L1(G)) is the

reduced C*-algebra of G; it is denoted by C∗r (G).The fullC*-algebra of G, denoted by C∗(G), is the enveloping C*-algebra of

L1(G) (seePalmer (2001,Definition 10.1.10) for the definition of the envelopingC*-algebra).

Theorem 9.1.11 of Wendel shows that the group algebra L1(G) determinesG uniquely, but the same is not true for the C*-algebras C∗(G) and C∗r (G) andthe von Neumann algebra V N (G), as is seen in Exercise 9.4.1.Among the unitary representations ofG themost important are the irreducible

ones. Every representationmay be decomposed into irreducible representations.

Definition 9.2.4 The unitary representation V of G on the Hilbert space His reducible if H is the direct sum H = H1 ⊕ H2 of proper subspaces such thatV (H j ) = H j for j = 1, 2. A unitary representation which is not reducible isirreducible.


Unitary representations of the groupG are studied via the∗-representations ofC∗r (G) and C

∗(G). It is the defining property of the enveloping C*-algebra thatthere is a correspondence between∗-representations of L1(G) (and therefore, byProposition 9.2.1, unitary representations ofG) and ∗-representations ofC∗(G).One of the basic theorems about C*-algebras asserts that they have sufficientlymany irreducible ∗-representations to separate every non-zero point from 0.Since the ∗-homomorphism R : L1(G)→ C∗r (G) is injective and has denserange, it follows that there are sufficiently many irreducible representations ofL1(G) to separate every non-zero point from 0.

Theorem9.2.5 Let G be a locally compact group. Then L1(G) is a semisimpleBanach algebra. �

It also follows that there are sufficiently many unitary representations of Gto separate every non-identity element from e.

Theorem 9.2.6 (Gelfand-Raikov) Let G be a locally compact group, and letx �= e be in G. Then there is an irreducible, continuous unitary representationV of G on a Hilbert space H such that V (x) �= IH. �

The ‘sufficiently many’ ∗-representations of L1(G) guaranteed by these the-orems are in fact representations of the reduced C*-algebra C∗r (G). It may bethat there are other irreducible ∗-representations and in this case the full C*-algebra C∗(G) is not the same as C∗r (G). This phenomenon is discussed furtherin Chapter 11; see Proposition 11.2.5.

Definition 9.2.7 Let V be a unitary representation of the locally compactgroup G, and letRV be the ∗-representation of L1(G) defined in (9.2.1). ThenV isweakly contained in the regular representation of G if ‖RV ( f )‖ ≤ ‖R( f )‖for every f ∈ L1(G).

One of the basic problems in harmonic analysis is to describe in some wayall the irreducible unitary representations of a given locally compact group G.There are no completely general techniques for doing this and each descrip-tion relies on particular properties of G. A great deal is known about unitaryrepresentations of semisimple Lie goups and nilpotent Lie groups but there isalso still much to learn. The class of groups for which the representation the-ory is most satisfactory is that of abelian groups. Representations of abeliansubgroups are also an important tool in the study of representations of otherclasses of groups.


9.3 The Fourier transform

When G is abelian, L1(G), C∗(G), and V N (G) are commutative Banach alge-bras. These algebras, which are discussed in Part I, are best understood withthe aid of the Fourier transform. In order to define this transform we must firstsay something about the irreducible representations of G.Every irreducible unitary representation of a locally compact abelian groupG

is one-dimensional. The irreducible representations thus correspond to boundedhomomorphisms from G to the multiplicative group T of non-zero complexnumbers, that is, to characters on G (see §3.4). The set � of all continuouscharacters is itself an abelian group with respect to pointwise multiplication:

〈x, γ1 + γ2〉 = 〈x, γ1〉〈x, γ2〉 (x ∈ G, γ1, γ2 ∈ �) ,

where we write 〈x, γ 〉 for γ (x). When equipped with the topology of uniformconvergence on compact subsets of G, � is also a locally compact group and iscalled the dual group of G. The dual group of G is sometimes denoted by G.For example, each character on Z has the form γz : n �→ zn for some z ∈ T,

and it is easily seen that in fact Z ∼= T (exercise). With a little more work itmay be shown that each continuous character onT has the form γn : z �→ zn forsome n ∈ Z, and that T ∼= Z. Similarly, each continuous character onR has theform γy : t �→ eiyt for some y ∈ R and R ∼= R. These basic examples and thestructure theory of locally compact abelian groups, Theorem 8.2.7, facilitatethe description of the dual group, that is, the set of irreducible representations,of any abelian group.LetG be a locally compact abelian group with dual group�. For f ∈ L 1(G),

the Fourier transform of f is the function f , defined on � by the formula

f (γ ) =∫Gf (s)〈−s, γ 〉 dm(s) (γ ∈ �) . (9.3.1)

For example, if G = R, then

f (y) =∫ ∞−∞

f (t) e−iyt dt (y ∈ R) .

You are asked to show in an exercise that the Fourier transform converts the con-volution product in L1(G) to pointwise product of functions on � and convertsthe adjoint in L1(G) to complex conjugation.Set

A(�) = { f : f ∈ L 1(G)} ,


the range of the Fourier transform, as in §3.4. The Riemann–Lebesgue lemma(see Rudin 1966, §5.14; or Helson 1983, §1.2) shows that A(�) is contained inC0(�). The Fourier transform is therefore a ∗-homomorphism from L1(G) intoC0(�).

Theorem 9.3.1 Let G be a locally compact abelian group. Then the Fouriertransform

F : f �→ f , L 1(G)→ A(�) ⊂ C0(�) ,

is continuous ∗-homomorphism with ‖F‖ = 1. �

In fact, the formula (9.3.1) when restricted to f ∈ C00(G) ⊂ L1(G) definesan isometry with respect to the L2-norm. This isometry extends by continuityto L2(G). The extended map is also known as the Fourier transform, and is alsodenoted by F .

Theorem 9.3.2 Let G be a locally compact abelian group. Then the Fouriertransform

F : g �→ g, L2(G)→ L2(�) ,

is an isometric isomorphism. It satisfies the identity(R( f )g)(γ ) = f (γ )g(γ ) ( f ∈ L1(G), g ∈ L2(G), γ ∈ �) . (9.3.2)

�

Equation (9.3.2) shows that the Fourier transform converts the regular rep-resentation of L1(G) on L2(G) to a representation of A(�) by multiplicationoperators on L2(�). Since A(�) is self-adjoint and separates points in �, theStone–Weierstrass theorem (Dunford and Schwartz 1958, Theorem IV.6.16)shows that A(�) is dense in C0(�), and it follows that C∗r (G) is isomorphicto C0(�). Since � is the set of all irreducible representations of G, equation(9.3.2) shows that every irreducible representation is weakly contained in theregular representation and soC∗(G) = C∗r (G) whenG is abelian. (The fact that� coincides with the set of all characters on L1(G) is Theorem 3.4.2 in Part I;there is a full proof in Dales (2000, Theorem 2.5.4).) Further reasoning showsthat V N (G) is isomorphic to L∞(�).


The Fourier transform of L1(G) to A(�) and the isomorphism of C∗(G)with C0(�) are special cases of the Gelfand transform of commutative Banachalgebras. This transform is discussed in Part I, Chapter 3.The convolution algebra M(G) is also a commutative Banach algebra. The

Fourier transform extends from L1(G) (identified with the ideal Ma(G)) toM(G). This extension is called the Fourier–Stieltjes transform, and denoted byµ �→ µ for µ ∈ M(G). Here µ is defined by

µ(γ ) =∫Gγ (−s) dµ(s) (γ ∈ �) . (9.3.3)

The Fourier–Stieltjes transform is a ∗-homomorphism from M(G) onto analgebra of continuous functions on �; the range of the transform is denoted byB(�) and called the Fourier–Stieltjes algebra.It is no longer the case however that the irreducible (one-dimensional) rep-

resentations of M(G) correspond to the irreducible representations of G. In theterms introduced in Part I, not every character on M(G) corresponds to a char-acter on G, and the Fourier–Stieltjes transform is not the same as the Gelfandtransform on M(G). Indeed, since M(G)/Mc(G) ∼= Md (G) (Proposition 9.1.3)and Md (G) ∼= 1(G), the character space of M(G) contains a copy of the dualgroup ofG with the discrete topology. TheBanach algebraM(G) has been stud-ied in some depth when G is abelian and many additional characters have beenidentified (Taylor 1973). The character space of M(G) even contains analyticdiscs (Brown and Moran 1976, 1978), which means that M(G) has quotientsisomorphic to algebras of analytic functions.

9.4 Exercises

1. Are the group algebras 1(C4) and 1(C2 × C2) isomorphic? What areC∗r (G), C

∗(G), and V N (G) when G is C4 and C2 × C2?2. Prove Proposition 9.1.8. Furthermore, show that ‖uλ � f − f ‖p λ→ 0 for

every f ∈ L p(G), where 1 ≤ p <∞, and show that ‖uλ � f − f ‖∞ λ→ 0for every f ∈ C0(G).

3. Complete the proof of Proposition 9.1.11 by verifying that ϑ is a homomor-phism and χ is a character.

4. (a) For ν ∈ M(G) and ϕ ∈ C0(G) define

ν · ϕ(x) =∫Gϕ(xy) dν(y) .


Verify that

〈ϕ, µ � ν〉 = 〈ν · ϕ, µ〉 (µ, ν ∈ M(G), ϕ ∈ C0(G)) .

(b) Let {uλ} be the bounded approximate identity for L1(G) defined inProposition 9.1.8. Show that

limλ‖uλ · ϕ − ϕ‖∞ = 0 (ϕ ∈ C0(G)) .

5. Show that the multiplierMµ on L1(G) satisfies ‖Mµ‖ = ‖µ‖. (Hint: if {uλ}is a bounded approximate identity for L1(G), then µ � uλ

w∗→ µ in M(G).)Complete the proof of Theorem 9.1.12 by verifying thatM(uλ) has a uniqueweak-∗ accumulation point µ and that M = Mµ.

6. Verify that the linear operatorRV : L1(G)→ B(H) defined in Proposition9.2.1 is a ∗-homomorphism, that is, show thatRV ( f � g) = RV ( f )RV (g)andRV ( f ∗) = RV ( f )∗.

7. Verify that the mapU : G → B(L2(G)) defined in 9.2.2 is a unitary grouprepresentation, that is, show that Uxy = UxUy and Ux−1 = U ∗x .

8. Identify the dual group of the finite cyclic group Cn .9. Show that Z ∼= T.10. Show that, for all f and g in L1(G) and γ in �, we have

f � g(γ ) = f (γ )g(γ ) and f ∗(γ ) = f (γ ) .


1. Wendel’s theorem (9.1.11) has been improved by N. J. Kalton amd G. V.Wood by weakening the requirement that T be an isometry to require onlythat it have small norm; see Kalton and Wood (1976) and Wood (1989).Galindo (2000) considers the still weaker condition that T is an isomorphismof topological algebras, but G1 and G2 are abelian and torsion-free.

2. Building on earlier work of Cohen (1960), Kepert (1997) has shown thathomomorphisms T : L1(G1)→ L1(G2), whereG1 andG2 are abelian, havea very precise description and that they automatically have closed range.

3. The factorization theorem 9.1.9 was proved by Cohen (1959) in order toanswer the question of whether every element of L1(G) is a product. Thishad previously been shown to be the case for certain locally compact groupsby Rudin (1957).

4. The theory of the decomposition of unitary representations of groups,or, equivalently, of non-degenerate representations of C*-algebras, into


irreducible representations is described in Palmer (2001, Chapter 9) andKadison and Ringrose (1983, 1986).

5. Theproofs ofTheorems9.2.5 and 9.2.6 rely on representations of the reducedgroup C*-algebra, C∗r (G), and thus on the left regular representation ofG on L2(G). The fact that locally compact groups have sufficiently manyirreducible unitary representations to separate points thus relies ultimatelyon the existence of Haar measure.Topological groupswhich are not locally compact donot have a translation

invariant measure, and it can happen that they do not have sufficiently many,or even any, irreducible unitary representations. Unitary representations ofnon-locally compact groups are the subject of current research with onerecent publication being Glockner and Neeb (2000).

6. Let G be a locally compact abelian group. Then each element x of G deter-mines a character ι(x) on the dual group G = � by

〈γ, ι(x)〉 = 〈x, γ 〉 (γ ∈ �) .

The map ι : G → G is in fact an isomorphism of topological groups. Thisis known as the Pontryagin duality theorem; see Morris (1977) and Rudin(1962).

7. Let � be any locally compact group. Define

A(�) := { f � g : f, g ∈ L2(�)} .

Then A(�) is a subalgebra of C0(�) and is called the Fourier algebra of �.It has a norm ‖ · ‖ defined by

‖h‖ = inf{‖ f ‖2‖g‖2 : h = f � g, f, g ∈ L2(�)} ,

which is in fact a Banach algebra norm on A(�). When � = G is abelian,A(�) is isomorphic to the convolution algebra L1(G). The Fourier algebrawas introduced by Eymard (1964). It carries a great deal of informationabout �; see, for example, Forrest (1992) and Dales (2000, 4.5.30).

10

Convolution operators

The structures of the various group algebras A associated with G, which isa large part of harmonic analysis, is best understood when G is abelian orcompact. In these cases the multiplication operators

Ma : f �→ a � f (a ∈ L1(G), f ∈ A) ,

are decomposable in the sense of Part IV, Definition 21.1.4.

10.1 Compact groups

If G is a compact group, then, for every a ∈ L1(G), the convolution operatorMa is a compact operator on L1(G) (Exercise 10.4.1) and on L2(G) (Exercise10.4.2). Therefore, as shown in Chapter 21 on decomposable operators, Ma isdecomposable. The decomposability of this operator may be used to analysethe ideal structure of L1(G). We begin by identifying elements in the centre ofL1(G).

Lemma 10.1.1 Let G be a compact group, and letU be an open neigbourhoodof e. Then there is a neighbourhood e ) V ⊂ U such that xVx−1 = V for everyx ∈ G.

Proof Since U is open, G \ U is compact. The function

(x, y) �→ xyx−1 : G × (G \ U)→ G \ {e}

is continuous. Hence its image is compact, and it is also clearly invariantunder conjugation by elements of G. Choose V to be the complement of thisimage. �

98

10 Convolution operators 99

If V is a neighbourhood of e which is invariant under conjugation, thenV ∩ V−1 is invariant and symmetric.

Lemma 10.1.2 Let V be a symmetric neighbourhood of e which is invariantunder conjugation, and put u = 1V/m(V). Then u is a self-adjoint element ofthe centre of L1(G).

Proof That u is self-adjoint follows because it is a real-valued function andV is symmetric.To see that u is in the centre, let f ∈ L1(G). Then, for each x ∈ G, we have

f � u(x) = 1

m(V)

∫Gf (y)1V (y−1x) dm(y)

= 1

m(V)

∫Gf (y)1V (xy−1) dm(y), because V is invariant under conjugation,

= 1

m(V)

∫Gf (z−1x)1V (z) dm(z), because m is left-invariant and unimodular,

= u � f (x) ,

and so u is in the centre. �

The group algebra L1(G) may now be analysed as a sum of full matrixalgebras.

Theorem 10.1.3 Let G be a compact group. Then

L2(G) =(⊕σ∈$

Hσ

)2

,

where eachHσ is a finite-dimensional subspace of L2(G) invariant under the left

regular representation U (see Definition 9.2.2). For each σ , there is a minimalcentral idempotent zσ in L1(G) such thatHσ = zσ � L2(G). Moreover, for eachσ , zσ � L1(G) := Iσ is a minimal two-sided ideal in L1(G) and is isomorphicto a full matrix algebra. The ideal

∑σ∈$ Iσ is dense in L

1(G).

Proof Let u = 1E/m(E), where E is a symmetric neighbourhood of e invariantunder conjugation. Then Mu is a compact, self-adjoint operator on L2(G).Hence L2(G) is the direct sum of the eigenspaces of Mu .Let H be an eigenspace of Mu corresponding to a non-zero eigenvalue. Then

there is a projection P in the algebra generated by Mu such thatH = P(L2(G)).


In fact, there is z in the algebra generatedbyu such that P = Mz . Sinceu belongsto the centre of L1(G), by Lemma 10.1.2, so does z. Hence for each x ∈ G wehave

Ux (H) = δx � H = δx � z � H = z � δx � H ⊂ H .

The eigenspace is thus G-invariant.It must be shown that there are sufficiently many eigenspaces corresponding

to non-zero eigenvalues for L2(G) to be their direct sum. For this, consider allcentral functions of the form u = 1E/m(E) := uE . Note that

U := {uE : e ∈ E = E−1, xEx−1 = E (x ∈ G)}is a commuting family of functions, and so the corresponding convolutionoperators on L2(G) have common eigenspaces. Suppose that f belongs to thezero eigenspace for all uE ∈ U. Then, since U contains an approximate identityfor L2(G) (by Exercise 9.4.2), f = 0. Hence L2(G) is the direct sum of finite-dimensional eigenspaces for U. For each of these eigenspaces there is a centralidempotent in L1(G) which projects onto it, and so it is invariant under U (G).Since G is compact, L2(G) ⊂ L1(G). The eigenspaces for U are thus

eigenspaces in L1(G) also and, since L2(G) is dense in L1(G), the sum ofthese eigenspaces is also dense in L1(G). Moreover, if z is a central idempo-tent projecting onto an eigenspace, then z � L2(G) is dense in z � L1(G) andso, since the first space is finite-dimensional, z � L2(G) = z � L1(G). It fol-lows that z � L1(G) is a finite-dimensional, two-sided ideal in L1(G) and thatz belongs to this ideal.Theorem9.2.5 shows that L1(G) is semisimple.Hence each ideal z � L1(G) is

a semisimple, finite-dimensional algebra, and is therefore isomorphic to a directsum of full matrix algebras. It follows that z is a sum of minimal idempotentswhich project onto the full matrix algebras. Let {zσ : σ ∈ $} be the collectionof all the minimal central idempotents arising in this way and define Hσ to bezσ � L2(G). �

The orthogonal decomposition of L2(G) may be further refined as follows.Let dσ be the integer such that the minimal idealHσ is isomorphic to the algebraMdσ of dσ × dσ matrices. Then zσ =

∑dσj=1 e j , where e j are minimal (no longer

central) idempotents. Put Kσ = Hσ � e1. Then Kσ is a dσ -dimensional Hilbertspace and U (Kσ ) = Kσ . Denote the restriction of U to Kσ by U (σ ). Then U (σ )

is an irreducible representation of G.Choose an orthonormal basis {kα} for Kσ , and define

cσαβ(x) =⟨U (σ )x kβ, kα

⟩


to be the matrix coefficient of U (σ )x with respect to this basis; see Hewitt andRoss (1979, Theorem 27.5). Then the functions cσαβ are continuous on G andsatisfy the orthogonality relations∫

Gcσαβ c

τγ δ dm =

{1/dσ , if σ = τ, α = γ and β = δ ,

0, otherwise .

See Hewitt and Ross (1979, Theorems 27.15, 27.19). These functions belongto Hσ , and {

d1/2σ cσαβ : α, β ∈ {1, 2, . . . , dσ }}

is an orthonormal basis for Hσ . Therefore the set of functions{d1/2σ cσαβ : σ ∈ $, α, β ∈ {1, 2, . . . , dσ }

}is an orthonormal basis for L2(G).These functions make the isomorphism ofHσ with Mdσ explicit because they

convolve as scaled matrix units

cσαβ ∗ cτγ δ ={cσαδ/dσ , if σ = τ and β = γ ,

0, otherwise .

See Hewitt and Ross (1979, Theorem 27.20(iii)). The translates of their conju-gates can be found by matrix multiplication:

δx ∗ cσαβ =dσ∑γ=1

cσγα(k)cσγβ .

See Hewitt and Ross (1979, Theorem 27.20(i)).We may now see that every irreducible unitary representation of G is con-

tained as a direct summand in the regular representation.

Proposition 10.1.4 Let G be a compact group. Then every irreducible con-tinuous unitary representation of G is equivalent to U (σ ) for a unique σ ∈ $.

Proof Let V be an irreducible unitary representation of G on the Hilbertspace HV . Then RV (zσ )HV �= {0} for at least one σ ∈ $ because other-wise Theorem 10.1.3 would show that RV (L1(G))HV = {0}. On the otherhand, if RV (zσ )HV �= {0} for two distinct central idempotents, z1 and z2 say,thenHV = RV (z1)HV ⊕RV (zσ )HV would reduce the representation V . Hence


RV (zσ )HV �= {0} for exactly one σ ∈ $. It is left as an exercise to show that Vis equivalent to U (σ ). �

Corollary 10.1.5 Let G be a compact group. Then C∗r (G) = C∗(G). �

10.2 Abelian groups

One of the basic theorems of topology is Urysohn’s lemma, which asserts that,if C and D are disjoint closed subsets of a normal space X , then there is acontinuous real-valued function f on X such that 0 ≤ f (x) ≤ 1 for all x ∈ X ,f (x) = 0 for all x ∈ C , and f (x) = 1 for all x ∈ D. (See Rudin (1963, 2.12)or Engelking (1977, Theorem 15.10)). The class of normal spaces includesall locally compact Hausdorff spaces, and Urysohn’s lemma is the basis ofall partition of unity arguments. In particular, it implies that multiplicationoperators on C0(X ) are decomposable.It was seen in Section 9.3 that, if G is abelian, then the Fourier (or Gelfand)

transform of f ∈ L1(G) belongs to C0(�), where � is the dual group. Thecontinuous functions whose existence is guaranteed by Urysohn’s lemma neednot belong to A(�), the algebra of Fourier transforms, however. The next resultshows that suitable functions for partition of unity arguments cannevertheless befound inside A(�). See Rudin (1962, Theorem 2.6.1) and Reiter and Stegeman(2000, Proposition 2.1.5).

Lemma 10.2.1 Let G be a locally compact abelian group. Let K and U besubsets of its dual group �, with K compact, U open, and K ⊂ U . Then thereis a function f ∈ L1(G) such that:

0 ≤ f (γ ) ≤ 1 (γ ∈ �); f (γ ) = 1 (γ ∈ K); and f (γ ) = 0 (γ �∈ U) .

Proof Let V be a neighbourhood of e such that VK ⊂ U (see Lemma 8.2.2).Let W be another neighbourhood of e which is symmetric and has compactclosure such thatW 2 ⊂ V (see Lemma 8.2.1).Define functions on �, ξ = 1W/m�(W) and η = 1WK. Then ξ and η belong

to L2(�) ∩ L1(�): 0 ≤ ξ � η(γ ) ≤ 1 for all γ ∈ �; ξ � η(γ ) = 1 for all γ ∈ K;and ξ � η(γ ) = 0 for all γ �∈ U .Let h and k be the inverse Fourier transforms of ξ and η, respectively. Then

h and k belong to L2(G), and so hk is in L1(G). Put f = hk. Then

f = hk = h � k = ξ � η .

Hence f has the desired values on K and U . �


The norm of f in L1(G) is usually bigger than 1, but it has the followingbound:

‖ f ‖1 ≤ ‖h‖2‖k‖2 = ‖ξ‖2‖η‖2 = (m�(WK)/m�(W))1/2 .

The lemma is sometimes stated as L1(G) has local units. One consequenceis that L1(G) is a regular Banach algebra; see Exercise 3.6.1.Local units and regularity are important in the study of spectral synthesis.

The question of spectral synthesis is as follows. Let F be a family of functionsin L1(G), and let g be a function such that g(γ ) = 0 whenever f (γ ) = 0 forevery f ∈ F . Can g be synthesized from F in the sense that it belongs tothe closed ideal generated by F? (Note that the stated condition is necessaryfor g to belong to this ideal.) The first result in this direction was the so-called‘Tauberian theorem’ ofWiener (1932),who showed that, if f ∈ L1(Rn) satisfiesf (γ ) �= 0 for every γ ∈ Rn , then f generates L1(Rn). Proposition 3.3.2 in Part Iis a similar result which applies when G is discrete and abelian. The Banachalgebraic proof of this result was an early application of Banach algebra theory.Wiener’s Tauberian theorem has been considerably strengthened by

V. A. Ditkin (Rudin 1972) For the statement of this theorem the definitionsof the hull of an ideal and the kernel of a set of maximal ideals must be recalledfrom Exercise 3.6.1 in Part I.

Theorem 10.2.2 (Ditkin) Let G be a locally compact abelian group, and letI be a closed ideal in L1(G). Suppose that the boundary of h(I ) contains noperfect set. Then I = k(h(I )). �

(A perfect set is an infinite compact set having no isolated points.)A closed subset X of�L1(G) such that k(X ) is the unique ideal with hull X is

said to be a set of synthesis. Wiener’s original theorem asserts that the empty setis a set of synthesis, and Theorem 10.2.2 asserts that any set whose boundarydoes not contain a perfect set is a set of synthesis. Every finite set is thereforea set of synthesis and so we have the following consequence.

Corollary 10.2.3 Let G be a locally compact abelian group, and let I be aclosed ideal with infinite codimension in L1(G). Then h(I ) is infinite. �

Lemma 10.2.1 implies that, for every γ ∈ �, there is an f ∈ L1(G) such thatf has compact support and f (γ ) �= 0. Hence the common zero set of the ideal

F = { f : f has compact support}

is empty. It follows by Wiener’s theorem that L1(G) is the closure of this ideal.


A function algebra such that the functions with compact support are a densesubalgebra is sometimes called a Wiener algebra.

Proposition 10.2.4 Let G be a locally compact abelian group, and let u ∈L1(G). Then the operator f �→ u � f, L1(G)→ L1(G), is decomposable.

Proof LetU1 andU2 be open subsets of C with C = U1 ∪U2. Without loss ofgenerality, it may be supposed that 0 �∈ C \U1. Then u−1(C \U1) is a compactsubset of G because u ∈ C0(G) and u−1(U2) is an open set with

u−1(C \U1) ⊂ u−1(U2).Choose g ∈ L1(G) such that g(γ ) = 1 for every γ ∈ u−1(C \U1) and such

that g(γ ) = 0 for every γ �∈ u−1(U2). Put X1 = ((δe − g) � L1(G))− and thenset X2 = (g � L1(G))−. Then L1(G) = X1 + X2, and the spectrum of Mu |X jis contained in Uj for each j . �

10.3 The free group

In contrast with the algebras of compact and abelian groups, convolution in 1(F2) need not be decomposable. This will be shown by the following calcula-tion, which is an improved exposition of an example taken fromWillis (1986b).Several further consequences will be deduced from this calculation in the nextchapter.

Proposition 10.3.1 Let F2 = 〈a, b〉 be the free group on two generators, andlet ta, tb ∈ C satisfy |ta| = 1 = |tb|. Then the linear operator T on c0(F2)definedby

Tϕ(w) = ϕ(w)+ taϕ(aw)+ tbϕ(bw) (ϕ ∈ c0(F2)) (10.3.1)

is surjective. More precisely, for each ϕ ∈ c0(F2), there exists ψ ∈ c0(F2) with‖ψ‖ ≤ ‖ϕ‖ such that ϕ = Tψ .The convolution operator (δe + taδa + tbδb)� on 1(F2) is the dual of T .

Hence this operator satisfies

‖(δe + taδa + tbδb) � f ‖1 ≥ ‖ f ‖1 ( f ∈ 1(F2))and has closed range.

Proof It ismore convenient to carry out the calculation for a related convolutionoperator on a larger group. This group will be the free product of three copiesof the cyclic group of order 2, C2 ∗ C2 ∗ C2 = 〈u0, u1, u2〉. The subgroup of


C2 ∗ C2 ∗ C2 generated by u0u1 and u0u2 is just the subgroup consisting ofall words of even length and has index 2. This subgroup is isomorphic toF2 = 〈a, b〉, with the isomorphism determined by a �→ u0u1 and b �→ u0u2.We identify F2 with the subgroup generated by u0u1 and u0u2, and so we see

that C2 ∗ C2 ∗ C2 is the union of cosets

C2 ∗ C2 ∗ C2 = F2 ∪ u0F2 = {words of even length} ∪ {words of odd length}.

Then c0(C2 ∗ C2 ∗ C2) is the direct sum c0(F2)⊕ c0(u0F2). Equation (10.3.1)then defines an operator T on c0(C2 ∗ C2 ∗ C2), and c0(F2) and c0(u0F2) areinvariant under T . Hence it will suffice to show that T is surjective on the spacec0(C2 ∗ C2 ∗ C2). Since T is the composite T = SR, where R is defined by

Rϕ(w) = ϕ(u0w)+ taϕ(u1w)+ tbϕ(u2w) (ϕ ∈ c0(C2 ∗ C2 ∗ C2))

and Sϕ(w) = ϕ(u0w), and since S is clearly invertible, it will suffice to showthat R is surjective. For ease of exposition we will suppose that ta = 1 = tb.Let ϕ ∈ c0(C2 ∗ C2 ∗ C2). We defineψ ∈ c0(C2 ∗ C2 ∗ C2) with Rψ = ϕ by

defining ψ(w) inductively on the length of w . Begin by setting ψ(e) = 0 andψ(w) = ϕ(e)/3 for w = u0, u1 or u2. Then Rψ(e) = ϕ(e).Assume that ψ(w) has been defined for all words w with |w | ≤ n such

that Rψ(w) = ϕ(w) for all w with |w | ≤ n − 1. Let w have length n. Then|uw | = n − 1 for u equal to one of u0, u1, and u2, and |uw | = n + 1 for uequal to the other two. Suppose, for instance, that |u0w | = n − 1 and that|u1w | = n + 1 = |u2w |. In this case, defineψ(u1w) = (ϕ(w)− ψ(u0w)) /2 =ψ(u2w). Then

Rψ(w) = ψ(u0w)+ ψ(u1w)+ ψ(u2w) = ϕ(w) .

The other cases are treated similarly to define ψ on all words of length up ton + 1 to achieve Rψ(w) = ϕ(w) for all w of length up to n. Continuing in thisway we construct ψ with Rψ = ϕ.It remains to show that ψ ∈ c0(C2 ∗ C2 ∗ C2). For this, define for each w the

sequence of words (w j )nj=0, where n = |w |, by removing letters one at a timefrom the left of w . Thus, for example, w0 = w , wn = e, and |w j | = n − j .(If the Cayley graph of C2 ∗ C2 ∗ C2 = 〈u0, u1, u2〉 is defined so that w isadjacent to u0w , u1w , and u2w , then (w j )nj=0 is the sequence of words on thepath from w to e.) Then it may be shown by induction that

ψ(w) = −n/2∑j=1

(−12

) jϕ(w2 j−1) (10.3.2)


if n is even, and

ψ(w) = 13

(−12

)(n−1)/2ϕ(e)−

(n−1)/2∑j=1

(−12

) jϕ(w2 j−1) (10.3.3)

ifn is odd. Since |ϕ(w)| → 0 as |w | → ∞, it follows thatψ ∈ c0(C2 ∗ C2 ∗ C2).�

The identities (10.3.2) and (10.3.3) may be used to show that the operatorT is surjective on p(F2) for every p > 2. By duality, (see Dunford Schwartz1958, §VI.6), T is injective with closed range on p(F2) for every p < 2. Themap T is injective and has dense range on 2(F2).

Proposition 10.3.2 The operator T defined in the previous proposition hasnon-trivial kernel. Hence the operator f �→ (δe + taδa + tbδb) � f is not sur-jective on 1(F2).

Proof It suffices to show that (δe + taδa + tbδb)� is not surjective on 1(F2) (seeDunford and Schwartz 1958, VI.6). For this, note that, since F2 is a free group,a �→ t−1a e2π i/3, b �→ t−1b e−2π i/3 determines a character on F2. Then the elementδe + taδa + tbδb belongs to the kernel of the corresponding representation of 1(F2), which is an ideal. Hence the range of convolution by this element iscontained in this proper ideal.We have shown that T ′ is not surjective. It follows that T itself is not

injective. �

It is not difficult to show that a surjective operator which is not injectivedoes not have the single-valued extension property (SVEP). Hence T does nothave property (β) and T ′ does not have (δ). (For definitions, see 21.2.1 and21.2.4)

Corollary 10.3.3 The operator on 1(F2) defined by

f �→ (δe + taδa + tbδb) � fis not decomposable. �

A ∗-algebra is symmetric if the spectrum of a∗a is contained in R+ ∪ {0} forevery a ∈ A (Dales 2000, §1.10). A locally compact group is symmetric if the∗-algebra L1(G) is symmetric. Abelian and compact groups are symmetric, butthere is no known characterization of symmetric groups. Free groups are notsymmetric however.


Proposition 10.3.4 F2 is not a symmetric group.

Proof The element (δe + δa + δb)∗ � (δe + δa + δb) is not invertible, but‖(δe + δa + δb)∗ � (δe + δa + δb) � f ‖1 ≥ ‖ f ‖1

for every f ∈ 1(F2). It follows that the unit disc is contained in its spectrum,and so 1(F2) is not symmetric. �

10.4 Exercises

1. LetG be a compact group. Show that the operatorMa : f �→ a � f on L1(G)is compact when a ∈ L1(G) by the following steps.(i) The function x �→ a � δx , G → L1(G), is continuous. Hence the set{a � δx : x ∈ G} is compact (in the norm topology on L1(G)).

(ii) Let f ∈ L1(G)+ satisfy ∫G f dm ≤ 1. Assuming thata � f =

∫Ga � δx f (x) dm(x) ,

show that a � f belongs to the closed convex hull of

{a � δx : x ∈ G} ∪ {0}.(iii) Deduce that {a � f : f ∈ L1(G)+, ∫G f dm ≤ 1} has compact closure,

and hence that {a � f : f ∈ L1(G), ‖ f ‖1 ≤ 1} has compact closure.2. (An alternative proof that f �→ a � f is compact.)Let G be a compact group. We show that for every a ∈ L1(G) the operatorf �→ a � f is a compact operator on L p(G) for 1 < p <∞ and on C(G).Since the set of compact operators is closed and since L1(G) ∩ L∞(G) isdense in L1(G), it may be supposed that a ∈ L1(G) ∩ L∞(G).(i) Let f ∈ L p(G). Show that a � f ∈ C(G) and ‖a � f ‖∞ ≤ ‖a‖q‖ f ‖p,where q is the conjugate index to p.

(ii) By using the fact that themap x �→ δx � a, G → Lq (G), is continuous,show that {a � f : f ∈ L p(G), ‖ f ‖p ≤ 1} is an equicontinuous andbounded subset of C(G). Conclude (with the aid of the Arzela–Ascolitheorem; see Dunford and Schwartz (1958, Theorem IV.6.7) that thisset is totally bounded.

(iii) Use the fact that the inclusion map C(G)→ L p(G) is continuous todeduce that {a � f : f ∈ L p(G), ‖ f ‖p ≤ 1} has compact closure inL p(G), and hence convolution by a is a compact operator.


3. Describe the functions cσαβ when G is the circle group T.4. Describe the functions cσαβ when G is the permutation group S3.5. Complete the proof of Proposition 10.1.4.6. Let G be a compact group. Show that G is isomorphic to a closed subgroupof a product

∏j U (n j ), where U (n) denotes the group of n × n unitary

matrices. Deduce that, if U is a neighbourhood of e, then there is a compactnormal subgroup K ⊂ U such that G/K is a Lie group.

7. Show that single points are not sets of synthesis for C (n)(I) when n ≥ 1.


1. Theorem 10.2.2 suggests that it is not easy to find subsets of � = G whichare not sets of synthesis for L1(G). However it was shown by Schwartz(1948) that the unit sphere in R3 is not a set of synthesis for L1(R3). Thisfact is the starting point of the construction in Grønbaek and Willis (1997).

2. By way of contrast, Herz (1958) showed that the unit circle in R2 is a set ofsynthesis for L1(R2). The question of synthesis is thus a very subtle one.

3. If G is compact, then � is discrete and so every set is a set of synthesis. IfG is not compact, then there is a set of non-synthesis in �, as was shown byMalliavin (1959). These ideas are presented very clearly in Rudin (1962).

11

Amenable groups

Many, but certainly not all, results about abelian groups and compact groupshave a common extension to the class of amenable groups. Some of these resultsare discussed in this chapter and it is seen that amenability may be regarded asa type of finiteness condition.

11.1 Definition and examples

Most of the definitions and results in this chapter may be found in the refer-ences (Hewitt and Ross 1979, §17; Greenleaf 1969; Reiter and Stegeman 2000,Chapter 8; Paterson 1988; Pier 1984). The first definition repeats 7.2.5.

Definition 11.1.1 The locally compact group G is amenable if there is a left-invariant mean on L∞(G). Thus G is amenable if there is a linear functionalM on L∞(G) such that:

1. M(ϕ) ≥ 0 whenever ϕ ≥ 0;2. M(1G) = 1; and3. M(xϕ) = M(ϕ) for every ϕ ∈ L∞(G) and x ∈ G.

Properties 1 and 2 say that M is a mean, and Property 3 says that M isinvariant under left translation.It may be shown that L∞(G) has a left-invariant mean if and only if it has

a right-invariant mean and that this is equivalent to L∞(G) having a two-sidedinvariant mean. It is also equivalent to L∞(G) having topologically invariantmean, that is, a mean M such that

M( f � ϕ) =(∫

Gf dm

)M(ϕ) ( f ∈ L1(G)) .

The mean property of M implies that it is continuous (with norm 1), but it is

109


shown inWillis (1988) that themean property and continuity are both redundantin the definition.

Proposition 11.1.2 The locally compact group G is amenable if and only ifthere is a non-zero, left-invariant functional (continuous or discontinuous) onL∞(G). �

Here are some basic examples of amenable groups and of non-amenablegroups.

Proposition 11.1.3 Compact groups and abelian groups are amenable.

Proof The Haar integral is an invariant mean on L∞(G) when G is compact.An indirect proof valid for abelian semigroups is given in Hewitt and Ross(1979, Theorem 17.5). We give a more direct proof for the case where G is thegroup of integers.For each n ∈ Z+, let m(n) ∈ 1(Z) be specified by

m(n) = 1

2n + 1n∑

j=−nδ j .

Then m(n) ≥ 0 and ‖m(n)‖1 = 1. Moreover, when n > k > 0, we have

m(n)k − m(n) = 1

2n + 1

(n+k∑j=n+1

δ j −k−n−1∑j=−n

δ j

),

and so ‖m(n)k − m(n)‖1 = 2k/(2n + 1) n→ 0 for every k ∈ Z. It follows that eachweak-∗ accumulation point of (mn)∞n=1 is an invariant mean on ∞(Z). �

Proposition 11.1.4 The free group on two generators is not amenable.

Proof Let ω = e2π i. Then the operator T ′′ ∈ B(∞(F2)) given by

T ′′ϕ(x) = ϕ(x)+ ωϕ(ax)+ ω2ϕ(bx) (ϕ ∈ ∞(F2))

is the second dual of the operator considered in Proposition 10.3.1, and so issurjective by that proposition and Dunford and Schwartz (1958, VI.6).Let M be a translation invariant functional on ∞(F2). Then, for every ele-

ment ϕ ∈ ∞(F2), we have ϕ = T ′′ψ for some ψ ∈ ∞(F2), and so

M(ϕ) = M(T ′′ψ) = M(ψ)+ ωM(ψa)+ ω2M(ψb).

11 Amenable groups 111

Since M is translation-invariant, it follows that

M(ϕ) = (1+ ω + ω2)M(ψ) = 0 .Hence there are no non-zero translation-invariant functionals on ∞(F2), andF2 is not amenable. �

Proofs of the following stability properties of amenability may be found inGreenleaf (1969) and Paterson (1988).

Theorem 11.1.5

(i) Let G be a locally compact group, and let N be a closed normal subgroupof G. Then G is amenable if and only if N and G/N are amenable.

(ii) Let G be an amenable group, and let ϕ : G → H be a homomorphismsuch that ϕ(G) = H. Then H is amenable.

(iii) Let G be an amenable group, and let H be a closed subgroup of G. ThenH is amenable. �

The above theorem enlarges the classes of groups shown to be amenable inProposition 11.1.3.

Corollary 11.1.6 (i) Solvable groups are amenable.(ii) Central groups are amenable. �

Proposition 11.1.4may also be combinedwith the theorem to show thatmanygroups are not amenable.

Corollary 11.1.7 Non-compact, semisimple Lie groups are not amenable.�

This corollary is provedby showing that non-compact, semisimpleLie groupshave closed subgroups isomorphic to F2. See Reiter and Stegeman (2000, §8.7).For a long time it was an open question, asked originally by von Neumann,

whether every non-amenable, group had a subgroup isomorphic to F2. Thisquestion was answered Ol’shanskiı (1980), where a non-amenable group isconstructed which has no non-abelian, free subgroup because every one of itsproper subgroups is cyclic.

Theorem 11.1.8 Suppose that G =⋃λ Gλ, where each group Gλ isamenable. Then G is amenable. �

Corollary 11.1.9 Locally finite groups, that is, discrete groups such that everyfinite subset generates a finite subgroup, are amenable. �


The class of groups containing abelian and compact groups and closed underextensions, dense homomorphic images, subgroups, and direct limits is theclass of elementary amenable groups. For some time these were the only groupswhich were known to be amenable. However amenable groups which are notelementary were constructed by Grigorchuk (1984, 1988).

11.2 Alternative characterizations of amenability

The amenability condition has many equivalent characterizations which re-flect the wide range of applications of amenability. Many of these aspects ofamenability show it to be a finiteness condition in some way.

Invariant means By definition, G is amenable if and only if there is a left-translation invariant mean on L∞(G). (The name was invented by M. M. Dayas a pun.) As already remarked, amenability is equivalent to the existenceof right-translation invariant, two-sided translation-invariant, and topologicallyinvariantmeans on L∞(G). It is also equivalent to the existenceof a left-invariantmean on C(G), the space of bounded continuous functions on G and on thespace of right uniformly continuous functions on G; a function f ∈ C(G) isright uniformly continuous if, for each ε > 0, there exists a neighbourhood Uof e such that | f (x)− f (y)| < ε whenever x, y ∈ G with xy−1 ∈ U .A characterizing property of compact groups is that their Haar measure is

finite, and in this case the Haar measure of the entire group is usually normal-ized to equal 1. Integration against Haar measure is then an invariant meanon spaces of bounded functions. If G is not compact, bounded functions can-not be integrated generally (because L∞(G,m) �⊂ L1(G,m)). The existence oftranslation-invariant means on L∞(G,m) therefore extends some features ofcompact (and in particular finite) groups to the class of amenable, locally com-pact groups. Indeed, it will be seen that certain results for finite and compactgroups which use the Haar measure in their proof have analogues for amenablegroups.Although the Haar measure is unique, invariant means are not. Rudin (1972)

showed that the compact abelian group T has invariant means other than Haarmeasure because it has invariant means which are not topological invariantmeans. The exact number of invariant means on a general locally compactgroup is found in Chou (1976).

Paradoxical decompositions A paradoxical decomposition of the group Gconsists of a finite partition G = A1 ∪ · · · ∪ Ak ∪ B1 · · · ∪ Bl into measurable


subsets and elements x1, . . . , xk , y1, . . . , yl such that G = x1A1 ∪ · · · ∪ xk Akand G = y1B1 ∪ · · · ∪ yl Bl are also partitions.It was shown by Tarski (1938) that G is amenable if and only if it does not

have a paradoxical decomposition. Non-amenability of G thus says that G isinfinite in a strong sense closely related to the algebraic structure: there is atwo-to-one map from G to itself which is a piecewise translation.Amenability first appeared in connection with paradoxical decompositions.

TheBanach–Tarski paradox shows that the sphere canbepartitioned intofinitelymany sets which may be rotated and reassembled to form two copies of theoriginal sphere. It follows that there is no finitely additive measure on the fam-ily of all subsets of the sphere which extends the familiar area measure onBorel sets. This is in contrast to the circle T or the line R or the planeR2, where the familiar Lebesgue measure can be extended to a translation-invariant, finitely additive measure on all subsets. It was observed by vonNeumann (1929) that the paradoxical decomposition of the sphere arises froma paradoxical decomposition of a free subgroup of the rotation group of thesphere. The symmetry groups of the line, plane and circle do not contain freesubgroups.

The Følner condition This condition says, roughly speaking, that amenablegroups have compact subsets almost invariant under translations. For example,ifG is compact, then the group itself is invariant under translation. For a secondexample, the proof in Proposition 11.1.3 that Z is amenable uses the fact that,for each n ∈ N, the subset

K = {−n,−n + 1, . . . , n − 1, n}

is almost invariant in a sense which we now make precise.

Definition 11.2.1 The locally compact group G satisfies Følner’s conditionif, for every ε > 0 and every compact subset C of G, there is a compact subsetK of G such that

m ((xK \ K ) ∪ (K \ xK )) /m(K ) < ε (x ∈ C) .

The locally compact group G is amenable if and only if it satisfies Følner’scondition.

Almost invariant probabilities Closely related to Følner’s condition are sev-eral formulations of almost invariance for probability measures on G. Denotethe set of non-negative, norm 1 functions in L1(G) by P(G). These are the


probability measures on G which are absolutely continuous with respect toHaar measure.

Proposition 11.2.2 The locally compact group G is amenable if it satisfiesany one of the following three equivalent conditions.

1. There is a net { fλ} in P(G) such that {x fλ − fλ} converges weakly to 0 forevery x ∈ G.

2. There is a net { fλ} in P(G) such that {‖x fλ − fλ‖1} converges to 0 for everyx ∈ G.

3. There is a net { fλ} in P(G) such that {‖g � fλ − fλ‖1} converges to 0 forevery g ∈ P(G). �

The proof that the first condition implies the second, due to M. M. Day,requires the functional-analytic fact (derived from the Hahn–Banach theorem)that the weak and strong closures of a convex subset of a Banach space are thesame. The argument does not show that weak convergence of {x fλ − fλ} to 0implies norm convergence. It in fact is shown in Rosenblatt and Willis (2001)that there is a net with {x fλ − fλ} weakly convergent but not norm convergent.This paper introduces the notion of configuration sets, which could be usefulfor further work on amenable groups.The properties in Proposition 11.2.2 are very close to the p = 1 case of the

following condition introduced by H. Reiter.

Definition 11.2.3 (Property Pp) Let 1 ≤ p <∞. The locally compact groupG satisfies Property Pp if, for every ε > 0 and every compact subset C of G,there exists f ∈ L p(G) such that f ≥ 0, ‖p‖p = 1, and

‖x f − f ‖p < ε (x ∈ C) .

Proposition 11.2.4 (Reiter and Stegeman 2000, Theorem 8.3.2) The condi-tions Pp for 1 ≤ p <∞ are all equivalent, and the group G is amenable if itsatisfies any one of these conditions. �

Unitary representations Reiter’s condition P2 implies that there is a net

{ fλ} of norm 1 elements in L2(G) such that ‖x fλ − fλ‖2 λ→ 0. In other words,

‖Ux ( fλ)− fλ‖2 λ→ 0, whereU is the left regular representation ofG on L2(G).The trivial representation of G is the one-dimensional representation definedby Vx = 1 (x ∈ G). Condition P2 therefore says that functions in the regularrepresentation approximate the trivial representation of G. That is the firststatement of the next characterization of amenability.


Proposition 11.2.5 Let G be a locally compact group. Then the followingconditions are each equivalent to G being amenable.

1. The trivial representation of G is weakly contained in the regular represen-tation.

2. Every unitary representation of G is weakly contained in the regular repre-sentation.

3. C∗r (G) = C∗(G).4. For every probability measure in M(G) (or in Ma(G) or Md (G)), the normof the operator f �→ µ � f, L2(G)→ L2(G), is equal to 1. �

The facts, proved in Chapters 9 and 10, that C∗r (G) = C∗(G) when G isabelian or compact may now be seen to be consequences of the amenability ofabelian groups and compact groups, respectively. Alternatively, they might beviewed as proofs that these groups are amenable.The unitary representation theory of finite groups shows that, if G is finite,

then CG is isomorphic to a direct sum of full matrix algebras and that everyirreducible unitary representation of G is equivalent to a direct summand of theregular representation. InTheorem10.1.3 the samewas seen to hold for compactgroups. Condition 2 of the above proposition extends this to amenable groups.It is no longer the case that every irreducible unitary representation is a directsummand in the regular representation; it is only weakly contained in it. Weakcontainment of an irreducible representation V in the regular representation Umeans that, for every ξ and η in HV , there are bounded nets { fλ} and {gλ} inL2(G) such that the ‘matrix coefficient’ x �→ 〈Vxξ, η〉 is approximated in thesense that

〈Vxξ, η〉 = limλ〈Ux fλ, gλ〉 ,

where the limit is uniform on compact subsets of G. Amenable groups areprecisely those for which we have this weak version of a theorem about finitegroups. Amenability thus appears once again as a finiteness condition.An immediate consequence of Condition 4 in Proposition 11.2.5 is that, if G

is not amenable, there are x1, . . . , xn in G and positive numbers c1, . . . , cn with∑nj=1 c j = 1 such that |

∑nj=1(c jδx j �)|L2(G) < 1. (J. Dieudonne (1960) made a

direct calculation in the case of F2 and showed that |δe + δa + δb|l2(G) =√8.)

It is an exercise to deduce the following result from this fact.

Corollary 11.2.6 Let G be a non-amenable group. Then there is no non-zero,translation-invariant functional on L2(G). �


The fixed point property This property means that every continuous andaffine action ofG on a compact, convex subsetC of a locally convex topologicallinear space has a fixed point. The next characterization was shown by Day andby Rickert (1967).

Proposition 11.2.7 The locally compact group G is amenable if and only ifit has the fixed point property. �

Cohomology of L1(G) One of the last formulations of amenability to beobtained concerns the cohomology of L1(G). Some definitions are required inorder to state it.

Definition 11.2.8 Let A be a Banach algebra.

(i) A Banach A-bimodule is a Banach space X together with left and rightactions (a, x) �→ a · x and (a, x) �→ x · a which are bilinear maps sat-isfying the associative laws

(ab) · x = a · (b · x), (a · x) · b = a · (x · b)and x · (ab) = (x · a) · b (a, b ∈ A, x ∈ X ) .

These maps must also satisfy the bound condition

‖a · x‖X ≤ ‖a‖A‖x‖X and ‖x · a‖X ≤ ‖a‖A‖x‖X (a ∈ A, x ∈ X ) .

(ii) A derivation from A to a bimodule X is a linearmap D : A→ X satisfyingthe Leibniz rule

D(ab) = a · D(b)+ D(a) · b (a, b ∈ A) .

(iii) Let x be in X. The map Dx : A→ X is an inner deriviation if

D(a) = a · x − x · a (a ∈ A) .

(iv) The set of continuous derivations is a linear subspace of B(A, X ), de-noted byZ1(A, X ). The set of inner derivations is a subspace ofZ(A, X ),denoted by N 1(A, X ). The quotient space

H1(A, X ) := Z1(A, X )/N 1(A, X )

is the first continuous cohomology group of A with coefficients in X.


The first and higher cohomology groups of Banach algebras are discussedin Part I, §6.2. For further information, see Dales (2000) and Helemskii(1989).The dual module of a Banach A-module is defined in Definition 6.1.5. The

dual actions of A on X ′ are defined so that the linear operator x ′ �→ x ′ · a isthe dual of the operator x �→ a · x and x ′ �→ a · x ′ is the dual of the operatorx �→ x · a. As part of his work on derivations on L1(G), B. E. Johnson provedthe following theorem; it gave rise to the definition of an amenable Banachalgebra. See Part I and also Johnson (1972, Theorem 2.5) or Dales (2000,Theorem 5.6.2).

Theorem 11.2.9 Let G be a locally compact group. Then G is amenable ifand only ifH1(L1(G), X ′) = {0} for every Banach L1(G)-bimodule X. �

In particular, since M(G) with the convolution action is the dualL1(G)-bimodule of C0(G) with appropriately defined actions, we have thatH1(L1(G),M(G)) = {0}.

Corollary 11.2.10 Let G be an amenable group, and let D : L1(G)→ L1(G)be a continuous derivation. Then there exists µ ∈ M(G) such that

D( f ) = f � µ− µ � f ( f ∈ L1(G)) . �

A proof of Theorem 11.2.9 in the case where G is discrete was presentedin Chapter 7. That this theorem of Johnson’s is yet another manifestation ofamenability as a finiteness condition can be seen by comparing it with a the-orem from purely algebraic cohomology theory. For each algebra A and A-bimodule X , the algebraic cohomology group H 1(A, X ) is defined just as inDefinition 11.2.8 except that there is no norm; see 7.1.2. Then it may be shownthat the algebra A satisfies H 1(A, X ) = {0} for every A-bimodule X if andonly if A is finite-dimensional and semisimple; see 7.2.1. It is conjectured thatthe same holds for Banach algebras and Banach bimodules but all proofs ofspecial cases assume that spaces concerned have the approximation propertyor compact approximation property at some point, (see Taylor 1972; Selivanov1976; Runde 1998). Vanishing of all cohomology groups of A with values indual A-bimodules is thus a weak version of a condition which, in the purelyalgebraic case at least, implies the finite-dimensionality of A.


11.3 Approximate identities in ideals

Amenability of G is also equivalent to the existence of (bounded) approximateidentities in certain ideals of L1(G). The first result is due to H. Reiter 1968.For this, note that the codimension 1 subspace

L10(G) :={f ∈ L1(G) :

∫Gf dm = 0

}is a closed ideal in L1(G). It is the kernel of the representation of L1(G) in-duced by the trivial representation of G; see equation (7.2.1). It is called theaugmentation ideal.

Proposition 11.3.1 The locally compact group G is amenable if and only ifL10(G) has a left bounded approximate identity.

For a full proof, see Dales (2000, Theorem 5.6.42). See the exercises for adirect proof that 10 (F2) does not have an approximate identity. It had previouslybeen shown by W. Rudin that, when G is compact, all ideals in L1(G) havebounded approximate identities. This was extended to the class of amenablegroups by Liu, van Rooij, and Wang (1973).

Proposition 11.3.2 Let G be an amenable group, and let I be a closed rightideal in L1(G). Then I has a left bounded approximate identity if and only if itis weakly complemented in L1(G).

(A subspaceY of aBanach space X isweakly complemented if the annihilatorof Y , which is defined by Y⊥ := {x ′ ∈ X ′ : 〈y, x ′〉 = 0 (y ∈ Y )}, has a Banachspace complement in X ′.) It is not difficult to see that any ideal which has abounded approximate identity must be weakly complemented. Amenability isessential for the ‘if’ direction of the proposition. In fact, the result is proved inLiu, van Rooij, and Wang (1973) for complemented ideals only. The strongerversion for weakly complemented ideals may be proved with the aid of thecohomological property shown in Theorem 11.2.9; for this, see Curtis and Loy(1989) and Dales (2000, Theorem 2.9.58).The previous proposition applies in particular to ideals with finite codimen-

sion in L1(G) because such ideals are complemented. The ‘if’ direction ofReiter’s theorem extends to these ideals (Willis 1982).

Proposition 11.3.3 Suppose that some closed right ideal of finite codimensionin L1(G) has a left bounded approximate identity. Then G is amenable. �


These results on approximate identities are related to another characteriza-tion of amenability in terms of random walks. Each probability measure onG determines a random walk on G; see for example Kaımanovic and Versik(1983). An important tool in the study of a random walk is its boundary. Theboundary is a measurable G-space imagined as consisting of points ‘at infinity’which the walker approaches as times proceeds.It is not directly relevant to make this precise here. For the purposes of this

chapter, we describe the boundary of the random walk corresponding to theprobability measure µ in Banach algebraic terms as follows. First, define

µJ := { f − µ � f : f ∈ L1(G)} .

Then µJ is a closed right ideal in L1(G) and has a left bounded approximateidentity consisting of functions of the form(

δe − 1n

n∑j=1

µ j

)� uE ,

where uE is a function as defined in Proposition 9.1.8. It may be shown thatthe quotient Banach space L1(G)/µJ is isometrically isomorphic to L1(, ν),where is the boundary of the random walk; see Willis (1990, Theorem 2.1).A random walk where the boundary consists of a single point is said to

be ergodic. It is clear that this occurs if and only if µJ has codimension 1 inL1(G). Since µJ ⊂ L10(G), it follows that the randomwalk is ergodic if and onlyif µJ = L10(G). As shown in the exercises, µJ has a left bounded approximateidentity. Hence, by Proposition 11.3.1, if G supports an ergodic random walk,thenG is amenable. The conversewas conjectured byH.Furstenberg andprovedindependently by Rosenblatt (1981) and byKaımanovic andVersik (1983); alsosee Willis (1990).

Theorem 11.3.4 The σ -compact locally compact group G is amenable if andonly if there is a probability measure µ on G such that L10(G) = µ J . �

Recall from the Kawada–Ito theorem 8.3.4 that, if µ is a probability mea-sure on the compact group G and if the support of µ generates G, then thesequence of probability measures

(∑nj=1 µ

j)nj=1 converges to Haar measure

(in the weak-∗ topology on M(G)). Theorem 11.3.4 is a weaker version of this,where compactness is replaced by amenability, so that once again amenabil-ity appears as a weak finiteness condition. There is an important differencehowever. Theorem 8.3.4 is a universal statement applying to every probability


measure µ having large enough support, but Theorem 11.3.4 is an existentialstatement and µ is quite difficult to construct. A probability measure on anamenable group need not be ergodic even if its support equals G; some exam-ples illustrating how subtle the boundary of a random walk can be are given inKaımanovic (1983).

11.4 Exercises

1. Show that, if G is amenable and N is a closed normal subgroup of G, thenG/N is amenable.

2. (An alternative proof that F2 is not amenable.) Let

Sa = {w ∈ F2 : w = ax as a reduced word} ,

and define subsets Sb, Sa−1 , and Sb−1 of F2 similarly. Note that

F2 = {e} ∪ Sa ∪ Sb ∪ Sa−1 ∪ Sb−1

is a partition. Show that F2 = a−1Sa ∪ Sa−1 is also a partition of F2, anddeduce that there is no translation invariant mean on F2.

3. Show that, if G has a paradoxical decomposition, then there is no left-invariant mean on L∞(G).

4. Let G be a group which satisfies Følner’s condition. Show that G satisfiesProperty P1 of Reiter and the second condition of Proposition 11.2.2. Deducethat G has a left-translation invariant mean.

5. Prove Corollary 11.2.6.6. Show that {(

δe − 1n

n∑j=1

µ j

)� uλ

}(n,λ)

is a left bounded approximate identity for the ideal Jµ. (See Proposition9.1.8 for the definition of uλ.)

7. Show that the distance between the elements

(δe + e2π i/3δa + e−2π i/3δb) � f and δe + e2π i/3δa + e−2π i/3δbis at least 1 for every f ∈ 10 (F2), and deduce that 10 (F2) does have any(even unbounded) approximate identity.

12

Harmonic analysis and automatic continuity

Knowledge of the structure of L1(G) can be used to show that derivations fromL1(G) are automatically continuous when G is abelian, compact, or is in oneof several other classes. However it is not known whether such derivations arecontinuous for every locally compact group G. Work on this problem leads tofurther questions in abstract harmonic analysis.

12.1 Automatic continuity of derivations

It was shown by Ringrose (1972) that every derivation from a C*-algebra A isautomatically continuous. Powerful general techniques for proving automaticcontinuity results have been developed from this and earlier work; see Part Iand the substantial account in Dales (2000). The following general method forproving automatic continuity of derivations was given by Jewell (1977).

Theorem 12.1.1 Let A be a Banach algebra such that:

1. every closed two-sided ideal with finite codimension in A has a boundedapproximate identity; and

2. if I is a closed two-sided ideal with infinite codimension in A, then there aresequences (an) and (bn) in A such that bna1· · · an−1 �∈ I , but bna1· · · an ∈ Ifor each n ≥ 2.

Then every derivation D : A→ X, where X is a Banach A-bimodule, is con-tinuous. �

Condition 2 of the theorem is used to show that the continuity ideal (seePart I and Exercise 12.4.1) of any derivation D : A→ X cannot have infinitecodimension. A variant of the Main Boundedness Theorem of Bade and Curtis

121


(see Theorem 5.2.2) is used to show this. The existence of a bounded approx-imate identity for the continuity ideal implies, via an application of Cohen’sfactorization theorem, that the restriction of D to its continuity ideal is contin-uous. Since the continuity ideal has finite codimension, it follows that D mustbe continuous.Theorem 12.1.1 applies to all C*-algebras A. The first condition is satisfied

because every ideal in a C*-algebra has a bounded approximate identity. Toverify the second condition, it is first observed that, if A/I has an infinite-dimensional commutative *-subalgebra C , then its character space �C (seeDefinition 3.1.1) is infinite, and C may be assumed to be C0(�C ). Then an infi-nite sequence of pairwise disjoint subsets of�C may be chosen and Urysohn’slemma used to write down a sequence of non-zero functions ( fn) such thatfm fn = 0 whenever m �= n. These functions show that the second condition issatisfied (exercise). If every ∗-subalgebra of A/I is finite-dimensional, then aseparate (and elementary) argument shows that A/I is itself finite-dimensional.(For details, see Dales (2000, Corollary 5.3.7).) No such elementary argumentshows that the conditions of Theorem 12.1.1 are satisfied by L1(G) for generallocally compact groups G. However, they can be shown for certain classes ofgroups, and Condition 1 can be replaced by a weaker hypothesis.

12.2 Finite-codimensional ideals in L1(G)

As we have already seen in Propositions 11.3.2 and 11.3.3, ideals with finitecodimension in L1(G) have a bounded approximate identity if and only ifG is amenable. Hence Theorem 12.1.1 can only apply to amenable groupsat best. However, Condition 1 of the theorem can be weakened. The boundedapproximate identity is needed only so that Cohen’s factorization theorem 9.1.9may be applied. The requisite factoring can be achieved in finite-codimensionalideals of general group algebras with the aid of the following result, which isproved in Willis (2001).

Proposition 12.2.1 Let G be a σ -compact, locally compact group, and let Ibe a closed two-sided ideal with finite codimension in L1(G).

(i) There is a closed left ideal L ⊂ I having a right bounded approximateidentity, and a closed right ideal R ⊂ I having a left bounded approximateidentity, such that

I = R + L . (12.2.1)

12 Harmonic analysis and automatic continuity 123

(ii) For every sequence ( fn) in I with ‖ fn‖1 n→ 0, there are elements a and band sequences (hn) and (kn) converging to 0 in I with

fn = hn � a + b � kn (n ∈ N) . (12.2.2)

�

The factoring of null sequences described in Equation (12.2.2) is just what isneeded to complete the automatic continuity argument behind Theorem 12.1.1.It is not possible to give the details of the proof of Proposition 12.2.1 here, but

we will compare the construction of the ideals R and L with the construction ofthe approximate identities in Proposition 11.3.2 in the cases where G is finite,where G is compact, and where G is amenable. It will be clearer to begin withthe case where I = L10(G).When G is a finite group, the element δe − mG is an identity for the ideal

L10(G). When G is compact but not finite, this ideal has a bounded approximateidentity of the form

{uλ � (δe − mG)} , (12.2.3)

where {uλ} is a bounded approximate identity for L1(G); see Proposition 9.1.8.If G is not compact, the Haar measuremG is not bounded and (12.2.3) does notmake sense, but whenG is amenablemG can be replaced by a net of probabilitymeasures { fν} as in Proposition 11.2.2(3) or, when G is also σ -compact, byconvolution powers of a single probability measure as in Proposition 11.3.4.Then L1(G) has a bounded approximate identity of the form

{uλ � (δe − fν)}(λ,ν) or{uλ �

(δe − 1

n

n∑j=1

µ j

)}(λ,n)

. (12.2.4)

Random walks, that is, convolution powers of a single probability measure,can also be used in the case where G is not amenable. Define

µJ := { f − µ � f : f ∈ L1(G)} and Jµ := { f − f � µ : f ∈ L1(G)} .

Then, as seen in the last chapter, µJ is a closed right ideal with a left boundedapproximate identity, and it may be seen in the same way that Jµ is a closedleft ideal with a right bounded approximate identity. These are the ideals R andL in (12.2.1) when I = L10(G).

Proposition 12.2.2 (Willis 1990) Let G be a σ -compact group, and letµ be aprobability measure on G which is absolutely continuous with respect to Haar


measure and with support equal to G. (The σ -compactness of G is requiredonly so that such a probability measure exists.) Then

L10(G) =µJ + Jµ . �

Note that the assertion holds for any probability measure satisfying the hy-pothesis; the probability measure is not specially constructed. (This hypothesiscan be weakened a little.) The proposition is in this way closer to the Kawada–Ito theorem than to Proposition 11.3.4. It has also been long known that, ifG is abelian and the support of the probability measure µ generates G, thenJµ = L10(G). This is known as the Choquet–Deny theorem (Choquet and Deny1961). Proposition 12.2.2 might therefore be regarded as a common general-ization of the Kawada–Ito and Choquet–Deny theorems. The proof representsthe quotient L1(G)/Jµ as L1() (where is the boundary of the random walkwith law µ), and applies the ergodic theorem to a certain operator on thisspace. A proof in more probabilistic terms has also been given by Kaımanovic(1992).Now we see how R and L are defined when I is any finite-codimensional

ideal. The quotient space L1(G)/I is finite-dimensional and the regular rep-resentation of G on L1(G) induces a representation V of G by isometries onthis quotient space. The closure of V (G) in B(L1(G)/I ) is a compact group,K say, and harmonic analysis on K plays an important part. The representationV factors through K as V = V ◦ ϕ, where ϕ : G → K is a group homomor-phism and V is a representation of K . Then V is the direct sum of irreduciblefinite-dimensional representations and there are corresponding minimal centralidempotents z1, . . . , zn in L1(K ). WhenG is finite or compact, I has a boundedapproximate identity of the form{

uλ �

(δe −

n∑j=1z j

)},

where {uλ} is a bounded approximate identity for L1(G). Compare this approx-imate identity with (12.2.3), noting that when I = L10(G), V is the trivial rep-resentation and {z1, . . . , zn} = {mG}. When G is not compact but is amenable,the functions z j do not belong to L1(G), but there are nets {z jν} of functions inL1(G) such that I has a bounded approximate identity{

uλ �

(δe −

n∑j=1z jν

)}(λ,ν)

.


Compare this with (12.2.4). The general case also relies on choices of nets offunctions in L1(G) to modify the bounded approximate identities of µJ and Jµ.This is fairly delicate however, and much of the work in Willis (2001) involvesmaking these choices. The closed ideals R and L are then defined to be the idealsfor which these nets are the respective left and right approximate identities.Proposition 12.2.1 implies that all derivations from L1(G) are continuous

whenever Condition 2 of Theorem 12.1.1 is satisfied.

12.3 Infinite-codimensional ideals in L1(G)

In this section we shall see that Condition 2 of Theorem 12.1.1 is satisfiedby L1(G) when G is abelian or compact. We shall also see that, when G isconnected, a similar condition is satisfied by M(G). This condition suffices toshow that derivations are continuous.

Abelian groups Suppose first of all thatG is an abelian groupwith dual group�, and let I be a closed ideal with infinite codimension in L1(G). Recall fromExercise 3.6.1 that the hull of I is

h(I ) = {γ ∈ � : f (γ ) = 0 ( f ∈ I )} .Then, by Corollary 10.2.3, h(I ) is infinite. Since � is Hausdorff, there is asequence (γn) of disjoint points in h(I ) and pairwise disjoint open sets (Un) withγn ∈ Un for each n ∈ N. Lemma 10.2.1 assures us that we can choose functionsfn ∈ L1(G) such that, for each n ∈ N, we have f n(γn) = 1 and f n(γ ) = 0 forγ �∈ Un . These functions then satisfy the following for each m and n in N:f 2n �∈ I because f 2n (γn) = 1, and fm fn = 0 whenever m �= n. It follows thatL1(G) satisfies Condition 2 of Theorem 12.1.1 when G is abelian.

Compact groups Suppose next that G is compact, and let I be a closedideal with infinite codimension in L1(G). Then infinitely many of the centralidempotents zσ identified in Theorem 10.1.3 do not belong to I . Since theyare minimal central idempotents, zσ zτ = zσ if τ = σ and zσ zτ = 0 if τ �= σ .These idempotents show that L1(G) satisfies Condition 2 of Theorem 12.1.1when G is compact.

Connected groups Finally, suppose thatG is connected, and let I be a closedideal with infinite codimension in L1(G). We aim to show that I cannot be thecontinuity ideal of a derivation. Theorem 8.2.4 tells us that G can be approx-imated by Lie groups and, after some argument, it may be supposed that G is


in fact a connected Lie group. Then G has many 1-parameter subgroups andthese are, in particular, abelian.Let H be a closed abelian subgroup of G. Then M(H ) is isomorphic to a

subalgebra of M(G), and L1(H ) acts as multipliers on L1(G) via

f � g(x) =∫Hf (y)g(y−1x) dmH (y) (a.e.x ∈ G, f ∈ L1(H ), g ∈ L1(G)) .

Define

IH := { f ∈ L1(H ) : f � g ∈ I (g ∈ L1(G))} .

Then IH is a closed ideal in L1(H ).Since G is a Lie group, it has abelian subgroups H1, . . . , Hr and a com-

pact connected subgroup K such that G = H1 · · · Hr K ; see Iwasawa (1949,Theorem 8). Since K is a Lie group, it has 1-parameter subgroups, sayL1, . . . , Ls , such that L1 · · · Ls contains a neighbourhood O of e. Since K iscompact and connected,On = K for some n ∈ N, and so it may in fact be sup-posed that there are closed abelian groups H1, . . . , Ht such thatG = H1 · · · Ht .(If some Hj is not closed, replace itwith its closure. The closure is still an abeliangroup.) If IHj had finite codimension in L

1(Hj ) for every j ∈ {1, . . . , t}, thenit would follow that I had finite codimension in L1(G) (see Willis 1992) incontradiction to our hypothesis. Hence IH has infinite codimension in L1(H )for some closed abelian subgroup H .Choose a closed abelian subgroup H so that IH has infinite codimension in

L1(H ). Then there are sequences of functions (an) and (bn) in L1(H ) satisfying:2′: (bna1 . . . an−1) � L1(G) �⊂ I , but (bna1 . . . an) � L1(G) ⊂ I for n ≥ 2.

These sequences may then be used, exactly as Condition 2 is used, to show thatI cannot be the continuity ideal of a derivation.The above comments show that the continuity ideal of a derivation D from

L1(G) has finite codimension in L1(G) when G is abelian, compact, or con-nected. Proposition 12.2.1 may then be used to show that the restriction of Dto its continuity ideal is continuous; see Exercise 12.4.3. Hence we have shownthe following result.

Proposition 12.3.1 Let G be a locally compact group which is either abelian,compact or connected, and let X be a Banach L1(G)-bimodule. Then everyderivation D : L1(G)→ X is continuous. �

It can also be shown that all derivations are continuous for solvable and forlocally finite groups and for groups satisfying certain finiteness conditions, suchas [FD]−-groups. Proofs of someof these cases can be found inDales (2000) and


Willis (1986a, 1992). Moreover, it can be shown that, if there is a discontinuousderivation from L1(G) for some locally compact group G, then there is onewhere G is discrete. Since every discrete group is the quotient of a free group,there would then be a discontinuous derivation from a free-group algebra.

Proposition 12.3.2 Assume that there is a locally compact group G witha discontinuous derivation D : L1(G)→ X. Then there is a discontinuousderivation

D : 1(F∞)→ ∞(F∞ × F∞) . �

A partial proof of this reduction is given in Willis (1986).Apparent progress can be made towards a proof that all derivations from

1(F∞) are continuous. Free groups have many infinite abelian subgroups be-cause every non-identity element generates a subgroup isomorphic to Z. Thesame argument as used above for abelian groups shows that, if D : 1(F∞)→ Xis a derivation, then I(D) ∩ 1(H ) has finite codimension in 1(H ) for everysuch abelian subgroup H . The continuity ideal is therefore very large, but itdoes not follow that it has finite codimension in 1(F∞). This apparent progressis misleading because there are groups for which our knowledge of automaticcontinuity techniques and of the group algebra are not adequate to show thatthe continuity ideal is non-zero.W. Burnside asked in 1902 whether every finitely generated group in which

every element satisfies xn = e for some fixed n ∈ N must be finite. A negativesolution was announced by P. S. Novikov in 1959 and a proof was published,for odd n ≥ 4381, by Novikov and S. I. Adian in 1968. Adian subsequentlyimproved the value of the exponent n, and established more results about thecounter-examples; seeAdian (1979). Let B(m, n) denote the free periodic groupwith m generators for which xn = e for every x in the group. Adian showedthat B(m, n) is infinite when m ≥ 2 and n ≥ 665 is odd. Furthermore, everyabelian subgroup of B(m, n) is cyclic of finite order (Adian 1979). Hence thegroup algebra of each abelian subgroup is finite-dimensional, and automaticcontinuity techniques give no information about the continuity ideal.The following problem is a weaker, functional-analytic version of Burnside’s

original question.

Problem 12.3.3 (i) Let f ∈ 1(B(m, n)). Is the spectrum of f always totallydisconnected?(ii) Let I be an ideal in 1(F∞) such that 1(H ) ∩ I has finite codimension

in 1(H ) for every abelian subgroup H of F∞. Is the spectrum of f + I totallydisconnected for every f ∈ 1(G)?


Apositive answer, or at least a positive answer for sufficientlymany f , wouldshow that the continuity ideal of a derivation has finite codimension in 1(G),and hence that every derivation from a group algebra is continuous.

12.4 Exercises

1. Let D : A→ X be a derivation, and let

I(D) := { f ∈ A : f · S(D) = {0}}

be its continuity ideal. Show that I(D) is a closed two-sided ideal in A.2. Let A be a Banach algebra, and suppose there are elements fn in A withfm fn = 0 ifm �= n and f 2n �= 0 wheneverm, n ∈ N. Construct sequences ofelements (an) and (bn) in A such that bna1 · · · an−1 �= 0, but bna1 · · · an = 0for n ≥ 2.

3. Let D : A→ X be a derivation, and let I(D) be its continuity ideal. Showthat, if I(D) has a bounded approximate identity, then the restriction of Dto I(D) is continuous.


1. It is easy to see that, if X is an A-bimodule, then A ⊕ X is a Banach algebraunder the product defined by

(a, x)(b, y) = (ab, a · y + x · b) (a, b ∈ A, x, y ∈ X ) ,

and that, if D : A→ X is a derivation, then themap� : A→ A ⊕ X definedby �(a) = (a, D(a)) is a homomorphism. If D is discontinuous, so is �.Hence techniques for showing that homomorphisms from an algebra arecontinuous can be expected to apply to derivations too.V. Runde has studied extensively the question of automatic continuity

of homomorphisms from group algebras. He has shown that, if G is an[FIA]−-group which for some integer n has an infinite number of inequiv-alent n-dimensional unitary representations, then (assuming the continuumhypothesis) there is a discontinuous homomorphism from L1(G) (Runde1994). Abelian groups fall into this class, and so there are indeed discon-tinuous homomorphisms from some group algebras. Runde conjectures thatthe existence of infinitely many inequivalent irreducible representations ofG is both a necessary and sufficient condition for there to be discontinuoushomomorphisms from L1(G).


This changes the emphasis ofworkon thehomomorphismsquestion some-what. Runde has shown that there are groups for which all homomorphismsare continuous, and has examined whether discontinuous homomorphismsfrom L1(G) correspond to discontinuous homomorphisms from its centre;see Runde (1994, 1996, 1997).

2. The question of the automatic continuity of derivations is a particular caseof the question of comparison of the continuous and algebraic cohomologytheories of Banach algebras. Let A be a Banach algebra, and let M be aBanach A-bimodule. Let Hq (A,M) denote the algebraic cohomology groupand Hq (A,M) the continuous group. Then there is a natural comparisonmap,

iq : Hq (A,M)→ Hq (A,M) .

When q = 1, this map is injective, and automatic continuity of derivationssays that i1 is surjective as well.M. Wodzicki (1991) has examined the comparison map for higher values

of q . He finds that iq = 0 for every Banach bimodule M provided that Ais an amenable Banach algebra and q ≥ n + 3, where the cardinality of Ais equal to ℵn . Thus there often is no relation between the continuous andalgebraic cohomologies for large values of q , in sharp contrast to the casewhere q = 1 when the two cohomology groups are often isomorphic to eachother.

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Part III

Invariant subspaces

JORG ESCHMEIERUniversitat des Saarlandes, Saarbrucken, Germany

13

Compact operators

13.1

Let T : V → V be a linear operator on a finite-dimensional, complex linearspace V . Then T possesses eigenvalues. This means that there is at least onecomplex numberλ such that ker(λ− T ) �= {0}. Since the latter space is invariantunder any linear operator A : V → V commuting with T , it follows that eitherT = λI or there is a non-trivial linear subspace ofV invariant under anyoperatorin the commutant of T . Here a linear subspace M of V is called non-trivial ifM �= {0} and M �= V .A famous question in operator theory, the invariant subspace problem, asks if,

more generally, for each continuous linear operator T on an infinite-dimensionalcomplex, separable Banach space (or Hilbert space) X there is a non-trivial,closed linear subspace of X that is invariant under T . This question has guideda considerable amount of research in operator theory. In the beginning of the1980s first counter-examples were given by C. J. Read and P. Enflo in the caseof operators on certain non-reflexive Banach spaces. In the Hilbert-space case,the question seems to be still open at the time of this writing.Of course, in the infinite-dimensional case, eigenvalues need no longer exist

as simple examples show such as the unilateral shift

S : 2→ 2, S(xk) = (0, x0, x1, . . . ),or the Volterra operator

V : C([0, 1])→ C([0, 1]), (V f )(t) =t∫0

f (s) ds ,


137

138 Part III Invariant subspaces, Jorg Eschmeier

which is described in Part I, Exercise 1.5.6 and Part IV, Example 22.2.6.A class of operators that is, in many respects, still close to operators onfinite-dimensional spaces is given by the compact operators. Any non-zerocomplex number λ in the spectrum of a compact operator is an eigenvalue forthe operator such that the corresponding eigenspace is finite-dimensional (seePart I, Theorem 2.2.5). Hence the invariant subspace problem for compact oper-ators is immediately reduced to the case of quasi-nilpotent compact operators,that is, compact operators T with spectrum σ (T ) = {0} (see Part I, 2.1.2).In 1954 it was shown by Aronszajn and Smith that each compact operator

on an infinite-dimensional Banach space X has a non-trivial, closed invariantsubspace. Results of Bernstein and Robinson (1966) and Halmos (1966) showthat each bounded operator T on X such that p(T ) is compact for some non-zeropolynomial p has a non-trivial closed invariant subspace.The aimof this chapter is to prove an invariant subspace result for compact op-

erators due toLomonosov (1973) that contains the above results as special cases.Throughout Chapters 13–20 we shall write B(X ) for the Banach algebra of

all bounded linear operators on a (complex) Banach space X . Here the normon B(X ) is the operator norm. We denote the closed unit ball of X by X [1].

Definition 13.1.1 A continuous linear operator K ∈ B(X ) on a Banach spaceX is compact if K (X [1]) is a relatively compact subset of X.

In 1973, Lomonosov proved the existence of non-trivial hyperinvariant sub-spaces for each operator T ∈ B(X ) \ C1X that commutes with a non-zero com-pact operator; we first describe the elegant proof of this theorem.

Definition 13.1.2 Let T ∈ B(X ), and letM ⊂ B(X ) be arbitrary.

(a) A closed invariant subspace for T is a closed linear subspace Y ⊂ X suchthat TY ⊂ Y . We write

Lat(T ) = {Y ; Y is a closed invariant subspace for T } .

(b) More generally, we define

Lat(M) =⋂{Lat(A); A ∈M} .

(c) The spaces contained in

Hyp(T ) =⋂{Lat(A); A ∈ (T )′} ,

where (T )′ = {A ∈ B(X ); AT = T A} is the commutant of T , are calledthe hyperinvariant subspaces for T .

The following is the result of Lomonosov.

13 Compact operators 139

Theorem 13.1.3 (Lomonosov) Let T ∈ B(X ) \ C1X . Suppose that T com-mutes with a non-zero compact operator. Then Hyp(T ) is non-trivial.

For the proof of Lomonosov’s theorem we shall need a well-known fixedpoint theorem.

Theorem 13.1.4 (Brouwer) Let K ⊂ Rn be a compact and convex set. Eachcontinuous map f : K → K has a fixed point, i.e., there is at least one pointx ∈ K with f (x) = x. �

Aproof of this result can be found in almost every book on algebraic topology.We are mainly interested in the following infinite-dimensional generalizationdue to Schauder.

Corollary 13.1.5 (Schauder) Let X be a normed space. Suppose that B ⊂ Xis convex and that C ⊂ B is a compact subset. Then each continuous mapf : B → C has a fixed point.

Proof We reduce Schauder’s result to Brouwer’s fixed point theorem. To provethe corollary, it suffices to find a sequence (xk) in B with

‖ f (xk)− xk‖ k−→ 0 .

Indeed, if y is the limit of a subsequence of ( f (xk)), then obviously f (y) = y.Let ε > 0. Since C is compact, there are c1, . . . , cr ∈ C with

C ⊂r⋃i=1Bε(ci ) .

Note that the convex hull K := 〈{c1, . . . , cr }〉 is homeomorphic to a convexcompact set in Rn . The functions fi : C → [0,∞) and F : C → K which aredefined by fi (x) = max(0, ε − ‖x − ci‖) and

F(x) =r∑i=1

fi (x)∑jf j (x)

ci

are both continuous with ‖F(x)− x‖ < ε for all x ∈ C .To complete the proof, observe that, by Brouwer’s fixed point theorem, the

map

T : K → K , T (x) = F ◦ f (x) ,


has a fixed point z ∈ K and that, for any such point z, we have

‖ f (z)− z‖ = ‖ f (z)− F( f (z))‖ < ε .

Hence f has a fixed point. �

As a consequence of Schauder’s fixed point theorem we obtain the so-calledLomonosov lemma.

Lemma 13.1.6 (Lomonosov) LetA ⊂ B(X ) be a subalgebra with 1 ∈ A, andsuppose that

Lat(A) = {{0}, X} .

Then, for any compact operator K ∈ B(X ) \ {0}, there are an operator A ∈ Aand a vector x ∈ X \ {0} such that K Ax = x.

Proof Choose a vector y ∈ X such that the compact and convex set C :=K B1(y) does not contain the zero vector 0 ∈ X . Here we write B1(y) for theopen ball with radius 1 and centre y in X . Since Lat(A) is trivial, it follows thatAx = X for all x ∈ X \ {0}. Hence the compactness of C allows us to choosefinitely many operators A1, . . . , An ∈ A with the property that

C ⊂n⋃i=1

A−1i (B1(y)) .

Fix any continuous map r : R+ → R+ with r−1(0) = [1,∞). Then the com-position F : C

f−→ B1(y)K−→ C , where

f (x) =n∑i=1

r (‖Ai x − y‖)∑jr (‖A j x − y‖) Ai x ,

has a fixed point x by Schauder’s fixed point theorem. But then we see imme-diately that x = F(x) = K Ax for a suitable operator A ∈ A. �

Lomonosov’s lemma and elementary linear algebra yield a proof ofLomonosov’s invariant subspace theorem.

Proof of Theorem 13.1.3 Under the hypotheses of Theorem 13.1.3, assumetowards a contradiction that the algebra A = (T )′ has no non-trivial invariant

13 Compact operators 141

subspace. Then, by Lomonosov’s lemma, there are A ∈ A and 0 �= x ∈ X with

ker(1− K A) �= {0} .

Since K A is compact and commutes with T , the space on the left is a finite-dimensional invariant subspace for T . But then T possesses eigenvalues. Thisis impossible, since any eigenspace for T belongs to Lat(A). �

A weaker version of Theorem 13.1.3 can be proved without using a fixedpoint argument. The spectral radius argument used in the following proof is dueto M. Hilden.

Theorem 13.1.7 Each compact operator K ∈ B(X ) \ {0} on an infinite-dimensional Banach space X has a non-trivial hyperinvariant subspace.

Proof Set A = (K )′. Assume towards a contradiction that Lat(A) is trivial.Choose y, C , and A1, . . . , An with respect to A and the compact operator Kexactly as in the proof of Lemma 13.1.6.Then Ky ∈ C . Hence Ai1Ky ∈ B1(y) for some 1 ≤ i1 ≤ n. But then

K Ai1Ky ∈ C , and again Ai2K Ai1Ky ∈ B1(y) for some 1 ≤ i2 ≤ n. Contin-uing in this way, we obtain vectors

yN = AiN · · · Ai1K N y ∈ B1(y) .

Define s = max(‖A1‖, . . . , ‖An‖). Then

r ≤ ‖yN‖ ≤ sN‖K N‖‖y‖ (N ∈ N)

holds with a suitable real number r > 0.We conclude that the spectral radius (see Proposition 2.1.3)

ν(K ) = limN→∞

‖K N‖1/N = max{|z|; z ∈ σ (K )}

is positive. Since any non-zero element in the spectrum of a compact operatoris an eigenvalue, we obtain as before the contradiction that Lat(A) is non-trivial. �

13.2 Exercises

1. Let X, Y be Banach spaces, and let T ∈ B(X, Y ). Show that T is compactif and only if T ′ ∈ B(Y ′, X ′) is compact.


2. Let K ∈ B(X ) be compact. Show that, if λ is a non-zero complex numbersuch that λ− K is not bounded below, then λ is an eigenvalue for K .Conclude that each non-zero point λ ∈ σ (K ) is an eigenvalue for K or forK ′. Deduce that on an infinite-dimensional Banach space a compact operatorwithout any non-trivial hyperinvariant subspace is quasi-nilpotent.

3. Let T ∈ B(X ) be compact. Show that each sequence (λn) of pairwise distincteigenvalues for T converges to zero. Conclude that each pointλ ∈ σ (T ) \ {0}is isolated in σ (T ) and is an eigenvalue for T .

4. UseLomonosov’s lemma to show that each norm-closed, unital subalgebraAof B(X ) that contains a non-zero compact operator K and possesses no non-trivial closed invariant subspaces contains a non-zero finite-rank operator.

5. Let T ∈ B(X ) \ C1X be a bounded operator such that p(T ) is a compactoperator for some non-zero polynomial p ∈ C[z]. Show that T has a non-trivial hyperinvariant subspace.


At the time when Lomonosov’s paper appeared, the question whether eachbounded operator on a complex and separable infinite-dimensional Banachspace possesses a non-trivial closed invariant subspace was still open. Counter-examples on certain non-reflexive Banach spaces were given by C. J. Read(1984) and P. Enflo (1987) (see also Beauzamy 1985). In the Hilbert-space casethe question is still open at the time of this writing. Theorem 13.1.3 is fromLomonosov (1973).Lomonosov’s result has been applied and improved in several directions. In

1974, Pearcy and Shields used Lomonosov’s result to show that each unital sub-algebra A of B(X ) without non-trivial closed invariant subspaces that containsa non-zero compact operator has to be strongly dense inB(X ). This gives a pos-itive answer to the so-called transitive algebra conjecture in a special case. In1975, Kim, Pearcy, and Shields proved that each non-scalar operator T ∈ B(X )for which there is a non-zero compact operator K such that im(T K − KT )is at most one-dimensional has a non-trivial invariant subspace. Other gener-alizations have been obtained by Fong, Nordgren, Radjabalipour, Radjavi andRosenthal.

14

Unitary dilations and the H∞-functionalcalculus

14.1

In Chapters 14–20 we shall describe and apply a second general method to con-struct invariant subspaces for certain classes of operators on Banach or Hilbertspaces. This method is due to Scott Brown, who used it in 1978 in his thesisto prove the existence of non-trivial, closed invariant subspaces for subnormaloperators. All applications of this so-called Scott Brown technique depend onthe existence of a suitable H∞-functional calculus for the operators under con-sideration. We begin by studying a classical case in which the existence of anH∞-functional calculus is well-known.Let H be a complex Hilbert space. Throughout Chapters 14–20, we shall

denote by [x, y] the inner product of given vectors x, y ∈ H . For an open setV in C, we write H (V ) for the Frechet space of all complex-valued analyticfunctions on V equipped with the topology of uniform convergence on allcompact subsets of V .A famous theorem of Sz.-Nagy from 1953 says that every contraction T on

H possesses a unitary dilation.

Theorem 14.1.1 Let T ∈ B(H ) be a contraction. Then there exists a Hilbertspace K containing H and a unitary operator U ∈ B(K ) such that

T k = PUk |H (k ∈ N) ,

where P denotes the orthogonal projection from K onto H.

Proof The orthogonal direct sum

H =⊕n∈Z

H ={(hn)n∈Z; hn ∈ H and ‖(hn)‖2 =

∑n∈Z

‖hn‖2 <∞}

143


is a Hilbert space relative to the inner product defined by

[(hn), (kn)] =∑n∈Z

[hn, kn] .

The operators D = √1− T ∗T and D∗ =√1− T T ∗ satisfy the relations

T D = D∗T and T ∗D∗ = DT ∗ .

Indeed, since T (1− T ∗T ) = (1− T T ∗)T , we obtain that

T p(1− T ∗T ) = p(1− T T ∗)T

holds for each polynomial p. But then the Stone–Weierstrass theorem impliesthat

T f (1− T ∗T ) = f (1− T T ∗)T

for each continuous function f ∈ C([0, 1]). Choosing f (t) = √t , we obtainthe first intertwining relation. The second follows analogously.As a consequencewe remark that the spacesD = DH andD∗ = D∗H satisfy

the inclusions

TD ⊂ D∗ and T ∗D∗ ⊂ D .

We regard H as a subspace of the Hilbert space

K = {(hn)n∈Z ∈ H; h0 ∈ H, hn ∈ D and h−n ∈ D∗ for all n ∈ N}

via the canonical isometry H → K defined by

h �→ (. . . , 0, [h], 0, . . . ) ,

where the square bracket indicates the zero-th position.We claim that the linear operator U : K → K defined by

Uh = (. . . , h−3, h−2, [D∗h−1 + Th0],−T ∗h−1 + Dh0, h1, h2, . . . )

is a unitary dilation of T . To see that U is isometric, note that all the terms in

‖h‖2 − ‖Uh‖2 = ‖h−1‖2 + ‖h0‖2

−[D∗h−1 + Th0, D∗h−1 + Th0]−[−T ∗h−1 + Dh0,−T ∗h−1 + Dh0]

14 Unitary dilations and the H∞-functional calculus 145

cancel each other. To check that U is surjective, note that the linear operatorV : K → K defined by

Vh = (. . . , h−1, D∗h0 − Th1, [T ∗h0 + Dh1], h2, h3, . . . )

yields a right inverse for U .Identifying H with a subspace of K , an elementary induction shows that

Unh = (. . . , 0, [T nh], DT n−1h, . . . , Dh, 0, . . . ) .

Thus we have shown that U is a unitary dilation of T . �

A unitary dilation U ∈ B(K ) of a contraction T ∈ B(H ) is called minimalif the only reducing subspace for U containing H is the space K itself or,equivalently, if

K =∨n∈Z

UnH.

Remark 14.1.2(a) It is elementary to check that the unitary dilation constructed in the proofof Theorem 14.1.1 is minimal.

(b) If U ∈ B(K ) is a unitary dilation of a given contraction T ∈ B(H ), thenthe restriction of U to K0 =

∨n∈Z

UnH is a minimal unitary dilation of T .

(c) If U ∈ B(K ) and V ∈ B(K ) are minimal unitary dilations of T ∈ B(H ),then the unique continuous linear map � : K → K with

�(Unh) = V nh (n ∈ Z, h ∈ H )

is a unitary operator with �U = V�.

As an important application of the dilation theorem, we obtain a continuityproperty for the polynomial functional calculus of a contraction.

Corollary 14.1.3 (von Neumann’s inequality) Let T ∈ B(H ) be a contrac-tion. Then

‖p(T )‖ ≤ |p|Dfor all polynomials p ∈ C[z].

Proof Let U be a unitary dilation of T . Using the fact that the C(σ (U ))–functional calculus of the normal operatorU is isometric (see Part I, Additional


notes, 4.5.2), we obtain the inequalities

‖p(T )‖ = ‖Pp(U )|H‖ ≤ ‖p(U )‖ = |p|σ (U ) ≤ |p|D . �

Since the restrictions of the complex polynomials to the unit disc D are uni-formly dense in the disc algebra A(D) (see Examples 1.2 (iii)), von Neumann’sinequality is equivalent to the existence of a contractive disc algebra functionalcalculus. For more on the functional calculus, see Chapter 4.

Corollary 14.1.4 Let T ∈ B(H ) be a contraction. Then there is a unique con-tractive algebra homomorphism & : A(D)→ B(H ) extending the polynomialfunctional calculus of the operator T . ✷

We want to describe conditions under which the disc algebra functionalcalculus of T extends to H∞(D).Let V ⊂ C be a non-empty, open set. Then L∞(V ) formed with respect to

the planar Lebesgue measure is the dual space of L1(V ), and hence carriesa weak-∗ topology, denoted by σ . Let H∞(V ) be the Banach algebra of allbounded analytic functions on V (see Examples 1.2 (xi)).

Lemma 14.1.5 The space H∞(V ) is a weak-∗ closed subspace of L∞(V ).

Proof Let B be the closed unit ball of L∞(V ). By the Krein–Smulian theorem(see Schaefer 1966, Theorem IV.6.4), it suffices to show that H∞(V ) ∩ B ⊂ Bis weak-∗ closed. Since L1(V ) is separable, (B, σ ) is metrizable. Let ( fk) be asequence in H∞(V ) ∩ B with σ − limk fk = f . ByMontel’s theorem, we may

suppose, after passing to a subsequence, that ( fk)k−→ g ∈ H∞(V ) uniformly

on compact subsets. Since C00(V ) is dense in L1(V ) (see Lang 1969, TheoremXII.6), the observation that∫

V

ϕgdλ = limk

∫V

ϕ fkdλ =∫V

ϕ f dλ (ϕ ∈ C00(V ))

implies that g = f almost everywhere. �

As a consequence of the previous result we can identify H∞(V ) with thedual space of the separable Banach space Q := L1(V )/⊥H∞(V ).


Lemma 14.1.6 A sequence ( fk) in H∞(V ) is a weak-∗ zero sequence if andonly if supk ‖ fk‖ <∞ and ( fk) converges to zero pointwise (or equivalently,uniformly on compact subsets of V ).

Proof Let ( fk) be a weak-∗ zero sequence. By the uniform boundedness prin-ciple, ( fk) is norm-bounded. By the argument given in the previous proof, eachsubsequence of ( fk) has a subsequence converging to zero in the Frechet spaceH (V ). Hence limk fk = 0 in H (V ).Conversely, suppose that ( fk) is norm-bounded and converges to zero point-

wise. By the dominated convergence theorem,∫V

ϕ fkdλk−→ 0 (ϕ ∈ C00(V )) .

Since ( fk) is bounded in L1(V )′, this implies that ( fk) is a weak-∗ zerosequence. �

Let X be a Banach space. Since the space H∞(V ) has a separable pred-ual, a linear map � : H∞(V )→ X ′ is weak-∗ continuous if and only if it issequentially weak-∗ continuous (Exercise 14.2.2). Thus in particular all pointevaluations

Eλ : H∞(V )→ C, f �→ f (λ) (λ ∈ V ) ,

are weak-∗ continuous.Let H be a Hilbert space. Then B(H ) can be identified isometrically with

the dual space of the Banach space C1(H ) of all trace-class operators on H viathe bilinear form

C1(H )× B(H )→ C, (A, T ) �→ Tr (AT ) .

For the definition of the trace and trace-class operators the reader is referred toConway (1991, Chapter 1). The weak-∗ topology of B(H ) with respect to thisduality is usually called the ultraweak operator topology. It is given by theseminorms

px,y(T ) =∣∣∣∣∣ ∞∑k=1[T xk, yk]

∣∣∣∣∣ ,where x = (xk) and y = (yk) run through all square summable sequences in H(see Stratila and Zsido 1979, Chapter 1). On the bounded subsets of B(H ), the


weak operator topology and the weak-∗ topology coincide. By definition theweak operator topology is the locally convex topology on B(H ) given bythe seminorms qx,y(T ) = |[T x, y]| where x, y ∈ H are arbitrary.

Remark 14.1.7 By the previous discussion a map � : H∞(V )→ B(H ) isweak-∗ continuous if and only if, for eachweak-∗ zero sequence ( fk) in H∞(V ),

limk[�( fk)x, y] = 0

for all x, y ∈ H .Using the same principle, we obtain a useful characterization of those maps

& : A(V )→ B(H ) that extend to a weak-∗ continuous linear map

� : H∞(V )→ B(H ) .

Here A(V ) denotes the closed subalgebra of C(V ) consisting of all functionsf ∈ C(V ) for which f |V is analytic; it is related to the uniform algebra A(V )of Examples 1.2(v).

Definition 14.1.8 Let V be a non-empty, open set in C.(i) A sequence ( fk) in A(V ) is a Montel sequence if ( fk |V ) is a weak-∗ zero

sequence in H∞(V ).(ii) The space A(V ) is pointwise boundedly dense in H∞(V ) if, for each func-

tion f in H∞(V ), there is a sequence ( fk) in A(V ) with ‖ fk‖ ≤ ‖ f ‖ suchthat ( fk) is weak-∗ convergent to f .

Let (Ak) be sequence of operators in B(H ). We shall write

WOT− limk→∞

Ak = A

if (Ak) converges to an operator A ∈ B(H ) in the weak operator topology, thatis, if limk→∞[Akx, y] = [Ax, y] for all vectors x, y in H .

Lemma 14.1.9 Suppose that A(V ) is pointwise boundedly dense in H∞(V ).Then a continuous linear operator & : A(V )→ B(H ) extends to a weak-∗continuous linear operator � : H∞(V )→ B(H ) if and only if

WOT− limk→∞

&( fk) = 0

for each Montel sequence ( fk) in A(V ).


Proof The necessity of the stated condition is obvious.Suppose that& satisfies the above continuity property. If f ∈ H∞(V ) and if

x, y are vectors in H , then, for each sequence ( fk) in A(V ) with weak-∗ limitf , the limit

fx,y = limk[�( fk)x, y]

exists and is independent of the choice of ( fk). Since we can achieve that‖ fk‖ ≤ ‖ f ‖, this argument gives an extension of& to a continuous linear map� : H∞(V )→ B(H ) with ‖�‖ = ‖&‖ and

[�( f )x, y] = fx,y ( f ∈ H∞(V ), x, y ∈ H ) .

Using the facts that the unit ball B of H∞(V ) is metrizable in the weak-∗topology and that B ∩ A(V ) is weak-∗ dense in B, one can easily show that,for each weak-∗ zero sequence ( fk) contained in B,

limk→∞[�( fk)x, y] = 0 (x, y ∈ H ) .

Therefore the extension � of & is weak-∗ continuous. �

Note that the extension constructed in the above proof satisfies ‖�‖ = ‖&‖.It is elementary (Exercise 14.2.3) to show that� remains multiplicative if& issupposed to be multiplicative.A contraction T ∈ B(H ) is called completely non-unitary if there is no non-

zero reducing subspace for T such that T restricted to this subspace is a unitaryoperator. A classical result of Sz.-Nagy and Foias shows that each completelynon–unitary contraction T possesses a weak-∗ continuous H∞(D)-functionalcalculus. A proof of this result can be found in Sz.-Nagy and Foias (1970,Chapter 3).We indicate briefly an alternative proof, which can be extended to more

general settings (see Eschmeier 1997).Let& : A(D)→ B(H ) be a contractive algebra homomorphism.By themax-

imum modulus principle we may regard A(D) as a closed linear subspace ofC(T) (see Part I, Exercise 1.5.4). By the Hahn–Banach and the Riesz represen-tation theorems, there is a family of measures µ(x, y) ∈ M(T) (x, y ∈ H ) suchthat

[&( f )x, y] =∫T

f dµ(x, y) ( f ∈ A(D))

and ‖µ(x, y)‖ ≤ ‖x‖ ‖y‖. We call (µ(x, y))x,y∈H a representing family for &.


Letm be the normalized linear Lebesgue measure on T. We call a measureµin M(T) absolutely continuous (singular) if µ, m (µ⊥m). By the Lebesguedecomposition theorem, each measure µ ∈ M(T) has a unique decomposition

µ = µa + µsinto an absolutely continuous part µa and a singular part µs .

Theorem 14.1.10 (F. and M. Riesz theorem) Let µ ∈ M(T) be a measurewith

µ(−n) =∫T

zn dµ = 0

for all n ∈ N. Then µ, m. In particular, each measure µ ∈ A(D)⊥ is abso-lutely continuous. �

Definition 14.1.11 A contractive algebra homomorphism& : A(D)→ B(H )is absolutely continuous (singular) if it possesses a representing family of mea-sures that are absolutely continuous (singular).

The Lebesgue decomposition for measures yields a corresponding decom-position for disc algebra representations.

Theorem 14.1.12 Let& : A(D)→ B(H ) be a contractive algebra homomor-phism. Then there are contractive algebra homomorphisms

&a, &s : A(D)→ B(H )

such that & = &a +&s , &a is absolutely continuous, &s is singular, and&a( f )&s(g) = 0 = &s(g)&a( f ) ( f, g ∈ A(D)) .

Proof Choose a representing family µ(x, y) of measures for &. Let

µ(x, y) = µa(x, y)+ µs(x, y)

be the Lebesgue decomposition. Then the map

H × H → M(T)/A(D)⊥, (x, y) �→ [µ(x, y)] ,

is sesquilinear. By the F. and M. Riesz theorem, the maps

H × H → M(T)/A(D)⊥, (x, y) �→ [µa(x, y)] ,

H × H → M(T), (x, y) �→ µs(x, y) ,


are sesquilinear. It follows easily that there are unique contractive linear maps&a, &s : A(D)→ B(H ) with

[&a( f )x, y] =∫T

f dµa(x, y), [&s( f )x, y] =∫T

f dµs(x, y) .

For f, g ∈ A(D) and x, y ∈ H , we have∫T

f g dµ(x, y) = [&( f )x, &(g)∗y] =∫T

f dµ(x, &(g)∗y) .

Again by the F. and M. Riesz theorem and the uniqueness of the Lebesguedecomposition, we have

gµ(x, y)− µ(x, &(g)∗y) = gµa(x, y)− µa(x, &(g)∗y) ∈ A(D)⊥ .

It follows that∫T

f g dµa(x, y) = [&a( f )x, &(g)∗y] =∫T

g dµ(&a( f )x, y) .

As above, we conclude that µs(&a( f )x, y) = 0 and thatf µa(x, y)− µa(&a( f )x, y) ∈ A(D)⊥ .

In this way, we obtain the multiplicativity of &a . Indeed,

[&a( f g)x, y] =∫T

g dµa(&a( f )x, y) = [&a(g)&a( f )x, y] .

Furthermore, for f, g ∈ A(D) and x, y ∈ H ,

[&s(g)&a( f )x, y] =∫T

g dµs(&a( f )x, y) = 0 .

The remaining parts of Theorem 14.1.12 can be obtained in a similar way.�

Definition 14.1.13 A measure µ ∈ M(T) is a Henkin measure if∫T

fk dµk−→ 0

for each Montel sequence ( fk) in A(D).


Since∫f dm = f (0) for all f ∈ A(D), the Lebesgue measure m is a Henkin

measure. If µ ∈ M(T) is a Henkin measure, then, by Henkin’s theorem (seeRudin 1980, Theorem 9.3.1), each measure ν , µ is a Henkin measure(cf. Exercise 15.2.4).It can easily be seen that the representations &a, &s in Theorem 14.1.12 are

uniquely determined by & (Exercise 14.2.4). By the preceding remarks, andby Lemma 14.1.9, the absolutely continuous part &a of & extends to a weak-∗continuous algebra homomorphism �a : H∞(D)→ B(H ).Since the family (µs(x, y))x,y∈H depends in a sesquilinear way on x and y

and satisfies ‖µs(x, y)‖ ≤ ‖x‖ ‖y‖, we can define a contractive linear extension�s : BM(T)→ B(H ) of &s by the formula

[�s( f )x, y] =∫T

f dµs(x, y) ( f ∈ BM(T), x, y ∈ H ) .

Here BM(T) is the Banach algebra of all bounded measurable functions on T.An argument similar to the one used in the proof of Theorem 14.1.12 showsthat �s is multiplicative. Since characteristic functions correspond to orthog-onal projections under �s , the map �s is even a C∗-algebra homomorphism(Exercise 14.2.5).If one denotes by Hs = &s(1)H the singular part of H , then

C(T)→ B(Hs), f �→ �s( f )|Hs ,

becomes a unital C∗-algebra homomorphism.

Corollary 14.1.14 (Sz.-Nagy–Foias) Let T ∈ B(H ) be a completely non-unitary contraction on a Hilbert space H. Then there is a unique weak-∗continuous algebra homomorphism � : H∞(D)→ B(H ) with ‖�‖ = 1 and�(1) = 1H , �(z) = T .

Proof If & : A(D)→ B(H ) is the disc algebra functional calculus of T then,with the above notations, T |Hs = �s(z)|Hs is a unitary operator on the reducingsubspace Hs for T . Hence Hs = {0} and & = &a has the claimed extension.

�

We call a contraction T ∈ B(H ) absolutely continuous if its disc algebrafunctional calculus extends to a weak-∗ continuous algebra homomorphism� : H∞(D)→ B(H ).


14.2 Exercises

1. Show that a minimal unitary dilation of a contraction is uniquely determinedin the sense of Remark 14.1.2 (c).

2. Let X, Y be Banach spaces with Y separable. Show that a linear mapS : Y ′ → X ′ is weak-∗ continuous if and only if it is weak-∗ sequentiallycontinuous.

3. Let V ⊂ C be a bounded open set, and let A be a weak-∗ dense subalgebraof H∞(V ). Let � : H∞(V )→ B(H ) be a weak-∗ continuous map. Showthat � is multiplicative if �|A is multiplicative.

4. Let & : A(D)→ B(H ) be a contractive algebra homomorphism. Show thatthere is atmost one pair (&1, &2) consisting of contractive algebra homomor-phisms &i : A(D)→ B(H ) with & = &1 +&2 such that &1 is absolutelycontinuous and &2 is singular.

5. Show that the map &s constructed in the proof of Theorem 14.1.12 extendsto a C∗-algebra homomorphism �s : BM(T)→ B(H ).


The idea of proving the existence of the Nagy–Foias functional calculus bydecomposing suitable representing measures into an absolutely continuous anda singular part goes back to Mlak (1969). The same idea can be used to de-compose representations of the ball algebra A(B) in the multivariable case(see Eschmeier 1997 or Chapter 20). In this case the Lebesgue decompositiontheorem has to be replaced by a more general decomposition theorem due toGlicksberg, Konig, and Seever (see Rudin 1980, Theorem 9.4.4). The classicproof and many other applications can be found in the book of Sz.-Nagy andFoias (1970, Chapter 3).

15

Hyperinvariant subspaces

15.1

Let T ∈ B(H ) be a contraction, and let & : A(D)→ B(H ), f �→ &( f ), beits disc algebra functional calculus. Recall that T is said to be of type C0·(respectively, C·0) if

(T k)SOT−→ 0 (respectively, (T ∗k)

SOT−→ 0).

By a result of Sz.-Nagy and Foias, a contraction which is neither of type C0·nor of type C·0 either is a scalar multiple of the identity operator or possessesnon-trivial hyperinvariant subspaces. The following observation will help us toextend this result to more general settings.

Lemma 15.1.1 A contraction T is of type C0· if and only if (&( fk))SOT−→ 0 for

each Montel sequence ( fk) in A(D).

Proof Fix a Montel sequence ( fk) with ‖ fk‖ ≤ 1 and a natural number N , andthen use the Taylor expansion of fk at zero to write

fk = pk + zN gk,

where pk is a polynomial of degree less than N . Then pk → 0 uniformly on D,and also, by the maximum modulus principle,

‖gk‖ ≤ ‖ fk‖ + ‖pk‖ ≤ 2

for k sufficiently large. But then, for any given vector x ∈ H , it follows that

lim supk→∞

‖&( fk)x‖ ≤ 2‖T N x‖.

154

15 Hyperinvariant subspaces 155

This shows that the stated condition is necessary for T to be of type C0·. Thesufficiency of the condition is obvious. �

If T is absolutely continuous, then an analogue of the last lemma for theH ∞(D)-functional calculus follows in the same way. Let V be a bounded openset inC and let& : A(V )→ B(H ) be a continuous algebra homomorphism.Weset V ∗ = {z; z ∈ V }, and we consider the continuous algebra homomorphism

& : A(V ∗)→ B(H ), f �→ &( f )∗(where f (z) = f (z)

).

Definition 15.1.2 A continuous algebra homomorphism & : A(V )→ B(H )is of type C0· if (&( fk))

SOT−→ 0 for each Montel sequence ( fk) in A(V ). We saythat & is of type C·0 if & is of type C0·.

For representations � : H ∞(V )→ B(H ), we define the C0·- and C·0-properties analogously, replacing Montel sequences by arbitrary weak-∗ zerosequences in H ∞(V ). Our generalization of the Sz.-Nagy–Foias result uses thefollowing property of Hankel-type operators.

Theorem 15.1.3 For each continuous function g ∈ C(V ), the operator

Sg : A(V )→ C(V )/A(V ), f �→ [g f ],

is compact. �

For a proof of this result, we refer the reader to Cole and Gamelin (1982,Theorem 6.3).

Corollary 15.1.4 Let g ∈ C(∂V ). For each Montel sequence ( fk) in A(V ),there is a Montel sequence (gk) in A(V ) such that

| fkg − gk |∂V k−→ 0.

Proof We may suppose that g ∈ C(V ). Let ( fk) be a Montel sequence. ByTheorem 15.1.3, each subsequence of ([g fk])k has a convergent subsequence inC(V )/A(V ). Since weak-∗ limk fk = 0, any such convergent subsequence haslimit zero. Hence limk[g fk] = 0 in C(V )/A(V ), and we can choose a sequence(gk) in A(V ) such that (g fk − gk) converges to zero uniformly on V . Clearly(gk) is a Montel sequence. �


Let λ : ∞ → C be a continuous linear form of norm 1 such that λ((sk)) =λ((sk+1)) for each sequence (sk) ∈ ∞ and such that λ((sk)) = limk sk for eachconvergent sequence of complex numbers. Then, for each bounded sequence(xk) in H , there is a unique vector L(xk) in H with

λ(([xk, y])) = [L(xk), y] (y ∈ H ) .

The resulting map

L : ∞(H )→ H, (xk) �→ L((xk)) ,

is continuous and linear, with norm 1, such that L((xk)) = (xk+1) for eachsequence (xk) in ∞(H ) and L((xk)) = x whenever (xk) converges weakly toa vector x in H . The very definition of L implies that L((Axk)) = AL((xk)) foreach operator A ∈ B(H ) and each bounded sequence (xk) in H .

Theorem 15.1.5 Let & : A(V )→ B(H ) be a continuous algebra homomor-phism. If & is not of type C·0, then there is a non-zero bounded linear mapj : C(∂V )→ H with &( f ) j = jM( f |∂V ) for all f ∈ A(V ).

Proof If& is not of type C·0, then there is a Montel sequence ( fk) in A(V ) anda vector x ∈ H such that ‖ fk‖ ≤ 1 and such that the limit ε = limk ‖&( fk)∗x‖exists and is non-zero. Define xk = &( fk)∗x/‖&( fk)∗x‖.Let g ∈ C(∂V ). By Corollary 15.1.4, there is aMontel sequence (gk) in A(V )

such that |g fk − gk |∂V k−→ 0. The mapping

j : C(∂V )→ H, j(g) = L(&(gk)xk) ,

is well-defined, continuous and linear with ‖ j‖ ≤ ‖&‖. Furthermore, by con-struction, we have [ j(1), x] = ε and &( f ) j = jM f |∂V for all f in A(V ). �

For sufficiently nice domains V in C (or even Cn), also the converse of thelast theorem holds (see Eschmeier 1997, Theorem 2.1).Let T ∈ B(H ), and let j : C(∂V )→ H be a bounded operator intertwining

Mz onC(∂V ) and T on H . Denote by I ( j) the largest closed ideal inC(∂V ) con-tained in the kernel of j , and define the support s( j) of j as the common zero setof the functions in I ( j). For f ∈ C(∂V ), let M f be the multiplication operator

M f : C(∂V )→ C(∂V ), g �→ f g.

Using continuous partitions of unity one can prove the next result.


Lemma 15.1.6 Let j and T be as explained above. Then:

(i) if z ∈ s( j) and if f ∈ C(∂V ) satisfies f (z) �= 0, then j ◦ M f �= 0;(ii) s( j ◦ M f ) ⊂ s( j) ∩ supp( f ) for all f ∈ C(∂V );(iii) if s( j) = {z}, then im( j) ∈ Lat(T ) is one-dimensional and z ∈ σp(T ). �

We leave the proof as an elementary exercise (Exercise 15.2.1). Using theabove observations,we can prove a generalization of the cited result of Sz.-Nagyand Foias.

Theorem 15.1.7 Let & : A(V )→ B(H ) be a continuous algebra homomor-phism. If & is neither of type C0· nor of type C·0, then either T = &(z) is ascalar multiple of the identity operator or Hyp(T ) is non-trivial.

Proof The hypothesis that& is neitherC·0 norC0· allows us to choose non-zerobounded linear maps

j : C(∂V )→ H and k : C(∂V )→ H ′

such that j intertwines Mz on C(∂V ) with T on H , and k intertwines Mz onC(∂V ) with the Banach-space adjoint T ′ ∈ B(H ′) of T . By Lemma 15.1.6, wemay suppose that s( j) and s(k) are disjoint. By Tietze’s extension theorem, themaps j and k extend to bounded intertwiners

J : C(s( j))→ H and K : C(s(k))→ H ′.

For each operator A commuting with T , the composition

X = X (A) : C(s( j))A◦J−→ H

K ′−→ M(s(k))

intertwinesMz onC(s( j)) andMz onM(s(k)). Since the last two operators havedisjoint spectra (their spectra are s( j) and s(k), respectively), a well-knownresult of Rosenblum (1956) shows that X = 0. Hence the space∨

{im AJ ; A ∈ (T )′} ⊂ ker(K ′)

is a non-trivial hyperinvariant subspace for T . �

If A(V ) is pointwise boundedly dense in H ∞(V ) and if & : A(V )→ B(H )isC0· orC·0, then, by Lemma 14.1.9, the map& extends to a weak-∗ continuousalgebra homomorphism � : H ∞(V )→ B(H ). It is elementary to check thatthe C0·- or C·0-property is inherited by � (Exercise 15.2.2).


We end this section with an application of Theorem 15.1.5 to subnormaloperators.

Corollary 15.1.8 Let S ∈ B(H ) be subnormal, and let& : A(V )→ B(H ) bea continuous unital algebra homomorphism with &(z) = S for some boundedopen set V in C. If& is not of type C·0, then there is a non-zero reducing spaceM for S such that S|M is normal with σ (S|M) ⊂ ∂V .

Proof Let N ∈ B(K ) be the minimal normal extension of the subnormal oper-ator S, and let j : C(∂V )→ H be a non-zero bounded linear map intertwiningMz on C(∂V ) and S on H . A Fuglede-type argument (Exercise 15.2.3) showsthat j intertwines Mz on C(∂V ) and N ∗ on K . Therefore the image of j iscontained in the space

H0 = {x ∈ H ; N ∗k x ∈ H for all k ∈ Z+},

which reduces N , and hence also S. But then S|H0 = N |H0 is normal with

j(C(∂V )) ⊂ XN |H0 (∂V ) .

Clearly the space on the right-hand side can be chosen as M . �

If in Corollary 15.1.8 the operator S is asked to be pure (or to possess nonormal part with spectrum in ∂V ), then & has to be of type C·0.

15.2 Exercises

1. Prove Lemma 15.1.6.2. Let V ⊂ C be a bounded open set in C such that A(V ) ⊂ H ∞(V ) is point-wise boundedly dense in H ∞(V ). Show that a weak-∗ continuous algebrahomomorphism � : H ∞(V )→ B(H ) is of type C0· if �|A(V ) has thisproperty.

3. Let K ⊂ C be compact and let N ∈ B(H ) be a normal operator. Suppose thatj : C(K )→ H is continuous linear with j(z f ) = N j( f ) for all f ∈ C(K ).Show that j(z f ) = N ∗ j( f ) for all f ∈ C(K ).

4. Let V ⊂ C be a bounded open set. Call a measure µ ∈ M(∂V ) a Henkinmeasure if limk→∞

∫∂Vfk dµ = 0 for each Montel sequence ( fk) in A(V ).

Show that if µ is a Henkin measure, then each measure ν ∈ M(∂V ) withν , µ is a Henkin measure. Hint: use Corollary 15.1.4.



The classic result of Sz.-Nagy and Foias saying that a contraction T which isneither of type C0· nor of type C·0 is either a scalar multiple of the identity orhas non-trivial hyperinvariant subspaces can be found in Sz.-Nagy and Foias(1970, Theorem II.5.4). Results similar to Theorem 15.1.5 and Theorem 15.1.7are contained in a paper of Apostol and Chevreau (1982). If in Corollary 15.1.8the operator S is only supposed to be subdecomposable, then, if& is not of typeC·0, it follows that HS(∂V ) �= {0}. Since hyponormal operators are subdecom-posable and since a hyponormal operator whose spectrum has planar Lebesguemeasure zero is normal, it follows that a hyponormal contraction is completelynon-unitary if and only if it is of type C·0.

16

Invariant subspaces for contractions

16.1

Let T ∈ B(H ) be a contraction. In 1979 Brown, Chevreau, and Pearcy provedthat T possesses non-trivial invariant subspaces if its spectrum is rich enoughclose to the unit circle. It is the aim of this chapter to give a proof of this resultand to explain, at the same time, the basic ideas of the Scott Brown factorizationtechnique.

Definition 16.1.1 Let V ⊂ C be open. A set σ ⊂ C is dominating in V if| f |V = | f |V∩σ for each f ∈ H∞(V ).

With this notion the cited invariant-subspace result is the following.

Theorem 16.1.2 (Brown, Chevreau, and Pearcy) Let T ∈ B(H ) be a con-traction such that σ (T ) is dominating in D. Then T has non-trivial invariantsubspaces.

For a subset M of a complex normed space E , we denote byC(M) and�(M)the convex and absolutely convex hull of M . We write C(M) and �(M) for theclosures of these sets.

Lemma 16.1.3 A set σ is dominating in an open set V in C if and only if

�({Eλ; λ ∈ σ ∩ V }) = Q[1] . �

This result is a direct consequence of the separation theorem (Exercise16.2.1). To prove Theorem 16.1.2, we are allowed to make certain reductions.First, according to Corollary 14.1.14 we may suppose that T has a weak-∗-continuous H∞-functional calculus � : H∞(D)→ B(H ). The second reduc-tion concerns the type of the spectrum of T .

160

16 Invariant subspaces for contractions 161

For an operator A ∈ B(X ) on a Banach space X , we consider the left essentialspectrum

σe(A) = {z ∈ C; im(z − A) is not closed or dim ker(z − A) = ∞}

and the right essential spectrum

σre(A) = {z ∈ C; dim(X/(z − A)X ) = ∞} .

The essential spectrum σe(A) = σe(A) ∪ σre(A) consists of all points z ∈ C

such that dim ker(z − A) = ∞ or dim(X/im(z − A)) = ∞. If wewant to provethe existence of a non-trivial invariant subspace for A, we may assume thatσ (A) = σe(A) = σre(A) (Exercise 16.2.2). Formore on the essential spectrum,see Chapter 24.In view of these remarks we may and shall suppose until the end of the proof

of Theorem 16.1.2 that T has a weak-∗ continuous H∞-functional calculus andthat σ (T ) = σe(T ).For x, y ∈ H , the linear functional

x ⊗ y : H∞(D)→ C, x ⊗ y( f ) = [�( f )x, y] ,

is weak-∗-continuous and defines an element in the predual spaceQ = L1(D)/⊥H∞(D) of H∞(D). Note that it suffices to find a point λ in D

and vectors x, y in H with

Eλ = x ⊗ y .

Indeed, in this case the space

M =∨{�( f )x ; f ∈ H∞(D) with f (λ) = 0}

is an invariant subspace for T different from the whole space H . If M = {0},then (λ− T )x = 0. So in any case Lat(T ) is non-trivial.Let L be the set of all elements L in Q with the property that, for any given

ε > 0 and any choice of vectors a1, . . . , as , b1, . . . , bs in H (with s arbitrary),there are vectors x, y in the closed unit ball of H with

(i) ‖ L − x ⊗ y‖ < ε,

(ii) ‖ x ⊗ bi‖ < ε, ‖ai ⊗ y‖ < ε (i = 1, . . . , s) .

Proposition 16.1.4 The set L is norm-closed and absolutely convex.


Proof Obviously, L is norm-closed.Let L1, L2 ∈ L and c1, c2 ∈ C with |c1| + |c2| ≤ 1 be given. We indicate

only how to find vectors x, y for L = c1L1 + c2L2 that satisfy condition (i).For any r > 0, there are x1, x2, y1, y2 in the closed unit ball of H with

‖Li − xi ⊗ yi‖ < r (i = 1, 2)

and such that x1 ⊗ y2, x2 ⊗ y1, x2 ⊗ x1, and y1 ⊗ y2 have norm less than r .Factor ci = ti si with complex numbers ti , si such that

|ti | = |si | = |ci |1/2 (i = 1, 2) .

Then x = t1x1 + t2x2, y = s1y1 + s2y2 satisfy max(‖x‖2, ‖y‖2) ≤ 1+ 2r aswell as

‖L − x ⊗ y‖ < 3r .

All that remains is to normalize x and y in a suitable way. �

By Theorem 15.1.7 and the remarks following Theorem 15.1.7, we maysuppose that � is of type C0· or C·0. The next observation shows how thesecontinuity properties of the Nagy–Foias functional calculus can be used. Recallthat a sequence (xk) in H is a weak zero sequence if limk→∞[xk, y] = 0 foreach vector y ∈ H .

Lemma 16.1.5 If� is of type C0· and (yk) is a weak zero sequence in H, then

limkx ⊗ yk = 0 (x ∈ H ) .

If � is of type C·0 and (xk) is a weak zero sequence in H, then

limkxk ⊗ y = 0 (y ∈ H ) .

Proof Fix x ∈ H . In the first case, by a normal family argument, the set

K = {�( f )x ; f ∈ H∞(D)[1]} ⊂ H

is compact. Hence, for each weak zero sequence (yk) in H ,

‖x ⊗ yk‖ = sup{|[z, yk]|; z ∈ K } k−→ 0 .

In the second case we use the same argument for � instead of �. �


A complex number z belongs to σe(T ) if and only if there is an orthonormal

sequence (xk) in H with ((z − T )xk) k−→ 0 (Exercise 16.2.3). The next resultshows why it is useful to have rich left essential spectrum.

Lemma 16.1.6 If (xk) is an orthonormal sequence with limk(z − T )xk = 0,then

limk→∞

xk ⊗ y = 0 (y ∈ H ) and limk→∞(Ez − xk ⊗ xk) = 0 .

Proof Let f ∈ H∞(D)[1], and let λ ∈ D. Then the unique function g ∈ H (D)with (z − λ)g(z) = f (z)− f (λ) for z ∈ D is boundedwith‖g‖∞ ≤ 2/(1− |λ|)(Exercise 16.2.4). For f ∈ H∞(D)[1], choose g as above, and note that

|xk ⊗ y( f )| ≤ |[�(g)(T − λ)xk, y]| + |[xk, y]| (k ∈ N) .

This proves the first part of the assertion. Similarly,

|(xk ⊗ xk − Eλ)( f )| = |[�( f )xk, xk]− f (λ)| = |[�(g)(T − λ)xk, xk]| k−→ 0

uniformly for f in the unit ball of H∞(D). �

Since Lat(T ) is non-trivial if and only if Lat(T ∗) is non-trivial, we maysuppose that � is of type C0·. Since σe(T ) is dominating in D, the previousresults imply that the closed unit ball of Q is contained in L. Now a standardprocedure, generally called the Scott Brown technique, can be used to factor allelements in Q.

Lemma 16.1.7 For L ∈ Q, x0, y0 ∈ H, and ε > 0, there are x, y ∈ H with‖L − x ⊗ y‖ < ε and ‖x − x0‖, ‖y − y0‖ ≤ ‖L − x0 ⊗ y0‖1/2 .

Proof Let d = ‖L − x0 ⊗ y0‖ > 0. By the above remarks, for each r > 0 thereare vectors a, b in the closed unit ball of H with

‖(1/d)(L − x0 ⊗ y0)− a ⊗ b‖ < r

and ‖x0 ⊗ b‖ < r , ‖a ⊗ y0‖ < r . In view of the estimate

‖L − (x0 + d1/2a)⊗ (y0 + d1/2b)‖ ≤ ‖L − x0 ⊗ y0 − d(a ⊗ b)‖ + 2d1/2r ,


it is clear that x = x0 + d1/2a and y = y0 + d1/2b will satisfy all assertions,provided that r is small enough. �

Theorem 16.1.8 For L ∈ Q, x0, y0 ∈ H, and ε > 0, there are x, y ∈ H withL = x ⊗ y and ‖x − x0‖, ‖y − y0‖ < ‖L − x0 ⊗ y0‖1/2 + ε.

Proof Define d0 = ‖L − x0 ⊗ y0‖ and choose dk > 0 (k ≥ 1) with∞∑k=1

d1/2k < ε .

Inductively, using in each step Lemma 16.1.7, we can choose sequences (xk),(yk) in H with ‖L − xk ⊗ yk‖ ≤ dk and

‖xk+1 − xk‖, ‖yk+1 − yk‖ ≤ d1/2k (k ≥ 0).

Clearly, x and y can be chosen as x = limk xk and y = limk yk . �

Applying the last result to the particular case that L is a point evaluation, weobtain the existence of non-trivial invariant subspaces for T . Thus the proof ofTheorem 16.1.2 is complete.

16.2 Exercises

1. Let M be a subset of a Banach space X . Show that �(M) is the closed unitball in X if and only if sup{|u(x)|; x ∈ M} = ‖u‖ for each u ∈ X ′.

2. Let T ∈ B(X ) (X a Banach space with dim X ≥ 2). Show that Lat(T ) isnon-trivial if σ (T ) �= σe(T ) or σ (T ) �= σre(T ).

3. Let X be a Banach space, and let T ∈ B(X ). Show that a complex numberλ belongs to σe(T ) if and only if each closed subspace M of finitecodimension in X contains a sequence (xk) of unit vectors such thatlimk→∞(λ− T )xk = 0.

4. Let V ⊂ C be a non-empty bounded open set, and let λ ∈ V . Show that themap Q : H∞(V )→ H∞(V ) associatingwith each function f ∈ H∞(V ) theunique function fλ ∈ H∞(V ) with (z − λ) fλ(z) = f (z)− f (λ) for z ∈ Vis weak-∗ continuous, maps polynomials to polynomials and satisfies theinequality ‖Q‖ ≤ 2/dist(λ, ∂V ).

5. Suppose that a compact set σ ⊂ C is dominating in D. Show that ∂D ⊂ σ .



The invariant subspace result stated asTheorem16.1.2 is fromBrown,Chevreauand Pearcy (1979). If T is a contraction such that σ (T ) is dominating in D,then σ (T ) contains the unit circle (Exercise 16.2.5). It was proved by Brown,Chevreau and Pearcy (1988) that the latter condition suffices to guarantee theexistence of a non-trivial invariant subspace. A very readable proof of this re-sult is contained in Bercovici (1990). An extension to the case of polynomiallybounded operators on Banach spaces is given in Ambrozie and Muller (2003).Results of Bercovici, Foias and Pearcy show that the H∞-functional calculus ofa completely non–unitary contraction whose essential spectrum σe(T ) is dom-inating in D allows the factorization of infinite matrices with coefficients inthe predual Q of H∞(D). It follows that contractions of this type possess anextremely rich invariant subspace lattice. For details on this class of contrac-tions, and many other results on the structure of the invariant subspace latticeof contractions obtained with the Scott Brown technique, the reader is referredto Bercovici, Foias and Pearcy (1985).

17

Invariant subspaces for subnormal operators

17.1

In 1978 Scott Brown proved the following general invariant-subspace result.

Theorem 17.1.1 (Brown) Each subnormal operator S on a Hilbert space Hof dimension at least 2 has a non-trivial invariant subspace.

The original proof of Brown (1978)was quite involved, and it was the startingpoint for a whole new area in operator theory. Later J. E. Thomson (1986) gavea simpler proof of the original Scott Brown result. We begin by explainingThomson’s proof of Theorem 17.1.1.Fix a non-zero vector x in H . To prove Theorem 17.1.1, wemay suppose that

H =∨{Skx ; k ∈ Z+} .

But then there is a positive measure µ on a compact set K in C such that Sis unitarily equivalent to Mz ∈ B(P2(µ)) (Exercise 17.2.4). Therefore we onlyconsider the case where S = Mz on P2(µ) as above. Clearly we may supposethat P2(µ) �= L2(µ). As before we denote by λ the two-dimensional Lebesguemeasure on C.

Lemma 17.1.2 Let g ∈ Lq (µ) \ {0} for some real number 1 < q < 2. Thenthere is a point w ∈ C such that

(i) g/(z − w) ∈ Lq (µ) and (ii)∫K

g(z)

z − w dµ(z) �= 0 .

Proof Fix R > 0 with |z| ≤ R for all z ∈ K . The function h defined on K × Kby

h(z,w) = |g(z)|q/|(z − w)|q (= 0 for z = w)

166

17 Invariant subspaces for subnormal operators 167

is measurable. Since, for each z in K , we have∫K

|z − w |−qdw ≤∫

DR (z)

|w |−qdw ≤∫

D2R (0)

|w |−qdw <∞ ,

Fubini’s theorem shows that the function h is (λ× µ)-integrable and that∫K

∫K

h(z,w) dµ(z)

dw <∞ ,

where the inner integral is finite for λ-almost every w in C.Let ϕ ∈ C100(C). In a similar way Fubini’s theorem shows that the function

(g/(z − w))∂ϕ(w) is µ-integrable for λ-almost every w in C and that∫C

∫K

g(z)

z − w ∂ϕ(w) dµ(z) dw = ∫

K

g(z)

∫C

∂ϕ(w)

z − w dw dµ(z)

= π

∫K

g(z)ϕ(z) dµ(z) .

The last equality follows from the complex version of Green’s formula. ByStone–Weierstrass, C100(C)|K is dense in C(K ). Hence the last integral is non-zero for some function ϕ as above.In particular, on a set of positive Lebesgue measure the integral in condition

(ii) exists and is non-zero. Hence there are complex numbers w satisfying bothconditions. ✷

In the case where g = 1, the above proof shows that the Cauchy transform

µ(w) =∫K

1

z − w dµ(z)

of µ exists λ-almost everywhere on C and is locally λ-integrable with

∂µ = −πµ (in the distributional sense).

For µ as above and 1 ≤ p <∞, we denote by P p(µ) the closure of the set ofall polynomials in L p(µ).

Theorem17.1.3 (Brennan) Let p > 2be a real numberwith P p(µ) �= L p(µ).Then there is a point w in C such that, for some positive constant c > 0,

| f (w)| ≤ c‖ f ‖p ( f ∈ C[z]) .


Proof Choose g ∈ Lq (µ) \ {0} with g⊥P p(µ), where q ∈ (1, 2) is the con-jugate exponent of p. By the previous lemma, there is a point w in C withg/(z − w) ∈ Lq (µ) and (gµ)(w) �= 0. But then from

0 =∫K

f (z)− f (w)

z − w g(z) dµ(z) ( f ∈ C[z])

we deduce that the map

f �→ f (w) = 1

(gµ)(w)

∫K

f (z)g(z)

z − w dµ(z)

on C[z] extends to a continuous linear functional on P p(µ). �

To prove the existence of an invariant subspace for S = Mz on P2(µ), wherewe require that P2(µ) �= L2(µ), it suffices to find x, y ∈ P2(µ) andw ∈ Cwith

[p(S)x, y] = p(w)

for all polynomials p. Note that the hypothesis that P2(µ) �= L2(µ) impliesthat we have P p(µ) �= L p(µ) for all p ≥ 2.

Proof of Theorem 17.1.1 Fix S and µ as above. Set p = 3 and q = 3/2.According to Theorem 17.1.3, there is a point w in C such that

C[z]→ C, p �→ p(w) ,

extends to a continuous linear functional L on P p(µ). ByHahn–Banach, there isa function h in Lq (µ)with‖h‖q = ‖L‖ such that h represents L on P p(µ). SinceP p(µ) is reflexive, there is a function r of norm 1 in P p(µ) with L(r ) = ‖L‖.Because we have equality in Holder’s inequality

‖h‖q = ‖L‖ = L(r ) =∫K

rh dµ ≤ ‖r‖p‖h‖q ,

there is a constant c > 0 (see Hewitt and Stromberg 1965, p. 190) with

|r |p = c3/2|h|q ,

or equivalently |r |2 = c|h| µ-almost everywhere.Since u = h/r (= 0 where r = 0) belongs to L2(µ) and since

[p(S)r, u]L2(µ) =∫K

pru dµ =∫K

ph dµ = p(w)

17 Invariant subspaces for subnormal operators 169

for each polynomial p, it follows that∨{p(S)r ; p ∈ C[z] with p(w) = 0} ∈ Lat(S)

is non-trivial or w ∈ σp(S). ✷

As a strengthening of the above result, we can prove (see Thomson 1986; orMartin and Putinar 1989):

Theorem 17.1.4 Each rationally cyclic subnormal operator S on a Hilbertspace of dimension at least 2 has a non-trivial hyperinvariant subspace. ✷

It is not known whether or not each subnormal operator, which is not amultiple of the identity has a non-trivial hyperinvariant subspace.A point w as in Theorem 17.1.3 is called a bounded point evaluation for

P p(µ). More recently, it was shown by Thomson (1991) that Theorem 17.1.3also holds in the case where p = 2.

17.2 Exercises

1. Let N ∈ B(H ) be a normal operator on a Hilbert space H , and let x ∈ H bea ∗-cyclic vector for N , that is, suppose that

H =∨{NkN ∗ j x ; k, j ∈ Z+} .

Show that there is a positive measure µ ∈ M(σ (N )) on the spectrum of Nand a unitary operator U : H → L2(µ) such that UN = MzU .

2. A normal extension N ∈ B(K ) of a subnormal operator S ∈ B(H ) is calledminimal if K is the only reducing subspace for N that contains H . Show thefollowing:

(a) a normal extension N is minimal if and only if K =∨{N ∗k H ; k ≥ 0};(b) if S1, S2 ∈ B(H ) are subnormal operators with minimal normal exten-sions Ni ∈ B(Ki ) (i = 1, 2), and if U ∈ B(H ) is a unitary operatorwith US1 = S2U , then there is a unitary operator V ∈ B(K1, K2) withV N1 = N2V and V |H = U ;

(c) any twominimal normal extensions of a subnormal operator are unitarilyequivalent.

3. Let N ∈ B(K ) be the minimal normal extension of a subnormal operatorS ∈ B(H ). Show that ∂σ (S) ⊂ σ (N ) ⊂ σ (S) and that a bounded connectedcomponent C of C \ σ (N ) either is contained in σ (S) or is disjoint to σ (S).


4. Let S ∈ B(H ) be a subnormal operator with a cyclic vector. Show that thereis a positive measure µ ∈ M(σ (S)) such that S is unitarily equivalent to Mz

on P2(µ) (= closure of C[z] in L2(µ)).


Using methods similar to the above, K. Yan (1988) proved that each subnormaltuple, that is, each commuting tuple S = (S1, . . . , Sn) ∈ B(H )n for which thereis a commuting tuple N = (N1, . . . , Nn) ∈ B(K )n of normal operators on alarger Hilbert space K ⊃ H with Ni H ⊂ H and Si = Ni |H , has a non-trivialjoint invariant subspace.Let T ∈ B(H ) be arbitrary, and let M ≥ 1 be a constant. A compact set K

in C containing σ (T ) is called M-spectral for T if ‖r (T )‖ ≤ M |r |K for eachrational function r with poles off K . For a subnormal operator S, its spectrum isa spectral (= 1-spectral) set (Exercise 18.2.2). It was shown by Stampfli (1980)that each operator T ∈ B(H ) for which the spectrum σ (T ) is an M-spectral setpossesses a non-trivial invariant subspace. Extensions of this result to the caseof operators on a Banach space are given in Prunaru (1996).

18

Invariant subspaces for subdecomposableoperators

18.1

Subnormal operators are examples of hyponormal operators. An operatorT ∈ B(H ) is hyponormal if ‖T ∗x‖ ≤ ‖T x‖ for all x ∈ H . The following gen-eralization of Theorem 17.1.1 was obtained in Scott Brown (1987).

Theorem18.1.1 (Brown) Each hyponormal operator with thick spectrum hasa non-trivial invariant subspace. �

Here a compact set K in C is called thick if there is a bounded open set V inC such that K is dominating in V . If a compact set K in C is not thick, thenR(K ) = C(K ) (see Brown 1987). The uniform algebra R(K ) is described inPart I, Examples 1.2(v). In particular, each subnormal operator with non-thickspectrum is normal (Exercise 18.2.3).To prove the above theorem, Scott Brown used a result of Putinar showing

that each hyponormal operator is the restriction of an operator with a C∞(C)-functional calculus (see Eschmeier and Putinar 1996, Chapter 6.4). Since oper-ators of the latter type are decomposable, each hyponormal operator is sub-decomposable, that is, it is the restriction of a decomposable operator. Thereforethe next result extends Theorem 18.1.1. For the definition and the theory ofdecomposable operators, see Part IV.

Theorem 18.1.2 (Eschmeier and Prunaru) Let S ∈ B(X ) be a subdecompos-able operator on a complex Banach space X.

(i) If σ (S) is thick, then Lat(S) is non-trivial.(ii) If σe(S) is thick, then Lat(S) is rich.

171


Here Lat(S) is called rich if it contains a sublattice which is isomorphic (as alattice) to the lattice of all closed linear subspaces of some infinite-dimensionalBanach space.One of the main difficulties in proving Theorem 18.1.2 is to extend the Scott

Brown technique to the Banach-space case. Belowwe indicate some of themainideas. First note that it suffices to prove the second part. Indeed, if σ (S) �= σe(S)and if dim(X ) > 1, then Lat(S) is non-trivial.To prove part (ii), it is enough to find a complex number z and sequences

(x j ) in X , (y j ) in X ′ such that

δ j,k p(z) = 〈p(S)x j , yk〉

for all natural numbers j, k and all polynomials p. In this case

M =∨{p(S)x j ; p ∈ C[z] and j ∈ N},N =∨{p(S)x j ; p ∈ C[z] with p(z) = 0 and j ∈ N}

are invariant for S such that N ⊂ M , (z − S)M ⊂ N , and dim(M/N ) = ∞. Ifπ : M → M/N is the quotient map, then

Lat(M/N )→ Lat(S), L �→ π−1(L),

defines a lattice embedding (Exercise 18.2.4).Since S is subdecomposable, S′ is the quotient of a decomposable operator.

More precisely, there is a decomposable operator T ∈ B(Z ) on some Banachspace Z and a surjective, continuous linear map q : Z → X ′ with qT = S′q.Let σe(S) be dominating in V . For a given compact set K in C, we denote

by ZT (K ) the local analytic subspace of the decomposable operator T ∈ B(Z )associated with K (see Chapter 22). The space

ZT (V ) =⋃{ZT (K ); K ⊂ V compact}

becomes an H (V )-module via

f z = f (T |ZT (K ))zwhenever K ⊂ V is a compact set with z ∈ ZT (K ). For x ∈ X and z ∈ ZT (V ),the map

x ⊗ z : H∞(V )→ C, f �→ 〈x, q( f z)〉,defines a weak-∗ continuous linear functional.The Scott Brown technique works best for the left essential spectrum. There-

fore the following observation is useful.

18 Invariant subspaces for subdecomposable operators 173

Lemma 18.1.3 Let S ∈ B(X ) be subdecomposable. Then

σe(S) = σre(S) = σe(S′) .

Proof The second equality follows from standard duality theory (Exercise18.2.5). If there were a point z in σe(S) \ σre(S), then

dim(X/(z − S)X ) <∞ = dim ker(z − S) .

In this case S would not even possess the single-valued extension property(see Finch 1975, proof of Corollary 11). �

By Lemma 18.1.3, the left essential spectrum of S′ is dominating in V . Letz ∈ σe(S′). Then each closed finite-codimensional subspace M ⊂ X ′ containsa sequence (yk) of unit vectors such that lim

k→∞(z − S′)yk = 0.

Question If (yk) is as above, is there a sequence (zk) in Z with q(zk) = ykand limk(z − T )zk = 0?In Eschmeier and Prunaru (1990), essential use of a lifting property of the

above type for approximate eigensequences ismade.However, simple examplesshow that the answer to the above question is in general negative. For instance,the bilateral shift

U : 2(Z)→ 2(Z), U ((xk)k∈Z) = ((xk+1)k∈Z) ,

is a decomposable lifting of the backward shift

S : 2→ 2, S((xk)k∈N) = ((xk+1)k∈N) .

Since σp(S) = D and σ (U ) = T, there is no way to solve the above liftingproblem in this concrete case.Fortunately it is possible to show that each operator that possesses a decom-

posable lifting at all has a canonical decomposable lifting which allows thelifting of approximate eigensequences in the above sense (see Eschmeier andPrunaru 1990, Proposition 1.4).As in Chapter 16, define L as the set of all elements L in the quotient

space Q = L1(V )/⊥H∞(V ) such that, for any given finitely many vectorsa1, . . . , as ∈ X , b1, . . . , bs ∈ ZT (V ) and any ε > 0, there are vectors x ∈ Xand z ∈ ZT (V ) with ‖x‖, ‖qz‖ ≤ 1 and

(i) ‖L − x ⊗ z‖ < ε ,

(ii) ‖x ⊗ bi‖ < ε, ‖ai ⊗ z‖ < ε (i = 1, . . . , s) .


It suffices to find a positive constant c such that

{L ∈ Q; ‖L‖ ≤ c} ⊂ L .

Then a standard factorization method due to Apostol, Bercovici, Foias, Pearcy(seeBercovici, Foias andPearcy1988) allowsone to factor infinitematriceswithcoefficients in Q (see Eschmeier and Putinar 1996, Chapter 6.5) in the followingsense. For each matrix L = (L jk) ∈ M(N, Q), we can prove the existence ofvectors xN in XN , zN in ZT (V )N (N ∈ N) such that the limits

x( j) = limN→∞

xN ( j) ∈ X, y( j) = limN→∞

q zN ( j) ∈ X ′

exist for each j ∈ N and such that, for all j, k ∈ N, we have

L jk = limN→∞

xN ( j)⊗ zN (k) .

Here xN ( j) and zN (k) are the components of the N -tuples xN and zN . Byapplying this result to the special case of a diagonal matrix of the special formL = (δ j,kEλ) j,k∈N with λ in V arbitrary, one obtains the relations

δ j,k p(λ) = limN→∞

〈xN ( j), q p(T )zN (k)〉 = 〈p(S)x( j), y(k)〉

for all polynomials p and all natural numbers j, k.Since L is norm-closed, it suffices to find a constant d > 0 such that, for

each absolutely convex combination L =∑ri=1 ciEλi of point evaluations at

points λ1, . . . , λr in σe(S′) ∩ V , the factorization problems (i) and (ii) canbe solved with factors x ∈ X and z ∈ ZT (V ) bounded by ‖x‖, ‖qz‖ ≤ d. Astraightforward decomposition into real and imaginary, positive and negative,parts reduces the problem to the case of convex combinations. To indicatea possible solution of the remaining problem we need two results from thegeometry of normed spaces.Denote by F(X ) the set of all finite-dimensional subspaces of X and by

Cof(X ) the set of all closed, finite-codimensional subspaces.

Lemma 18.1.4 Let δ > 0. For each space M in F(X ), there is a space N inCof(X ) with ‖m + n‖ ≥ (1− δ)‖m‖ for m ∈ M and n ∈ N. �

Lemma 18.1.5 (Zenger) Let x1, . . . , xr ∈ X be linearly independent, and letc1, . . . , cr be non-negative real numbers with

∑i ci = 1. Then there are a linear

combination∑r

i=1 µi xi in the closed unit ball of X and a functional y in theclosed unit ball of X ′ with 〈µi xi , y〉 = ci (i = 1, . . . , r ). �


Lemma 18.1.4 is proved in Singer (1981, Lemma III.1.1). A proof of thesecond result can be found in Bonsall and Duncan (1973, p. 20).

Exercise 18.1.6 Let I be an index set. For i ∈ I , let Pi be a property that a unitvector in X might have. Suppose that, for each i ∈ I , each space N ∈ Cof(X )contains unit vectors with property Pi . Then, for each M in F(X ) and eachfinite set of indices i1, . . . , ir , there are unit vectors x1, . . . , xr such that xν hasproperty Piν and such that, with L = lin{x1, . . . , xr }, we have

(i) the projection from M + L onto L along M has norm less than 3,(ii) max

1≤i≤r|αi | ≤ 3

∥∥∑ri=1 αi xi

∥∥ for all α1, . . . , αr ∈ C.

By Zenger’s lemma, for c1, . . . , cr ≥ 0 with∑r

i=1 ci = 1, there are a linearcombination

∑ri=1 µi xi in the closed unit ball of X and a functional y on X

with norm less than 3 such that 〈µi xi , y〉 = ci for all i and such that y|M = 0.Fix δ > 0. Let K ⊂ V be compact. For λ in K , we say that a unit vector

y ∈ X ′ has property Pλ if there are z ∈ Z and w ∈ ZT (K ) with

y = qz, ‖(λ− T )z‖ < δ, ‖z − w‖ < δ.

Sincewe are supposing that approximate eigensequences for S′ can be lifted andsince σ (T/ZT (K )) ∩ int(K ) = Ø, it follows that, for each λ in σe(S′) ∩ int(K ),each space N ∈ Cof(X ′) contains unit vectors with property Pλ.Hence, for λ1, . . . , λr in σe(S′) ∩ int(K ), M ∈ F(X ′) and c1, . . . , cr ≥ 0

adding up to 1, there are unit vectors y1, . . . , yr in X ′ and associated vectors zi ,wi as above such that there is a linear combination $µi yi in the unit ball of X ′

and a vector x in ⊥M of norm less than 3 with |µi | ≤ 3 and

〈x, µi yi 〉 = ci (i = 1, . . . , r ) .

For f ∈ H∞(V ) and λ ∈ V , let fλ ∈ H∞(V ) be the function with

(z − λ) fλ(z) = f (z)− f (λ)

for z ∈ V . Define

z =r∑i=1

µiwi .


Then ‖qz‖ ≤ 1+ 3r‖q‖δ and, when δ approaches zero, the right-hand side of∣∣∣∣∣x ⊗ z( f )− r∑i=1

ci f (λi )

∣∣∣∣∣ ≤∣∣∣∣∣ r∑i=1

µi 〈x, q fλi (T |ZT (K ))(T − λi )wi 〉∣∣∣∣∣

+∣∣∣∣∣ r∑i=1

µi 〈x, q f (λi )(wi − zi )〉∣∣∣∣∣

tends to zero uniformly for f in the unit ball of H∞(V ). Choosing M and Nappropriately one can achieve that x and z also satisfy the remaining conditionsin the following result.

Theorem 18.1.7 Let c1, . . . , cr ≥ 0 with∑r

i=1 ci = 1, and let

λ1, . . . , λr ∈ σe(S′) ∩ V .

If a1, . . . , as ∈ X, b1, . . . , bs ∈ ZT (V ) and ε > 0 are arbitrary, then there arevectors x ∈ X, z ∈ ZT (V ) with ‖x‖ ≤ 3, with ‖qz‖ ≤ 2, and with(i) ‖∑r

i=1 ciEλi − x ⊗ z‖ < ε,(ii) ‖x ⊗ bi‖ < ε, ‖ai ⊗ z‖ < ε (i = 1, . . . , s). �

Thus we have summarized the main steps in a possible proof of Theorem18.1.2. A more detailed proof, including also the case of quotients of decom-posable operators, can be found in Eschmeier and Prunaru (1990).

18.2 Exercises

1. Show that each subnormal operator is hyponormal.2. Let S ∈ B(H ) be subnormal. Show that, for each function f ∈ H (σ (S)),‖ f (S)‖ ≤ | f |σ (S) (using Exercise 17.2.3, as well as the spectral radius andthe analytic spectral mapping theorem, one can even show that equalityholds here). Conclude that S possesses a contractive functional calculus& : R(σ (S))→ B(H ).

3. Let S ∈ B(H ) be subnormalwith R(σ (S)) = C(σ (S)). Show that S is normalHint: Exercise 15.2.3.

4. Let T ∈ B(X ) be a bounded operator on a Banach space. Suppose that, forsome point z ∈ C, there are sequences (x j ) in X , (yk) in X ′ with

δ j,k p(z) = 〈p(T )x j , yk〉 ( j, k ∈ Z+) .

Show that Lat(T ) is rich.


5. For T ∈ B(X ) (X a Banach space) show that σre(T ) = σe(T ′).6. Solve Exercise 18.1.6 as formulated in the main text.


The first applications of the Scott Brown technique to Banach-space operatorswere given by Apostol (1981). Theorem 18.1.2 is from Eschmeier and Prunaru(1990). Special cases of Theorem 18.1.2 for operators acting on quotients ofclosed subspaces of p (1 < p <∞) have been obtained by Albrecht andChevreau (1987). A multivariable version of Theorem 18.1.2 is contained inEschmeier andPutinar (1996).Usingdualitymethodsone canprove an analogueof Theorem 18.1.2 for quotients of decomposable operators (see Eschmeierand Prunaru 1990, Theorem 2.1). Intrinsic characterizations of subdecompos-able operators and of quotients of decomposable operators have been givenin Albrecht and Eschmeier (1997). For instance, a continuous linear operatorS ∈ B(X ) on a Banach space X is subdecomposable if and only if it satisfiesBishop’s property (β), that is, all multiplication operators

H (U, X )→ H (U, X ), f �→ (z − S) f (U ⊂ C open),

are continuous with closed range. For a discussion of property (β), seeChapter 23. A proof of the first part of Theorem 18.1.2 which does not makeuse of canonical extensions and liftings was given more recently in Eschmeierand Prunaru (2002).

19

Reflexivity of operator algebras

19.1

A subalgebra A ⊂ B(H ) is called reflexive if A = Alg Lat(A), where the spaceon the right-hand side consists of all operatorsC ∈ B(H )withLat(A) ⊂ Lat(C).A single operator T ∈ B(H ), or more generally, an n-tuple T = (T1, . . . , Tn)inB(H )n is called reflexive if the smallestWOT-closed subalgebraWT ofB(H )containing T1, . . . , Tn and the identity operator is reflexive. Equivalently, onecan ask that

Alg Lat(T ) =WT ,

where Alg Lat(T ) consists of all operators C ∈ B(H ) with Lat(T ) ⊂ Lat(C).Sarason proved in 1966 that each normal operator and each analytic Toeplitz

operator on H 2(T) is reflexive. In 1971 it was shown by Deddens that allisometries are reflexive. Since all these operators are subnormal, the followingresult of Olin and Thomson (1980) contains the above observations as specialcases.

Theorem 19.1.1 (Olin and Thomson) Every subnormal operator is reflexive.

In the following we indicate some of the main steps leading to a proof of theOlin and Thomson result. For simplicity, we shall restrict ourselves to the caseof pure subnormal operators.Let S ∈ B(H ) be subnormal. It is easy to see that the set

V =⋃{U ⊂ C bounded open; σ (S) is dominating in U }

is the largest bounded open set inC in which σ (S) is dominating. For a compactset K inC, let R(K ) and A(K ) be as above and as in Chapter 1. As usual R(K )

178

19 Reflexivity of operator algebras 179

is called a Dirichlet algebra if

{Re( f |∂K ); f ∈ R(K )} ⊂ CR(∂K )

is uniformly dense (see Conway 1991, Chapter V.14).If V is empty, then one can show (Brown 1987) (this result is non-trivial!) that

R(σ (S)) = C(σ (S)). In this case, the operator S is normal (Exercise 18.2.3). Ifthe set V is non-empty, then it possesses very nice properties.

Theorem 19.1.2 Let σ ⊂ C be compact. Suppose that the largest boundedopen set V in C in which σ (S) is dominating is non-empty. Then:

(i) ∂V = ∂V and V = int(V ) ⊂ σ ;(ii) R(V ) is Dirichlet, R(V ) = A(V ) = A(V ), and the components of V are

simply connected;(iii) R(V ) = A(V ) is pointwise boundedly dense in H∞(V );(iv) the polynomials are weak-∗ dense in H∞(V ). �

A proof of Theorem 19.1.2 is given in the appendix to Part III.In the following let S ∈ B(H ) be subnormal such that the largest bounded

open set V in C in which σ (S) is dominating is non-empty.

Lemma 19.1.3 If σ (S) is not contained in V , then S has a non-trivial normalpart.

Proof Let D be a closed disc with D ∩ V = Ø and centre λ ∈ σ (S). ThenD ∩ σ (S) is dominating in no open set anymore. Thus by the above remarks itfollows that R(D ∩ σ (S)) = C(D ∩ σ (S)). By a result of Clancey and Putnam(1972), the operator S has a non-trivial normal part. �

The results ofChapter 15 can be used to obtain an H∞(V )-functional calculusfor pure subnormal operators.

Theorem 19.1.4 Let S ∈ B(H ) be a pure subnormal operator, and let V bethe largest bounded open set in C in which σ (S) is dominating. Then S has aunique contractive weak-∗ continuous functional calculus

� : H∞(V )→ B(H ) .

This functional calculus is isometric and of type C·0.


Proof The uniqueness follows from Theorem 19.1.2(iii).Since σ (S) ⊂ V , the subnormal operator S possesses a contractive functional

calculus over R(V ) = A(V ). By Corollary 15.1.8, this functional calculus is oftype C·0. By Theorem 19.1.2(iii) and the remarks following Theorem 15.1.7, itextends to a contractive H∞(V )-functional calculus of type C·0. To see that �is isometric, note that

f (V ∩ σ (S)) ⊂ σ (�( f ))

for each f ∈ H∞(V ). This spectral inclusion follows from the identity

(S − λ)�( f − f (λ)

z − λ)= �( f )− f (λ) ,

which is valid for all f ∈ H∞(V ) and λ ∈ V . �

Since � is isometric and weak-∗ continuous, its range is weak-∗ closed. ByTheorem 19.1.2(iv), the range of � coincides with the smallest unital, weak-∗closed subalgebra AS of B(H ) containing S. Note that� : H∞(V )→ AS is anisometric isomorphism and a weak-∗ homeomorphism if H∞(V ) and AS areregarded as the dual spaces of Q and C1(H )/⊥AS , respectively. In this situation,� is called a dual algebra isomorphism.Write V as the disjoint union of its connected components

V =⋃{Vk ; 0 ≤ k < N } (N ∈ Z+ ∪ {∞}) .

Let ek ∈ H∞(V ) be the characteristic function of Vk . The operators Pk = �(ek)are orthogonal projections such that the spaces Hk = PkH are pairwise orthog-onal reducing subspaces for im(�) = AS . Furthermore,

SOT−∑k

Pk = 1H .

For f ∈ H∞(Vk), let f be the trivial extension of f onto all of V . Then�k : H

∞(Vk)→ B(Hk), f �→ �( f )|Hk ,

is an isometric weak-∗ continuous functional calculus for Sk = S|Hk of typeC·0. Since the polynomials are weak-∗ dense in H∞(Vk), the algebra homomor-phisms �k : H∞(Vk)→ ASk are dual algebra isomorphisms again. Note that

AS = {C ∈ B(H ); Hk ∈ Lat(C) and C |Hk ∈ ASk for all k}

(Exercise 19.2.1). We abbreviate the space on the right by⊕0≤k<N

ASk .


Fix k and a conformal map ϕ : Vk → D. The composition

�(k) : H∞(D) −→ H∞(Vk)�k−→ ASk ,

where the first map is given by substituting the argument by ϕ, is a dual algebraisomorphism (of type C·0). A standard argument (Exercise 19.2.3) shows thateach operator in ASk is subnormal again. Hence �

(k) is an isometric weak-∗continuous H∞-functional calculus of the subnormal operator Tk = �k(ϕ). Inparticular, we have ATk = ASk .Thus we have proved the following decomposition theorem.

Theorem 19.1.5 (Conway and Olin) Let S be a pure subnormal operator.Then there is an orthogonal decomposition H =⊕0≤k<N Hk into countablymany reducing spaces for S such that, with Sk = S|Hk,

AS =⊕0≤k<N

ASk

and such that H∞(D) ∼= ASk as dual algebras for all k. ✷

To prove that a pure subnormal operator is reflexive it therefore suffices toconsider the following special case.

Theorem 19.1.6 Let T ∈ B(H ) be subnormal with a weak-∗ continuousisometric functional calculus � : H∞(D)→ B(H ). Then T is reflexive andAlg Lat(T ) = AT . ✷

Indeed, suppose that Theorem 19.1.6 holds and thatC ∈ Alg Lat(S) for somepure subnormal operator S. Then, with the notations fixed above,

C |Hk ∈ Alg Lat(Sk) = Alg Lat(Tk) = ATk = ASk

for each k. Hence C ∈ AS , and the reflexivity of S is proved.Before we indicate some of the ideas leading to a proof of Theorem 19.1.6,

we consider an elementary, but typical, example.

Example 19.1.7 Let T = Mz on H = H 2(D). Since H is a Hilbert space ofanalytic functions, it possesses a reproducing kernel. More precisely, there is aconjugate analytic function k : D → H with

[ f, k(λ)] = f (λ) ( f ∈ H, λ ∈ D)


such that H is the closed linear span of all the kλ. It follows that

(λ− T )∗k(λ) ≡ 0 .

If C ∈ Alg Lat(T ), then C∗ ∈ Alg Lat(T ∗) acts on the eigenspaces of T ∗ as ascalar multiple of the identity (Exercise 19.2.4). Hence C∗k(λ) = g(λ)k(λ) forλ ∈ D with a bounded function g. Since

[C f, k(λ)] = g(λ)[ f, k(λ)] = (g f )(λ) ,

we conclude thatC = �(g) ∈ AT , where� is the H∞-functional calculus of T .The above example motivates us to give the next definition.

Definition 19.1.8 Let T ∈ B(H ). A space M ∈ Lat(T ) is an analytic (or D-analytic) invariant subspace for T if there is a non-zero conjugate analyticfunction k : D → M with (λ− T |M)∗k(λ) ≡ 0.

In the situation of Theorem 19.1.6 one cannot expect the whole space H tobe an analytic invariant subspace for T (as in Example 19.1.7). Neverthelessthe proof of Theorem 19.1.6 depends on the observation that there is a denseset of vectors x in H such that the spaces

Mx =∨{T kx ; k ≥ 0}

are analytic invariant subspaces for T . As before, for x, y ∈ H , regard

x ⊗ y : H∞(D)→ C, x ⊗ y( f ) = [�( f )x, y] ,

as an element in the predual Q of H∞(D). For k ∈ Z+, define functionals E (k)in Q by

〈E (k), f 〉 = f (k)(0)/k! ( f ∈ H∞(D)) .

Suppose that x, yk are vectors in H such that x ⊗ yk = E (k) for all k ∈ Z+ andsuch that k(λ) =∑ j y jλ

jconverges onD. Then Eλ = x ⊗ k(λ) for λ ∈ D, and

the function

D → Mx , λ �→ PMx k(λ) ,

turns Mx into an analytic invariant subspace for T .Thus to prove that there is a dense set of vectors x generating an analytic

invariant subspace for T , it suffices to prove the second part of the next result.


Theorem 19.1.9 Let T be as in Theorem 19.1.6.

(i) There is a constant r > 0 such that, for each L ∈ Q and any given vectorsa, b ∈ H, there are x, y ∈ H with L = x ⊗ y and

‖x − a‖ ≤ r‖L − a ⊗ b‖1/2, ‖y‖ ≤ r (‖L − a ⊗ b‖1/2 + ‖b‖) .

(ii) For each ε > 0, there is a constant c = c(ε) > 0 such that, for each se-quence (Lk)k≥1 in Q and each vector a ∈ H, there are vectors x, yk ∈ Hwith ‖x − a‖ < ε and such that, for all k ∈ N,

Lk = x ⊗ yk, ‖yk‖ ≤ ck12‖Lk‖ . ✷

For a proof of Theorem 19.1.9 that even works in the multivariable case (forsubnormal n-tuples with isometric H∞-functional calculus over the Euclideanunit ball in Cn), the reader is referred to Eschmeier (1999, Theorem 1.10 andTheorem 2.3).Let Mx be an analytic invariant subspace for T via a conjugate analytic

function k : D → Mx . Then, as in Example 19.1.7, it follows (Exercise 19.2.5)that, for each operator C ∈ Alg Lat(T ), there is a function g = gC,x in H∞(D)with

(C |Mx )∗k(λ) = g(λ)k(λ)

for λ ∈ D. Since by Theorem 19.1.9 the set

C = {x ∈ H ; Mx is a D–analytic invariant subspace for T }

is dense in H , to complete the proof of Theorem 19.1.6, it suffices to showthat, for each operator C ∈ Alg Lat(T ) and each vector x ∈ C, we have

�(gC,x )|Mx = C |Mx

and that all the functions gC,x (x ∈ C) coincide.

For a complete proof of Theorem 19.1.6 along these lines, we refer the readerto Section 3 in Eschmeier (1999, Theorem 3.6 and Theorem 3.7).

19.2 Exercises

Throughout Exercises 19.2.1 to 19.2.3, let S ∈ B(H ) be a subnormal operatorwith minimal normal extension N ∈ B(K ).


1. Let� : H∞(V )→ AS be a dual algebra isomorphism. For k = 0, . . . ,N−1,denote by ek ∈ H∞(V ) the characteristic functions of the connected com-ponents of V . Set Hk = im�(ek), Sk = S|Hk . Show:(a) AS = {A ∈ B(H ); Hk ∈ Lat(A) and A|Hk ∈ ASk for all k};(b) AS →

⊕∞

ASk , A �→ (A|Hk)k , is an isometric isomorphism;(c) if L : AS → C is weak-∗ continuous, then ‖L‖ =∑k ‖L|ASk‖.

2. Suppose that f : σ (N )→ C is bounded andmeasurable with f (N )H ⊂ H .Show that σ ( f (N )) is equal to the spectrum of theminimal normal extensionof f (N )|H . Deduce that ‖ f (N )‖ = ‖ f (N )|H‖.

3. Let µ be a scalar spectral measure for N , and let � : L∞(µ)→ W ∗(N ) bethe associated isomorphism of von Neumann algebras (= weak-∗ continuous∗-isomorphism). Show that � induces a dual algebra isomorphism

& : P∞(µ)→ AS, f �→ �( f )|H .

Hint: use Exercise 19.2.2 to show that & is isometric.4. Let T ∈ B(H ) and let C ∈ Alg Lat(T ). Show that C restricted to theeigenspaces of T acts as a scalar multiple of the identity operator.

5. Let M be an analytic invariant subspace for T via a conjugate analytic func-tion k : D → M . Show that, for each C ∈ Alg Lat(T ), there is a uniquebounded analytic function g : D → C with (C |M)∗k(λ) = g(λ)k(λ) forλ ∈ D.Aweak-∗ closed subalgebraA ⊂ B(H ) is said to possessproperty (A1(r )),

where r ≥ 1, if, for any weak-∗ continuous linear functional L : A → C

and any s > r , there are x, y ∈ H with L = [x ⊗ y] in C1(H )/⊥A and with‖x‖ ‖y‖ ≤ s‖L‖.

6. Let N ∈ B(H ) be normal. Show that AN has property (A1(1)).7. Let T ∈ B(H ) be such thatAT has property (A1(r )). Show thatAT = WT andthat the WOT-topology and the weak-∗ topology coincide on this algebra.

8. Show that the dual algebra generated by a pure subnormal operator hasproperty (A1(r )) for some r ≥ 1. Hint: use Theorem 19.1.5 to reduce theassertion to Theorem 19.1.9(i).

9. Suppose that the reflexivity of normal and pure subnormal operators hasbeen shown. Prove that each subnormal operator is reflexive.


Let T be a contraction mapping with a weak-∗ continuous functional calculusmap � : H∞(D)→ B(H ). If σ (T ) is dominating in D, then � is isometric


(cf. the proof of Theorem 19.1.4). Brown and Chevreau (1988) proved that eachcontraction T with an isometric weak-∗ continuous H∞-functional calculus isreflexive. Using this result Conway and Dudziak (1990) (and independently,Prunaru) proved that each operator T ∈ B(H )whose spectrumσ (T ) is a spectralset for T is reflexive. In the multivariable case it was shown by Bercovici(1994) that each commuting family of isometries is reflexive. Theorem 19.1.6was generalized in Eschmeier (1999) to subnormal n-tuples with a weak-∗continuous H∞-functional calculus over the unit ball in Cn .

20

Invariant subspaces for commuting contractions

20.1

A tuple T = (T1, . . . , Tn) ∈ B(H )n of commuting bounded operators on aHilbert space H is called a spherical contraction if

∑ni=1 T

∗i Ti ≤ 1. We know

from Chapter 16 that a contraction T ∈ B(H ) with dominating spectrum in theopen unit disc has non-trivial invariant subspaces. Therefore it seems naturalto ask whether, for every spherical contraction T ∈ B(H )n with dominatingjoint spectrum in the open Euclidean unit ball B = {z ∈ Cn; |z| < 1}, the jointinvariant subspace lattice of T

Lat(T ) =n⋂i=1Lat(Ti )

contains non-trivial spaces.In Chapter 14, we used the Sz.-Nagy dilation theorem to prove von

Neumann’s inequality for contractions T ∈ B(H ). Since the polynomials areuniformly dense in the ball algebra A(B) = { f ∈ C(B); f |B is analytic}, vonNeumann’s inequality on the unit ball, that is, the inequality

‖p(T )‖ ≤ |p|B (p ∈ C[z1, . . . , zn]) ,

is equivalent to the existence of a contractive ball algebra functional calculus forT . However, it is well known (see Drury 1978; or Arveson 1998) that, for eachn > 1, there are spherical contractions T ∈ B(H )n for which the polynomialfunctional calculus is unbounded, that is, for which

sup{‖p(T )‖; p ∈ C[z1, . . . , zn] with |p|B ≤ 1} = ∞ .

Our aim in the following is to indicate that some of the invariant-subspaceresults valid for contractions with rich spectrum in the unit disc possess

186

20 Invariant subspaces for commuting contractions 187

multivariable analogues, at least for spherical contractions that admit normalboundary dilations.

Definition 20.1.1 Let n ∈ N, let H be a Hilbert space, and let T ∈ B(H )n bea commuting tuple.

(i) A spherical unitary on H is a commuting tuple U ∈ B(H )n of normaloperators such that

∑ni=1U

∗i Ui = 1.

(ii) A spherical dilation of T is a spherical unitary U ∈ B(K )n on a largerHilbert space K containing H such that

T k = P Uk |H (k ∈ (Z+)n) ,where P is the orthogonal projection from K onto H and T k = T k11 · · · T knn .

(iii) A spherical dilation U ∈ B(K )n of T is minimal if the only reducingsubspace for U containing H is the space K itself.

A commuting tuple T = (T1, . . . , Tn) ∈ B(H )n is called a spherical isometryif∑n

i=1 T∗i Ti = 1. Let T ∈ B(H )n be a spherical contraction with a spherical

dilationU ∈ B(K )n . By the spectral theorem for commuting normal operators,the polynomial functional calculus of U extends to a unique C∗-algebra ho-momorphism π : C(∂B)→ B(K ). Thus T satisfies von Neumann’s inequalityand possesses a contractive ball algebra functional calculus& : A(B)→ B(H ).We call &, or also T , absolutely continuous if & extends to a weak-∗ contin-uous algebra homomorphism � : H∞(B)→ B(H ). If one wants to show thatT has non-trivial joint invariant subspaces, then one may suppose that T iscompletely non-unitary, that is, possesses no non-zero reducing subspace Msuch that T |M is a spherical unitary. In this case, T is absolutely continuous.Indeed, the following decomposition theorem can be proved.

Theorem 20.1.2 Let & : A(B)→ B(H ) be a contractive algebra homomor-phism. Then there are contractive algebra homomorphisms

&a, &s : A(B)→ B(H )

with & = &a +&s such that &a is absolutely continuous, &s extends to aC∗-algebra homomorphism �s : C(∂B)→ B(H ), and

&a( f )&s(g) = 0 = &s(g)&a( f ) ( f, g ∈ A(B)) . �

This theorem can be proved in the same way as Theorem 14.1.12. It sufficesto replace the Lebesgue decomposition theorem and the F. and M. Riesztheorem by suitable multivariable generalizations. To be more precise, letM(∂B) be the space of all regular, complex Borel measures on ∂B. Denote


by M0 its weak-∗ compact, convex subset consisting of all probability mea-sures ρ ∈ M(∂B) with

f (0) =∫∂B

f dρ ( f ∈ A(B)) .

Definition 20.1.3 Let µ ∈ M(∂B). Then µ is called absolutely continuous ifµ, ρ for some measure ρ ∈ M0. The measure µ is called singular if it isconcentrated on an Fσ -set N ⊂ ∂B with ρ(N ) = 0 for all ρ ∈ M0.

By the Glicksberg–Konig–Seever decomposition theorem (for example, seeRudin 1980, Theorem 9.4.4), every measure µ ∈ M(∂B) has a unique decom-positionµ = µa + µs into an absolutely continuous partµa and a singular partµs . By Henkin’s theorem (Exercise 20.2.1) and the Cole–Range theorem (seeRudin 1980, Theorem 9.6.1), a measureµ ∈ M(∂B) is absolutely continuous ifand only if it is a Henkin measure in the sense that, for everyMontel sequence( fk) in A(B), we have

limk→∞

∫fk dµ = 0 .

Since every measure µ ∈ A(B)⊥ is obviously a Henkin measure, we obtainin particular that the only singular measure µ ∈ A(B)⊥ is the trivial measureµ = 0. Now, to prove Theorem 20.1.2, it suffices to apply the Glicksberg–Konig–Seever theorem to an arbitrary family of representing measures of &,and then to argue exactly as in the one-variable case.As a useful byproduct of the above proof of Theorem 20.1.2, one obtains:

Corollary 20.1.4 Let U ∈ B(K )n be a minimal spherical dilation of a spher-ical contraction T ∈ B(H )n. If T is absolutely continuous, then so is U. �

Let T ∈ B(H )n be a commuting tuple with a continuous ball algebra func-tional calculus & : A(B)→ B(H ). We say that T , or also &, is of type C0· if&( fk)

SOT−→ 0 for each Montel sequence ( fk) in A(B). We say that T , or &, isof type C·0 if & : A(B)→ B(H ) defined by

&( f ) = &( f )∗ ( f (z) = f (z))

is of type C0·. Exactly as in the one-variable case, it is possible to prove that Thas always non-trivial invariant subspaces if T is neither C0. nor C·0. A proofcan be based on the ball analogue of Theorem 15.1.3.


Theorem 20.1.5 For each function g ∈ C(B), the operator

Sg : A(B)→ C(B)/A(B), f �→ [g f ] ,

is compact. �

Theorem 20.1.5 follows from well-known compactness properties of suit-able Hankel-type operators (see Rudin 1980, Theorem 6.5.4). A proof valid inmore general situations can be found in Didas (2002; see also Cole andGamelin1982). A commuting tuple S ∈ B(H )n is called subnormal if there is a commut-ing tuple N ∈ B(K )n of normal operators on a Hilbert space K ⊃ H such thatH ∈ Lat(N ) and S = N |H . Exactly as in the one-variable case (Chapter 15),one can use Theorem 20.1.5 to prove the following corollaries.

Corollary 20.1.6 Let& : A(B)→ B(H ) be a continuous algebra homomor-phism. Then& is not of type C·0 if and only if there is a non-zero bounded linearmap j : C(∂B)→ H with &( f ) j = jM( f |∂B) for all f ∈ A(B). �

Corollary 20.1.7 Let& : A(B)→ B(H ) be a continuous algebra homomor-phism, and let T = (&(z1), . . . , &(zn)). If T is neither of type C0· nor oftype C·0, then either all components of T are scalar multiples of the identityoperator, or the hyperinvariant subspace lattice Hyp(T ) = Lat((T )′) of T isnon-trivial. �

Corollary 20.1.8 Let S ∈ B(H )n be a subnormal tuple that possesses a con-tinuous A(B)-functional calculus. Then S is completely non-unitary if and onlyif S is of type C·0. �

Using the above results, it is possible to prove an invariant-subspace resultfor spherical contractions that contains the corresponding one-variable result ofBrown, Chevreau, and Pearcy (Theorem 16.1.2) as a special case. To formulateit, we need a suitable notion of joint spectrum. Let T ∈ B(H )n be a commutingtuple on H . The Harte spectrum σH(T ) of T consists of those points λ ∈ Cn

for which the map

H → Hn, x �→ ((λi − Ti )x)ni=1,

is not bounded below or for which the map

Hn → H, (xi )ni=1 �→

n∑i=1(λi − Ti )xi ,


is not surjective. The Harte spectrum is a compact subset of Cn which, forn = 1, reduces to the usual spectrum of an operator T ∈ B(H ). For this andmore results on joint spectra, see Eschmeier and Putinar (1996).

Theorem20.1.9 Let T ∈ B(H )n be a commuting tuple that possesses a spher-ical dilation. Suppose that the Harte spectrum σH(T ) of T is dominating in B.Then T has non-trivial joint invariant subspaces. �

Here a set σ ⊂ Cn is, of course, called dominating in B if | f |B = | f |B∩σ forall f ∈ H∞(B). The results preceding Theorem 20.1.9 can be used to reduce theassertion to the case where T possesses a weak-∗ continuous H∞(B)-functionalcalculus of type C·0. Then an application of the Scott Brown factorization tech-nique leads to aproof ofTheorem20.1.9.Anadditional complication arises fromthe fact that one is not allowed to assume that the Harte spectrum of T coincideswith the left essential spectrum of T , as we did in the one-dimensional case.To overcome this difficulty, it can be shown that, in the above situation, the

tuple T has an absolutely continuous co-isometric extension, that is, there isan absolutely continuous spherical isometry V ∈ B(K )n on a Hilbert space Kcontaining H such that H ∈ Lat(V ∗) and T = V ∗|H . The tuple V has a Wold-typedecompositionV = S ⊕ R∗ into a completely non-unitary subnormal tupleS and a spherical unitary R∗. By Corollary 20.1.8, the adjoint S∗ is of type C0·,andT is the restriction of S∗ ⊕ R.Nowa standard applicationof theScottBrownfactorization technique, using the C·0-property of T and the C0·-property of S∗,allows one to complete the proof of Theorem 20.1.9. For details, we refer thereader to Eschmeier (1997).An extension of the methods explained in Chapter 19 was used in Eschmeier

(1999) to prove a reflexivity result for algebras of subnormal operators on theunit ball which extends Theorem 19.1.6.

Theorem 20.1.10 Let T ∈ B(H )n be subnormal spherical contraction witha weak-∗ continuous isometric functional calculus� : H∞(B)→ B(H ). ThenT is reflexive and AlgLat(T ) = AT . �

It is an open question whether the invariant-subspace result formulated inTheorem 20.1.9 remains true when the Harte spectrum σH(T ) is replaced by theTaylor spectrum of T . The Taylor spectrum always contains theHarte spectrum,and it is, in many respects, the more natural joint spectrum (see Eschmeier andPutinar 1996). The answer to this question is positive in dimension n = 2 (seeDidas 2002).


20.2 Exercises

1. Use Theorem 20.1.5 and the ball analogue of Corollary 15.1.4 to prove that,if µ ∈ M(∂B) is a Henkin measure, then every measure ν ∈ M(∂B) withν , µ is a Henkin measure. This result is usually referred to as Henkin’stheorem.

2. Complete the proof of Theorem 20.1.2 as outlined in this chapter.3. Let & : A(B)→ B(H ) be a contractive algebra homomorphism. Show thatthere is at most one pair (&1, &2) consisting of contractive algebra homo-morphisms &i : A(B)→ B(H ) with & = &1 +&2 such that &1 has a rep-resenting family of measures that are absolutely continuous, and &2 has arepresenting family of measures that are singular on ∂B.

4. Let & : A(B)→ B(H ) be a continuous algebra homomorphism. Show thatthere is an extension of & to a weak-∗ continuous algebra homomorphism� : H∞(B)→ B(H ) if and only if& has a representing family of measureson ∂B that are absolutely continuous, or equivalently, if each representingfamily of measures for& on ∂B consists of absolutely continuous measures.

5. Use Rudin (1980, Theorem 6.5.4) to prove Theorem 20.1.5.6. Let T ∈ B(H )n be a spherical contraction. Show that there is a unique or-thogonal decomposition H = H0 ⊕ H1 with reducing subspaces H0, H1 forT such that T |H0 is completely non-unitary and T |H1 is a spherical unitary.


Henkin’s theorem and the Cole–Range theorem imply that on ∂B the set ofall Henkin measures coincides with the band of measures generated by theset M0 of all measures that represent the point evaluation at 0 on A(B). Al-though Henkin’s theorem fails on the unit polydisc Dn , one can still show (seeEschmeier 2001a) that the band of measures generated by the set M0(Tn) ofall measures on Tn that represent the point evaluation at 0 on A(Dn) is thelargest band in M(Tn) consisting entirely of Henkin measures. This result isused in Eschmeier (2001a, 2001b) to show that Theorem 20.1.9 remains trueif the open unit ball is replaced by the open unit polydisc, and to prove ananalogue of Theorem 20.1.10 on Dn . Since by Ando’s dilation theorem everypair T = (T1, T2) of commuting contractions has a dilation to a commutingpair U = (U1,U2) of unitary operators, we obtain in particular that every pairT = (T1, T2) of commuting contractions with dominating Harte spectrum inD2 has a non-trivial joint invariant subspace. It is an open question whetherthe Harte spectrum can be replaced by the Taylor spectrum in this result. Thisquestion has a positive answer, even in arbitrary dimension n, if the tuple T is


supposed to consist of doubly commuting contractions (see Albrecht and Ptak1998).Theorem 20.1.10 remains true if the condition that T possesses an isometric

weak-∗ continuous H∞(B)-functional calculus is replaced by the hypothesisthat the Taylor spectrum of T is dominating in B (see Eschmeier 1999). Reflex-ivity results for spherical contractions that extend the above mentioned resultsfor subnormal spherical contractions can be found in Eschmeier (2001c). InDidas (2002), many of the above results are shown to hold for commuting tu-ples T ∈ B(H )n which possess an isometric,weak-∗ continuous H∞-functionalcalculus � : H∞(D)→ B(H ) and a normal boundary dilation over a strictlypseudoconvex domain D inCn , or more generally, a strictly pseudoconvex openset D in a Stein submanifold X of Cn . In particular, every subnormal n-tuplewith dominating Taylor spectrum in D, or with isometric weak-∗ continuousH∞(D)-functional calculus, is shown to be reflexive.By a result of Bercovici (1994) at least each commuting family of isometries

on a Hilbert space is reflexive. Azoff and Ptak (1995) extended this result to thecase of jointly quasinormal tuples. The question whether each spherical isome-try is reflexive seems still to be open. Some partial results have been obtained inMuller and Ptak (1999) and Eschmeier (2001c). The general question, whetherevery subnormal tuple S ∈ B(H )n on a Hilbert space is reflexive, is one of themajor open questions in this area at the time of this writing.

Appendix to Part III

It is the purpose of this appendix to indicate a possible proof of Theorem 19.1.2.For each open set U in C, we denote by P∞(U ) the weak-∗ closure of

the polynomials in L∞(U ) ∼= L1(U )′, where L1(U ) and L∞(U ) are formedwith respect to the planar Lebesgue measure. Then P∞(U ) is a weak-∗ closedsubalgebra of H∞(U ).Let σ be a compact set in C. We say that σ is dominating for P∞(U ) if

| f |U = | f |U∩σ ( f ∈ P∞(U )) .

Let us assume that σ is dominating for P∞(U ) for some non-empty, boundedopen set U in C. Then

V =⋃{U ; U ⊂ C is bounded and open, and σ is dominating for P∞(U )}

is the largest bounded open set such that σ is dominating for P∞(V ).Define K = σ (Mz, P∞(V )). If λ �∈ K , then 1/(λ− z) ∈ P∞(V ). In par-

ticular, V ⊂ int(K ) and R(K )|V ⊂ P∞(V ). A standard result in rational ap-proximation theory (cf. the proof of Theorem 3 in Brown (1987)) shows thatthere is a family (Li )i∈I of bounded components of C \ K such that the setC = K ∪⋃{Li ; i ∈ I } has the following properties:(i) K is dominating in int(C) (i.e., for H∞(int(C))),(ii) R(C) is a Dirichlet algebra.

By Conway (1991, Theorem VI.4.8) the algebra R(C) is pointwise bound-edly dense in H∞(int(C)). Hence H∞(int(C))|V ⊂ P∞(V ). We claim that therestriction map

H∞(int(C))→ P∞(V ), f �→ f |V ,

193


is isometric. Otherwise, |g|int(C) > |g|V for some g ∈ H∞(int(C)), and therewould be a point a ∈ int(C) ∩ K with |g(a)| > |g|V . But then a �∈ V and be-cause of the fact that

|(z − a) f |V ≥ dist(a, V )| f |V ( f ∈ P∞(V ))

the proper ideal (z − a)P∞(V ) ⊂ P∞(V ) is weak-∗ closed. By the Hahn–Banach theorem this ideal is annihilated by some weak-∗ continuous linearfunctional ρ : P∞(V )→ C with ρ(1) = 1. Since on polynomials ρ acts as thepoint evaluation at a, the functional ρ is multiplicative (cf. Exercise 14.2.3). Bythe cited result in Conway (1991), there is a sequence (rn) of rational functionswith no poles on C which is uniformly bounded on C and converges to gpointwise on int(C). We obtain the contradiction

|g(a)| = | limnrn(a)| = | lim

nρ(rn|V )| = |ρ(g|V )| ≤ |g|V .

The observation that

| f |int(C) = | f |V = | f |V∩σ ≤ | f |int(C)∩σfor all f ∈ H∞(int(C)) shows that σ is dominating in int(C). The maximalityof V implies that V = int(C). It follows that

K = C, H∞(V ) = P∞(V ) ,

and that V coincides with the largest bounded open set in C such that σ isdominating in V for H∞(V ).Using for a third time Conway (1991, Theorem VI.4.8), we see that R(V ) is

pointwise boundedly dense in H∞(V ). Using the fact that ∂V ⊂ ∂V ⊂ ∂K andthat R(K ) is a Dirichlet algebra, we can easily deduce that R(V ) is a Dirichletalgebra. By Gamelin (1969, Corollary II.9.3), we have R(V ) = A(V ) = A(V ).The last equality follows because V = int(V ), or equivalently, ∂V = ∂V . SinceR(V ) is aDirichlet algebra, the connected components ofV = int(V ) are simplyconnected (see Conway 1991, Theorem VI.5.2).To see that V is contained in the polynomially convex hull σ of σ (see

Definition 4.1.1) it suffices to observe that, for each point z ∈ V and eachpolynomial p, we have

|p(z)| ≤ |p|V = |p|V∩σ ≤ |p|σ .

Thus the proof of Theorem 19.1.2 is complete.

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Part IV

Local spectral theory

KJELD BAGGER LAURSENUniversity of Copenhagen, Denmark

21

Basic notions from operator theory

21.1 Introduction

This part of the book is intended as an invitation to the subject of local spectraltheory. It contains the basics and some indications of the way the subject hasdeveloped. I would like to thank Garth Dales and Michael Neumann for theirnumerous good comments and suggestions. The entire story of the fascinatingsubject that Chapters 21–25 deal with may be found in Laursen and Neumann(2000), and I hope that after having been through these chapters you will wantto go for more in that book, which also contains a full bibliography.The phrase local spectral theory carries many connotations. Among the ones

that are appropriate here you should expect to find concepts such as spectralsubspaces, that is, invariant subspaces on which the restricted operator hasa spectrum consisting of a chunk of the original spectrum. The archetypal con-ceptual framework is provided by the spectral theorem for normal operatorson a Hilbert space, which specifies how this decomposition of the underly-ing space and of the spectrum is supposed to look. Another similar exam-ple is provided by the spectral theorem for compact operators on a Banachspace.Both of these examples may be traced back to what is often a high point

of a first course in linear algebra, namely a result on diagonalizing symmetricmatrices such as the following. (A symmetricmatrix [ai j ] satisfies the relationsai j = a ji for all i, j , while for a symmetric operator T on a finite-dimensional,real inner-product space V with inner product [·, ·], it is true that we have[T x, y] = [x, T y] for all x, y ∈ V .)


201

202 Part IV Local spectral theory, Kjeld Bagger Laursen

Theorem 21.1.1 A linear operator T : V → V, where V is a finite-dimensional real inner-product space, is symmetric if and only if V has anorthonormal basis consisting of eigenvectors for T . �

For a Hilbert space this may be turned into the spectral theorem for normaloperators. To appreciate this from our chosen vantage point we need to knowwhat a spectral measure is. An operator N ∈ B(H ) is normal when it satisfiesNN ∗ = N ∗N and self-adjoint when N = N ∗. Here T ∗ is the adjoint of theoperator T ∈ B(H ) defined by the formula

[x, T ∗y] = [T x, y] for all x, y ∈ H ,

as in §3.5.

Definition 21.1.2 Let B denote the collection of Borel subsets of C, and letH be a Hilbert space. Then a spectral measure E is a map from B to the setof projections on H (i.e. all self-adjoint idempotents in B(H )) which has theseproperties:

� E(∅) = 0, E(C) = IH , the identity operator on H;� E(B1 ∩ B2) = E(B1)E(B2) for all Borel sets B1, B2;� if {Bj }∞j=1 is a sequence of pairwise disjoint Borel sets, then

E(⋃∞

j=1Bj)h =∑∞

j=1E(Bj )h ,

for every element h ∈ H, the sum converging in the Hilbert space norm (theterms are pairwise orthogonal).

Of course, in the statement to follow, the proper interpretation of the integralneeds attention; we shall not dwell on it, because this point will not be neededelsewhere. The spectrum of an operator is defined in Part I, §2.1.

Theorem 21.1.3 Let N be a normal operator on the Hilbert space H. Thenthere is a unique spectral measure E on the Borel subsets of the spectrum σ (N )of N for which

N =∫z dE(z) .

Moreover, for every Borel set � ⊆ C the subspace E(�)H is N-invariant and

σ (N |E(�)H ) ⊆ � ∩ σ (N ). �

21 Basic notions from operator theory 203

It is in this light that themodern definition of a decomposable operator shouldbe viewed.Throughout these Chapters 21–25, we shall be considering complex Banach

spaces (as well as Banach algebras), and it will be assumed that, unless some-thing else is explicitly stipulated, the term X refers to a complex Banach space,and the term A refers to a commutative complex Banach algebra.

Definition 21.1.4 A bounded linear operator T : X → X is decomposable if,for every open cover {U, V } of C, there are closed T -invariant subspaces Y, Zof X for which:

� X = Y + Z ;� σ (T |Y ) ⊆ U and σ (T |Z ) ⊆ V .

Chapter 22 will look at examples inmore detail (but it is an easy consequenceof Theorem 21.1.3 that normal operators are decomposable – see Exercise21.3.1). Here we shall concentrate on introducing the basic concepts.If U ⊆ C is open, then H (U ) denotes the algebra (pointwise operations) of

analytic functions defined on U. We give this algebra the topology of locallyuniform convergence, that is, of uniform convergence on all compact subsets ofU. It may be shown that this topology is locally convex and that it is induced bya complete and translation-invariant metric; thus H (U ) is a Frechet algebra (seePart I, Examples 1.2 (x)). Among the elements of H (U ) we find the constantfunction 1 and the identity function Z .The analytic functional calculus is defined in Chapter 4. We recall it this

way: if T ∈ B(X ) has spectrum σ (T ) andU is an open neighbourhood of σ (T )on which the analytic function f is defined, then the map

� : f → f (T ) := 1

2π i

∫�

f (λ)(λ− T )−1 dλ

(where � is a contour in U surrounding σ (T ), as explained in Chapter 4) is analgebra homomorphism from the Frechet algebra H (U ) into B(X ) for which�(1) = IX and �(Z ) = T .There are connections between functional calculi and decomposability, al-

though in the case of the analytic functional calculus this connection is limitedto the situations in which the spectrum is disconnected. To display an extremecase: this calculus may be used to show that, if σ (T ) is totally disconnected,then T is decomposable (Exercise 21.3.2). (Recall from general topology thata topological space is said to be totally disconnected if none of its connectedsubsets contains more than one point.) The reason the argument goes through is


that a totally disconnected compact Hausdorff space has a base for its topologyconsisting of clopen sets, as shown, for instance, in Appendix A.7 of Rudin(1991). For more on totally disconnected spaces, see Engelking (1977).A class that displays a better connection is that of the generalized scalar

operators. To explain: look at the algebra C∞(C) of complex-valued functionsthat have continuous partial derivatives of all orders with respect to the variablesx and y (as usual for a complex number z, we write z = x + iy, where x andy are real). This algebra may be equipped with the topology determined by theseminorms

‖ f ‖k,Q :=∑|α|≤k

1

α!sup {|D α f (λ)| : λ ∈ Q} for all f ∈ C∞(C) ,

where Q ranges over all compact subsets of C, k over Z+, and α = (α1, α2) isan arbitrary double index of integers which specifies the order of differentiation:

D α f (λ) = ∂α1x ∂α2y f (λ) ,

|α| := α1 + α2, and α! := α1!α2!; the factors 1/α! are there to ensure submul-tiplicativity of the seminorm ‖ · ‖k,Q .With the topology of this countable familyof seminorms, C∞(C) is a Frechet algebra. Obviously the set H (C) of entirefunctions is a subalgebra of C∞(C).

Definition 21.1.5 Let T ∈ B(X ) be given. If there is a continuous algebrahomomorphism� from C∞(C) into B(X ) for which�(1) = I and�(Z ) = T ,then T is a generalized scalar operator.

Note that the homomorphism of this definition will necessarily be an exten-sion from H (C) of the homomorphism that appears in the analytic functionalcalculus.A generalized scalar operator is decomposable. This is a consequence of the

fact that in C∞(C) we have partitions of unity (cf. Exercise 21.3.3).

21.2 The properties (β) and (δ)

The main goal for the rest of this chapter is to introduce two seemingly merelytechnical conditions, and to relate them to decomposability. Our aim is Theorem21.2.8, which shows that these two conditions together are equivalent to de-composability. It will become clear in the course of the next few chapters thatboth conditions have some truly astounding relations, which go well beyond


this first result. Both properties (which are called properties (β) and (δ)) goback to Errett Bishop’s PhD thesis (1959).

Definition 21.2.1 An operator T ∈ B(X ) has property (β) (or just: T has (β))if, for every open subset U of C and every sequence ( fn) of analytic functionsfn : U → X with the property that (T − λ) fn(λ)→ 0 as n→∞, locally uni-formly onU, it follows that fn(λ)→ 0 as n→∞, also locally uniformly onU.

This property may be conveniently rephrased in terms of an operator TU , de-fined on the Frechet space H (U, X ) of X -valued analytic functions, topologizedby locally uniform convergence on U. This operator is specified by

(TU f )(λ) := (T − λ) f (λ) for every λ ∈ U .It may then be shown that an operator T has (β) if and only if TU is injectiveand has closed range, for every open set U in C (Exercise 21.3.4).Incidentally, this last observation is also a possible lead-in to the notion of

the single-valued extension property SVEP, which is as follows.

Definition 21.2.2 An operator T ∈ B(X ) has SVEP if, for every open subsetU of C, the equation (T − λ) f (λ) = 0 has only one analytic solution, namelyf = 0.

Thus T has SVEP if and only if TU is injective for every open U. It is clearthat (β) implies SVEP. The converse is not true, although youmay have to thinkhard to come up with a counter-example (Exercise 21.3.5). We shall displaysome later. For more on SVEP, see Part V, Chapter 27.To introduce property (δ) we need the glocal subspaces.

Definition 21.2.3 Let T ∈ B(X ) be given and suppose that F ⊆ C is closed.Then the glocal spectral subspace is

XT (F) := {x ∈ X : there is an analytic function f : C \ F → X

for which (T − λ) f (λ) = x for all λ ∈ C \ F}.

In other, and more formal, terms,

XT (F) = {x ∈ X : x ∈ TC\F H (C \ F, X )}.On the resolvent set ρ(T ) we obviously have that every x ∈ X may bewritten

as x = (T − λ)(T − λ)−1x for every λ ∈ ρ(T ), which shows immediately thatX = XT (σ (T )). Also, it is an easy consequence of Liouville’s theorem thatXT (∅) = {0} (Exercise 21.3.6).


Definition 21.2.4 An operator T ∈ B(X ) has property (δ) (or just (δ)) if

X = XT (U )+ XT (V )

for every open cover {U, V } of C.

If T : X → Y is a continuous linear operator between the Banach spaces Xand Y, then T induces a linear map T ' : H (U, X )→ H (U, Y ), defined by thecomposition

(T ' f )(λ) := T f (λ) for all λ ∈ U and f ∈ H (U, X ) .You may want to convince yourself that T ' is continuous; this is not hard(Exercise 21.3.7).Here we shall need this map only for open discs. We use D(a; b) to denote

the closed disc in C centred at a and of radius b, and D(a; b) to denote thecorresponding open disc. The open unit disc is called just D. If U = D(λ 0; r ),then every f ∈ H (U, X ) is given by a power series

f (λ) =∞∑n=0

an(λ− λ 0)n for all λ ∈ U ,

where the coefficients an are elements of X. Since the convergence of this seriesis locally uniform, the continuity of T ' implies that

(T ' f )(λ) =∞∑n=0

T (an)(λ− λ 0)n for all λ ∈ U .

We shall need the fact that surjectivity is transferred from T to T '.

Proposition 21.2.5 Let T : X → Y be a continuous linear surjection from theBanach space X onto the Banach space Y. Then, for every open disc U ⊆ C,

the induced map T ' : H (U, X )→ H (U, Y ) is a continuous and open linearsurjection.

Proof Let U = D(λ 0; r ), where λ 0 ∈ C and r > 0. Then, for every functiong in H (U, Y ), we have a power series expansion g(λ) =∑∞

n=0 bn(λ− λ 0)n,valid for everyλ ∈ U.The radius of convergence of this power series is at least r.The open mapping theorem tells us there is a constant c > 0 for which, for

every y ∈ Y, there is an x ∈ X such that T x = y and ‖x‖ ≤ c ‖y‖ .So, for everyn ∈ N,wemay choose an element an ∈ X so that Tan = bn and ‖an‖ ≤ c ‖bn‖ .The estimates

lim supn→∞

‖an‖1/n ≤ lim supn→∞

c1/n ‖bn‖1/n = lim supn→∞

‖bn‖1/n ≤ 1r,


tell us that the power series f (λ) :=∑∞n=0 an(λ− λ 0)n converges for all λ in

the disc U, hence defines a function f ∈ H (U, X ). Since

(T ' f )(λ)=∞∑n=0T (an)(λ− λ 0)n =

∞∑n=0bn(λ− λ 0)n = g(λ) for all λ ∈ U,

T ' is surjective, hence also, by the open mapping theorem, open. �

To obtain the proof of Theorem 21.2.8, we need first some technicalobservations.Given T ∈ B(X ), for a T -invariant closed linear subspace Y of X, let

T | Y ∈ B(Y )

denote the operator given by the restriction of T to Y, and let

T/Y ∈ B(X/Y )

denote the operator induced by T on the quotient space X/Y. The subscript f(f as in full) refers to σ (T ) with all its holes filled in. This is the polynomiallyconvex hull of σ (T ), as defined in Definition 4.1.4.

Proposition 21.2.6 Let T ∈ B(X ) be an operator, and suppose that Y and Zare T -invariant closed linear subspaces of X with the property that X = Y + Z .Then

σ (T/Y ) ⊆ σ (T ) ∪ σ (T | Y ) ⊆ σf(T ) and σ (T/Z ) ⊆ σf(T | Y ) ⊆ σf(T ) .

Proof We know that (T/Y )Q = Q T, where Q : X → X/Y is the naturalquotient map. For arbitrary λ ∈ ρ(T ) ∩ ρ(T | Y ), T/Y − λ is surjective be-cause both Q and T − λ are. Moreover, if (T/Y − λ)Qx = 0 for some x ∈ X ,then (T − λ)x ∈ Y, and hence x ∈ Y because λ ∈ ρ(T | Y ); thus T/Y − λ isinvertible on X/Y. This shows that σ (T/Y ) ⊆ σ (T ) ∪ σ (T | Y ).Next, the classical identity

(T − λ)−1 = −∞∑n=0

λ−n−1 T n for all λ ∈ C with |λ| > ‖T ‖ ,

shows that (T − λ)−1Y ⊆ Y for all such λ and, since C \ σf(T ) is connected,the identity theorem for vector-valued functions tells us that (T − λ)−1Y ⊆ Yfor all λ ∈ C \ σf(T ), hence that σ (T | Y ) ⊆ σf(T ).Finally, the assumption that X = Y + Z provides us with a canonical

surjection from the space Y onto the quotient X/Z with kernel Y ∩ Z .If the corresponding isomorphism is called R : Y/(Y ∩ Z )→ X/Z , then


(T/Z )R = R (T | Y )/(Y ∩ Z ), and so σ (T/Z ) = σ ((T | Y )/(Y ∩ Z )). Fromthe work already done, we know that

σ ((T | Y )/(Y ∩ Z )) ⊆ σ (T | Y ) ∪ σ (T | (Y ∩ Z )) ⊆ σf(T | Y ) ⊆ σf(T ) ,

and hence σ (T/Z ) ⊆ σf(T | Y ) ⊆ σf(T ) . �

Proposition 21.2.7 Suppose that Y and Z are closed linear subspaces of Xand that U ⊆ C is an open disc. Then the following properties hold.

(i) The quotient map from X onto X/Y induces a canonical topological iso-morphism H (U, X/Y ) ∼= H (U, X )/H (U, Y ).

(ii) If X = Y + Z , then the operator � : H (U, Y )× H (U, Z )→ H (U, X ),given by �( f, g)(λ) := f (λ)+ g(λ) for all f ∈ H (U, Y ), g ∈ H (U, Z ),and λ ∈ U, is a continuous and open linear surjection. In particular, ifX = Y ⊕ Z (direct sum), then the map� yields a canonical identificationH (U, X ) ∼= H (U, Y )⊕ H (U, Z ).

Proof (i) This is immediate from Proposition 21.2.5: let T : X → X/Y be thecanonical quotient map and observe that ker T ' = H (U, Y ).For (ii) first note that the definition &( f, g)(λ) := ( f (λ), g(λ)) for f in

H (U, Y ), g in H (U, Z ), and λ in U yields a topological linear isomorphism& from H (U, Y )× H (U, Z ) onto H (U, Y × Z ). Next, let T : Y × Z → Xbe the canonical continuous linear surjection from the product space Y × Zonto X, given by T (u, v) = u + v for all u ∈ Y and v ∈ Z . Then Proposition21.2.5 tells us that the corresponding map T ' : H (U, Y × Z )→ H (U, X ) iscontinuous, surjective, open, and linear. This implies our claims for �, since� = T ' ◦&. �And now the result itself.

Theorem 21.2.8 Decomposability implies (β) and (δ) – and vice versa.

Proof The first step is to show that decomposability implies (β). Let U ⊆ C

be open, and consider a sequence of analytic functions fn : U → X for which(T − λ) fn(λ)→ 0 as n→∞, locally uniformly onU. It is enough to show thatfn → 0 as n→∞ uniformly on any closed disc in in U. So take an arbitraryclosed disc D ⊆ U and choose an open disc E such that D ⊆ E ⊆ E ⊆ U.Apply the definition of decomposability of T to the open cover {E,C \ D}of C. This gives us T -invariant closed linear subspaces Y, Z ⊆ X for whichσ (T | Y ) ⊆ E, σ (T | Z ) ∩ D = ∅, and X = Y + Z . Proposition 21.2.7 then


yields functions gn ∈ H (U, Y ) and hn ∈ H (U, Z ) such thatfn(λ) = gn(λ)+ hn(λ) for all λ ∈ U and n ∈ N .

Additionally, by Proposition 21.2.6, σ (T/Z ) ⊆ σ f (T | Y ) ⊆ E, so, for everyλ in the boundary ∂E of E, the operator T/Z − λ is invertible on the quo-tient space X/Z . By compactness and continuity, we obtain a c > 0 such that‖ (T/Z − λ)−1 ‖ ≤ c for all λ ∈ ∂E . Consequently, if Q : X → X/Z is thenatural quotient map, then

Qgn(λ) = Q fn(λ) = (T/Z − λ)−1Q(T − λ) fn(λ) for all λ ∈ ∂E,and therefore ‖ Qgn(λ) ‖ ≤ c ‖ (T − λ) fn(λ) ‖ for all λ ∈ ∂E . This estimate,and our assumption on the functions fn, then imply that the analytic functionsQ ◦ gn in H (U, X/Z ) converge to zero uniformly on ∂E and therefore, by themaximum modulus principle, uniformly on E . Now, by Proposition 21.2.7,H (E, X/Z ) may be identified with H (E, X )/H (E, Z ), and consequently weobtain functions kn ∈ H (E, Z ) for which gn + kn → 0 as n→∞, locally uni-formly on E, hence uniformly on D.Now, fn = gn + hn = (gn + kn)+ (hn − kn) on D for all n ∈ N, and so it re-

mains to see that hn − kn converges to 0 uniformly on D.Observe that, becauseσ (T | Z ) ∩ D = ∅, there is a constant c1 > 0 such that ‖ (T | Z − λ)−1 ‖ ≤ c1for all λ ∈ D. Since both hn and kn map into the space Z and hn = fn − gn,we conclude that

‖ (hn − kn)(λ) ‖ ≤ c1 ‖ (T − λ)(hn − kn)(λ) ‖≤ c1 ‖ (T − λ) fn(λ) ‖ + c1 ‖ (T − λ)(gn + kn)(λ) ‖

for all λ ∈ D and n ∈ N. We have now shown that hn − kn → 0 uniformlyon D.That decomposability implies (δ) is immediate: if C = U ∪ V and Y, Z are

chosen in accordance with the definition of decomposability, then the inclusionσ (T |Y ) ⊆ U implies that Y ⊆ XT (U ). Similarly, Z ⊆ XT (V ). This establishes(δ).Finally, if T has both (β) and (δ) then decomposability will follow once we

have seen that eachXT (F) is closed. And this follows from (β).Here is a sketchof the argument. SinceXT (F) := {x ∈ X : x ∈ TC\F H (C \ F, X )},we see that

XT (F) = TC\F H (C \ F, X ) ∩ X.We leave it as an exercise (21.3.8) that the closedness of XT (F) follows fromthis. Next, if U ∪ V = C is an open cover, then choose open sets U1 and V1,


still covering C, so that U1 ⊆ U and V1 ⊆ V .We have

X = XT (U1)+ XT (V1) .

It is then not too difficult, using the open mapping theorem, to see that

σ (T |XT (U1)) ⊆ U1 ⊆ U ,

and also that

σ (T |XT (V1)) ⊆ V1 ⊆ V

which establishes the decomposability of T (Exercise 21.3.9). �

21.3 Exercises

1. Use Theorem 21.1.3 to show that normal operators are decomposable.2. Show that, if σ (T ) is totally disconnected, then T is decomposable.3. Show that a generalized scalar operator is decomposableHint: inC∞(C) wehave partitions of unity.

4. Show that an operator T has (β) if and only if TU is injective and has closedrange, for every open set U in C.

5. Find an example of a bounded linear operator which has SVEP, but not (β).6. Use Liouville’s theorem to show that XT (∅) = {0}.7. Prove that the map T ' introduced after Definition 21.2.4 is continuous.8. Show that, sinceXT (F) = TC\F H (C \ F, X ) ∩ X,XT (F) is closed providedthat T has (β). (cf. the proof of Theorem 21.2.8.)

9. Suppose that XT (F) is closed. Show that σ (T |XT (F)) ⊆ F . Hint: use theopen mapping theorem.


Decomposability may be described entirely in terms of spectral capacities(subspace-valued functions from the Borel sets ofCwhich behave pretty muchlike the ranges of spectral projections, cf. Definition 21.1.2), which brings itconceptually even closer to the spectral theorem. The class of decomposableoperators is but one of the natural extensions based in the spectral theorem. Weshall define a couple of them in the next chapter, but there are others: spectraland scalar operators, going back to Dunford, as well as various strengtheningsof the decomposability requirement. Chapter 1 of Laursen andNeumann (2000)tells more. Of course, Hilbert space operator theory has a history and dynamics


of its own, displayed by the appearance of several classes containing the normalones, such as sub- or hyponormal operators.The history of decomposability is quite interesting. The subject has generated

many classes of operators, as well as a conceptual evolution, for instance of thevery definition of decomposability, which was quite complicated when Foiasfirst proposed it, but now, thanks to Albrecht and Lange, reads very simply.Among the leading classical practitioners of this branch of operator theory wefind people such as Dunford, Bishop, Foias, Vasilescu, Frunza and Albrecht,while themore recent developments have been strongly influenced by, inter alia,Albrecht, Lange, Eschmeier and Putinar. Chapter 1 of Laursen and Neumann(2000) contains an account of all this.In Gleason (1962), it was shown that Proposition 21.2.5 actually holds for

arbitrary open subsetsU ofC. The proof of this general version is considerablymore involved, and such a high level of generality is not needed in the presentwork.

22

Classes of decomposable operators

22.1 Local properties of operators

In this chapter we shall take a closer look at super-decomposable and at gen-eralized scalar operators. Along the way we also obtain some examples ofoperators with (β), namely all isometries.The companion concept of the glocal subspace is the local analytic spectral

subspace. Here is first the definition of local spectrum.

Definition 22.1.1 Let T ∈ B(X ) and x ∈ X. The local resolvent ρT (x) of T atx is the set of λ ∈ C for which there is an open neighbourhood N (λ) on whichthe equation (T − µ) f (µ) = x has an analytic solution f : N (λ)→ X. Thelocal spectrum is

σT (x) := C \ ρT (x) ,

and, given a closed subset F of the complex plane, the local analytic subspace is

XT (F) := {x ∈ X : σT (x) ⊆ F} .

It follows directly from the definition that XT (⋂Fα) =

⋂XT (Fα) for arbi-

trary collections {Fα} of closed subsets of C.There are some immediate connections between the glocal and the local

spectral subspaces. In fact, it is clear that, if x ∈ XT (F), then C \ F ⊆ ρT (x),and hence XT (F) ⊆ XT (F) for every closed set F ⊆ C. As a matter of fact itis not difficult to tell when these two classes of subspaces coincide.

Proposition 22.1.2 For T ∈ B(X ),we haveXT (F) = XT (F) for every closedset F ⊆ C if and only if XT (∅) = {0}, and this happens precisely when T hasSVEP.

212

22 Classes of decomposable operators 213

Proof Exercise 22.4.1. �

The surjectivity spectrum is defined as

σsu(T ) := {λ ∈ C : T − λ is not surjective} ,while the approximate point spectrum is

σap(T ) := {λ ∈ C : there is a sequence {xn} of unit vectorsfor which (T − λ)xn → 0 as n→∞} .

Among the basic properties of the surjectivity spectrum, we note the fol-lowing (the proofs are left as exercises – think of the open mapping theoremwhen you solve them). For every λ ∈ C \ σsu(T ), there is a c > 0 for whichX = XT (C \ D(λ; c)) (cf. Exercise 24.3.3). In particular, σ (T ) \ σsu(T ) ⊆σp(T ), and, if T has SV EP, then σ (T ) = σsu(T ). The most immediate con-nection to local spectral theory is that

σsu(T ) =⋃ {σT (x) : x ∈ X} .

Similarly, for the approximate point spectrum it is a basic fact that it is a closedset which contains the topological boundary of the spectrum.

Proposition 22.1.3 Let T ∈ B(X ) be decomposable. Then

σ (T ) = σap(T ) = σsu(T ) =⋃ {σT (x) : x ∈ X} .

Proof We know from Theorem 21.2.8 that T has (β), hence SVEP, so theremarks just made show us that σ (T ) = σsu(T ) =

⋃ {σT (x) : x ∈ X} . To seethat σap(T ) = σ (T ), we may argue this way: given any λ ∈ σ (T ) and ε > 0,let U ⊆ C be an open set for which λ /∈ U and U ∪ D(λ; ε) = C. By decom-posability, there are T -invariant closed linear subspaces Y, Z ⊆ X for whichX = Y + Z , σ (T | Y ) ⊆ U and σ (T | Z ) ⊆ D(λ; ε). The subspace Z must benon-trivial, since otherwise X = Y, and hence σ (T ) = σ (T | Y ) ⊆ U, whichwould contradict that λ /∈ U. Since Z is non-trivial and since

∂σ (T | Z ) ⊆ σap(T | Z ) ⊆ σap(T ) ,

it follows from the inclusion σ (T | Z ) ⊆ D(λ; ε) that D(λ; ε) ∩ σap(T ) �= ∅.Thus we have λ ∈ σap(T ) because this set is closed. �

Here is another useful observation:


Lemma 22.1.4 Let U be an open subset of C. If T ∈ B(X ), x ∈ X, andf ∈ H (U, X ) are chosen so that TU f ≡ x on U, then f (λ) ∈ TU H (U, X ) forall λ ∈ U. Moreover, σT (x) = σT ( f (λ)) for all λ ∈ U.

Proof Given an arbitrary λ ∈ U, we define g : U → X by

g(λ) := f ′(λ) and g(µ) := ( f (µ)− f (λ))/(µ− λ) for all µ ∈ U \ {λ} .

Then g ∈ H (U, X ). Since (T − µ) f (µ) = x for all µ ∈ U, a short calculationreveals that (T − µ)g(µ) = f (λ) for all µ ∈ U \ {λ}, and, by continuity, also(T − λ)g(λ) = f (λ). That settles the first part of the lemma. It also shows thatU ⊆ ρT ( f (λ)).On the other hand, for any ω ∈ ρT (x) \U, we may choose an open neigh-

bourhood W of ω such that λ /∈ W and an analytic function h : W → X forwhich (T − µ)h(µ) = x, identically on W. If we then define

k(µ) := (h(µ)− f (λ))/(µ− λ)

for all µ ∈ W, we have obtained an analytic function k : W → X for which

(T − µ)k(µ) = f (λ) for all µ ∈ W .

This shows that ω ∈ ρT ( f (λ)), and hence that ρT (x) ⊆ ρT ( f (λ)).For the opposite inclusion, take any ω ∈ ρT ( f (λ)), and consider an an-

alytic function h : W → X on an open neighbourhood W of ω such that(T − µ)h(µ) = f (λ) for all µ ∈ W. Substitution of this leads to

(T − µ)(T − λ)h(µ) = (T − λ)(T − µ)h(µ) = (T − λ) f (λ) = x

for allµ ∈ W,which shows that ω ∈ ρT (x), and hence that ρT ( f (λ)) ⊆ ρT (x).All in all, σT (x) = σT ( f (λ)) for all λ ∈ U. �

22.2 Super-decomposable operators

Here is our first class of examples of decomposable operators. The super-decomposable operators are based on the idea of operator-valued partitionsof unity. More specifically, we are considering a class of operators for whichthe condition of decomposability is implemented by certain operator ranges.

Definition 22.2.1 An operator T ∈ B(X ) is super-decomposable if, for everyopen cover of C by two subsets U and V, there is an operator R ∈ B(X ) which


commutes with T and for which

σ (T | RX ) ⊆ U and σ (T | (I − R)X ) ⊆ V .

This definition makes sense because, by the condition R T = T R, the spacesRX and (I − R)X are T -invariant. It is obvious that super-decomposableoperators are decomposable.

Proposition 22.2.2 Every operator T ∈ B(X )with totally disconnected spec-trum is super-decomposable.

Proof For any open cover {U, V } of C, σ (T ) ∩U is a relatively open neigh-bourhood of the closed subset σ (T ) \ V of σ (T ).Hence a standard compactnessargument provides us with a clopen subset G of σ (T ) for which G ⊆ U andσ (T ) \ G ⊆ V . We may then choose disjoint open sets U1, V1 ⊆ C so thatU1 ⊆ U, V1 ⊆ V, U1 ∩ σ (T ) = G, and V1 ∩ σ (T ) = σ (T ) \ G.Subsequently we pick a contour � in U1 ∪ V1 that surrounds σ (T ), and let

R := 1

2π i

∫�

f (λ) (λ− T )−1 dλ ,

where now f ∈ H (U1 ∪ V1) is the characteristic function of U1. The operatorR ∈ B(X ) is a projection, it commutes with T, and it is not difficult to showthat

σ (T | RX ) = G ⊆ U1 ⊆ U and σ (T | (I − R)X ) = σ (T ) \ G ⊆ V1 ⊆ V ;(see, e.g., Theorem 7.3.20 of Dunford and Schwartz 1958; or Part I, Theorem4.3.1). This shows that T is super-decomposable. �

This result applies to all operators with countable spectrum, and hence, inparticular, to all quasi-nilpotent operators (cf. Part I, Chapter 2). It also covers allalgebraic operators T ∈ B(X ), those for which p(T ) = 0 for some non-trivialpolynomial p (since the spectra of such operators are finite). Further examplesare: compact operators and Riesz operators (where an operator T ∈ B(X ) is aRiesz operator if, for each λ ∈ C \ {0}, the spaces ker(T − λ) and X/(T − λ)Xare both of finite dimension; in other words, T is Riesz if T − λ is Fredholm,for every λ ∈ C \ {0}). See also Part V.We have touched on the analytic functional calculus (cf. the account in Part

I, Chapter 4). We now show that non-analytic functional calculi tend to gen-erate super-decomposable operators. To be specific, let A be an algebra ofcomplex-valued functions, defined on an arbitrary non-empty subset of C,


so that the following two conditions are fulfilled. First, we suppose that A isrichly endowed, which is to mean that A contains the restrictions to of allpolynomials, and that, for every f ∈ A and every λ ∈ C \ supp f, there exists afunction g ∈ A for which (Z − λ)g = f.Here it is natural to define the supportof a function f ∈ A as the set

supp f := {λ ∈ : f (λ) �= 0} ,

where the closure is taken inC, not just in the relative topology of. Secondly,we suppose that A admits partitions of unity, in the sense that, for an arbitraryfinite open cover {U1, . . . ,Um} of , there are functions f1, . . . , fm ∈ A forwhich supp fk ⊆ Uk for k = 1, . . . ,m and f1 + · · · + fm ≡ 1 on .Obvious examples of such algebras are the algebras Cm() of all m times

continuously differentiable functions defined on an open set ⊆ C, for ar-bitrary m ∈ {0, 1, . . . ,∞}. Moreover, if, additionally, is bounded, then ex-amples are provided by the algebras Cm() of all functions in Cm() which,together with all their partial derivatives up to order m, have continuous exten-sions to the closure .

Definition 22.2.3 Suppose that the algebra A is richly endowed and admitspartitions of unity. An operator T ∈ B(X ) is an A-scalar operator if thereexists an algebra homomorphism � : A→ B(X ) for which �(1) = IX and�(Z ) = T .

Here IX is the identity operator on X . There is no continuity assumptionon � in this definition (although it may be shown that such homomorphismshave quite remarkable automatic continuity properties). The map � is calleda non-analytic functional calculus for T . An important special case occurswhen A = C∞(C), and we also require that � be continuous with respect tothe natural Frechet algebra topology on C∞(C): in this case, we have alreadyintroduced the terminology that the operator T is generalized scalar. We shallgive more details on these operators below.It is immediate that the spectrum of any A-scalar operator is contained in

, where is the domain of the functions in the algebra A. But our ambitionnow is to show that A-scalar operators are super-decomposable and to give anexplicit description of the local spectral subspaces.

Theorem 22.2.4 Let A be an algebra of functions on an arbitrary non-empty set ⊆ C, and suppose thatA is richly endowed and admits partitionsof unity. Also, let the A-scalar operator T ∈ B(X ) be given by an algebra


homomorphism� : A→ B(X ) for which�(1) = I and�(Z ) = T . Then T issuper-decomposable, and the local spectral subspace XT (F) for a closed setF ⊆ C is given by

XT (F) =⋂{ker�( f ) : f ∈ A and supp f ∩ F = ∅} .

Proof The idea of the proof is, for arbitrary closed F ⊆ C, to establish enoughproperties of the right-hand set

E(F) :=⋂{ker�( f ) : f ∈ A and supp f ∩ F = ∅} .

to see that it is indeed XT (F). Evidently, E(F) is a closed linear subspace of Xand E(F) is invariant under�(g) for all g ∈ A, hence, in particular, under T .Anargument involving both partitions of unity and the fact thatA is richly endowedthen shows that σ (T | E(F)) ⊆ F. It is now clear that E(F) ⊆ XT (F). The nextstep then establishes that T is super-decomposable: let {U, V } be an open coverof C, and choose f ∈ A for which supp f ⊆ U and supp (1− f ) ⊆ V . Theoperator R := �( f ) commutes with T, and

RX ⊆ E(supp f ) and (I − R)X ⊆ E(supp (1− f )) .

By the spectral inclusions from the preceding paragraph, it follows that T issuper-decomposable.Finally, the inclusion E(F) ⊇ XT (F) is a bit of a technical tour de force:

we first establish that E(⋂∞n=1Fn) =

⋂∞n=1E(Fn) for every countable family

{Fn} of closed sets in C. Here the inclusion ⊆ is obvious. For the other one,let x ∈⋂∞n=1E(Fn), and let f ∈ A be any function for which supp f ∩ F = ∅,where F :=⋂∞n=1Fn. By compactness, there exists k ∈ N such that

σ (T ) ∩ supp f ⊆ (C \ F1) ∪ · · · ∪ (C \ Fk) .If we supplement this union with the complement of σ (T ) ∩ supp f, we havea finite open cover of C. Consequently, we may choose functions f0, . . . , fk∈ A for which f0 + · · · + fk is 1 on , σ (T ) ∩ supp f ∩ supp f0 = ∅, andsupp f j ∩ Fj = ∅ for j = 1, . . . , k. It is clear that

�( f f0)X ⊆ E(supp ( f f0)) ⊆ XT (supp ( f f0) ∩ σ (T )) = {0},and hence�( f f0)x = 0.Moreover, since x ∈ E(Fj ) and supp f j ∩ Fj = ∅,weobtain �( f j )x = 0 for j = 1, . . . , k. All in all, we conclude that

�( f )x =k∑j=0

�( f f j )x =k∑j=1

�( f )( f j )x = 0 ,

and therefore x ∈ E(F).


With this knowledge about E(F) we are now in a position to show thatE(F) ⊇ XT (F). To do this, we fix an arbitrary open neighbourhood U of F,and choose an open set V ⊆ C so thatU ∪ V = C and V ∩ F = ∅. From whatwe did before, we see that X = E(U )+ E(V ). Since

E(U ∩ V ) = E(U ) ∩ E(V ) ,

the quotient spaces X/E(U ) and E(V )/E(U ∩ V ) are seen to be canonicallyisomorphic. This implies that

σ (T/E(U )) ⊆ σ (T | E(V )) ∪ σ (T | E(U ∩ V )) ⊆ V ∪ (U ∩ V ) = V ,

by Proposition 21.2.6. On the other hand, we know that σ (T | XT (F)) ⊆ F .Now, if Q : X → X/E(U ) denotes the canonical quotient map, then clearly

(T/E(U )) Q | XT (F) = Q (T | XT (F)) .

By the analytic functional calculus, it follows that

f (T/E(U )) Q | XT (F) = Q f (T | XT (F))

for every function f analytic on some open neighbourhood of V ∪ F. SinceV ∩ F = ∅, there is an analytic function f for which f ≡ 1 on an open neigh-bourhood of V and f ≡ 0 on an open neighbourhood of F. For this func-tion, it follows that f (T/E(U )) is the identity operator on X/E(U ), whilef (T | XT (F)) is the zero operator on XT (F).We conclude that Q | XT (F) = 0,and hence that XT (F) ⊆ E(U ). Since F may be written as an intersection ofcountablymany open neighbourhoods, and E preserves countable intersections,we have finally shown that XT (F) ⊆ E(F). �

Example 22.2.5 This example is based on Laursen and Neumann (2000,Example 1.4.11). Consider a commutative Banach algebra A with totally dis-connected character space �A and let � denote an algebra homomorphismfrom A to some B(X ).We claim that �(a) ∈ B(X ) is super-decomposable forevery element a ∈ A. Once this is established, we see (taking � to be the leftregular representation of A) that all multiplication operators on such an algebraare super-decomposable. This covers the case of the group algebra L1(G) of acompact abelian group G. (Since the character space of L1(G) may be identi-fied with the dual group of G, and since the dual group of a compact abeliangroup is discrete, we have that, in this case,�L1(G) is discrete, and hence totally


disconnected.) For a brief discussion of the algebras L1(G), see Part I, §3.4 andPart II.The proof uses the Silov idempotent theorem (see Part I, Additional note

4.5.4). If A is as stipulated, and we also suppose (to cover the more compli-cated case) that A has no unit element, let A# := A ⊕ C be the unitization of A,and let�1 : A#→ B(X ) denote the canonical extension of the given�, so that�1(a + λ) := �(a)+ λI for all a ∈ A and λ ∈ C . An operator T ∈ B(X ) issuper-decomposable if and only if this holds for T − λ, for any fixed λ ∈ C,

so we may replace a by a − λ, if necessary, and hence may add the assump-tion that a ∈ A# be invertible in A#. To establish that �1(a) ∈ B(X ) is super-decomposable, let {U, V } be an open cover of C. If µ ∈ C is the scalar forwhich a − µ ∈ A, we may suppose that µ ∈ V . The Gelfand transform ofa − µ vanishes at infinity on �A, so the set

K := {ϕ ∈ �A : ϕ(a) ∈ C \ V }

is compact, and it is clear that K is contained in the open set

L := {ϕ ∈ �A : ϕ(a) ∈ U } .

By local compactness and total disconnectedness of �A, its Gelfand topologyhas a base consisting of compact and open sets. Consequently, there is a com-pact and open set C ⊆ �A for which K ⊆ C ⊆ L . Now we invoke the Silovidempotent theorem: there is an idempotent r ∈ A for which ϕ(r ) = 1 for allϕ ∈ C and ϕ(r ) = 0 for all ϕ ∈ �A \ C. Evidently, R := �(r ) commutes withT . Moreover, for the spectrum σ (ar ) of ar ∈ A, elementary Gelfand theorytells us that

σ (ar ) = {ϕ(ar ) : ϕ ∈ �A} ∪ {0} ⊆ U ∪ {0} .

Hence ar − λ is invertible in A# if 0 �= λ ∈ C \U. This implies that, for everysuch λ, there is an element aλ ∈ A# for which (ar − λ)aλ = r, and therefore(a − λ)aλr = r.This equality also holds forλ = 0 ifwe choosea0 := a−1. Nowapply the homomorphism�1: we see that (T − λ)�1(aλ)R = R, and hence that(T − λ)�1(aλ)x = x for all x ∈ RX and λ ∈ C \U. Since �(aλ) commuteswith T and leaves RX invariant, this establishes that σ (T | RX ) ⊆ U. An anal-ogous argument gives σ (T | (I − R)X ) ⊆ V . So T is super-decomposable.

�

It is interesting that for many super-decomposable operators it is possible togive a description of their local spectral subspaces in purely algebraic terms.


This is important in applications of local spectral theory to automatic continuityquestions, where we want topological conclusions, but not topological assump-tions. The way to do this is to relate the local spectral subspaces to spaces(dating back to Johnson and Sinclair (1969) which are called the algebraicspectral subspaces.For a subset F of C and a linear map T on a vector space X , consider the

collection of all linear subspacesY of X with the property that (T − λ)Y = Y forevery λ ∈ C \ F. The linear span E T (F) of all such subspaces Y will evidentlyitself have the property that (T − λ)E T (F) = E T (F) for every λ ∈ C \ F.Thisshows that E T (F) is the largest linear subspace with respect to this surjectivitycondition.A simple example: if the map T is injective then

E T (C \ {0}) =∞⋂n=1

T nX.

This intersection is often called the generalized range and denoted by T ∞X.Indeed, the inclusion⊆ is immediate, and the other one follows from the obser-vation that, by injectivity,

⋂∞n=1 T

nX = T (⋂∞n=1 T nX ).The space E T (C \ {0})is also studied in Part V, where it is called the algebraic core (Definition 26.1.5;the above formula for E T (C \ {0}), when T is injective, is a special case ofProposition 26.1.10).The space E T (∅) is of particular interest. It is the largest linear subspace Y

of X for which (T − λ)Y = Y for all λ ∈ C, and is called the largest divisiblesubspace for T . In the theory of automatic continuity, it is often crucial toexclude the existence of non-trivial divisible subspaces, because their presencetends to preclude the desired continuity conclusions. For this sort of reason, weare particularly interested in classes of operators T for which E T (∅) = {0}.Now getting back to the case of a bounded linear operator T ∈ B(X ), it is

easy to see that E T (F) = E T (F ∩ σ (T )) for all sets F ⊆ C. Furthermore, it isclear that XT (F) ⊆ E T (F) for all sets F ⊆ C . In general, these inclusions willbe strict, even when F = ∅. Indeed, XT (∅) = {0} precisely when T has SVEP,but even super-decomposable operators may well have non-trivial divisiblesubspaces.For an example of this, let T ∈ B(X ) be both quasi-nilpotent and injective.

By Proposition 22.2.2, T is super-decomposable. Thus XT (∅) = {0}, while itfollows from the observations we have just made that

E T (∅) = E T ((C \ {0}) ∩ {0}) = E T (C \ {0}) = T ∞X .The generalized range may be quite large, as illustrated, for instance, by theVolterra operator:


Example 22.2.6 The Volterra operator (cf. Part I, Exercise 1.5.6) We defineX := C([0, 1]) , and let T ∈ B(X ) be given by

(T f )(t) :=∫ t

0f (s) ds for all f ∈ X and t ∈ [0, 1] .

Then T is injective, compact, and quasi-nilpotent. In particular, it follows thatT is a super-decomposable operator with XT (∅) = {0}.On the other hand, fromthe identity E T (∅) = T ∞X , we may conclude that E T (∅) is the set of all in-finitely differentiable functions on [0, 1] which, along with all their derivatives,vanish at 0. Thus E T (∅) is strictly larger than XT (∅). �

As another typical example (where E T (∅) is small) we mention the caseof decomposable multiplication operators on semisimple commutative Banachalgebras. They will be discussed in Chapter 25, but if T is any multiplicationoperator on a commutative Banach algebra A, then it is easily seen that theintersection

⋂ {(T − λ)A : λ ∈ C} is contained in the radical of A, and hencemust be trivial when X is semisimple.Here is a simple special case in which we can readily give an explicit, al-

gebraic description of the local spectral subspaces. We shall mention a similarone a bit later, for generalized scalar operators (Theorem 22.3.2).

Proposition 22.2.7 Let T ∈ B(X ) be super-decomposable and suppose that⋂λ∈C

(T − λ)p X = {0} for some p ∈ N .

Then, for every closed set F ⊆ C,

XT (F) = E T (F) =⋂

λ∈C\F(T − λ)p Xx .

Proof The idea of proof is similar to that of Theorem 22.2.4: given a closed setF ⊆ C, let GT (F) be the intersection of the spaces (T − λ)p X where λ rangesover C \ F. The containments XT (F) ⊆ E T (F) ⊆ GT (F) are immediate and,to show the converse, it is enough to show thatGT (F) ⊆ XT (U ) for an arbitraryopen neighbourhood U of F.By definition of super-decomposability and properties of the spectral sub-

spaces, there is an operator R ∈ B(X ) forwhich T R = R T, RX ⊆ XT (C \ F),and such that (I − R)X ⊆ XT (U ). Hence to establish that GT (F) ⊆ XT (U ),it remains to be seen that RGT (F) = {0}. By the assumption on T, for this itsuffices to show that RGT (F) ⊆ (T − λ)p X for every λ ∈ C. If λ ∈ C \ F,


this inclusion follows from the very definition of GT (F) and the fact thatthe operators R and T commute. On the other hand, if λ ∈ F, the inclusionRGT (F) ⊆ (T − λ)p X follows from

RX ⊆ XT (C \ F) = (T − λ)XT (C \ F) .

Thus RGT (F) = {0}, and consequently GT (F) ⊆ XT (U ). �

22.3 Generalized scalar operators

We have mentioned already that an operator T ∈ B(X ) is generalized scalar ifT = �(Z ) for some continuous algebra homomorphism � : C∞(C)→ B(X )with�(1) = IX .We have also observed that, asA-scalar operators, generalizedscalar operators are super-decomposable, hence also decomposable.Muchmoreis true.The continuity of � means that there exist a compact subset Q of C, a

constant c ≥ 0, and k ∈ Z+ such that

‖�( f )‖ ≤ c ‖ f ‖k,Q for all f ∈ C∞(C) ,

where C is canonically identified with R2, and

‖ f ‖k,Q :=∑|α|≤k

1

α!sup {|D α f (λ)| : λ ∈ Q} for all f ∈ C∞(C) .

To start off, here is a specific example of a generalized scalar operator.

Example 22.3.1 Every surjective isometry T ∈ B(X ) is generalized scalar: afunctional calculus on C∞(C) for such an operator is given by

�( f ) :=∞∑

n=−∞f (n) T n for all f ∈ C∞(C) ,

where f (n) is the nth Fourier coefficient of the restriction of the function f tothe unit circle T, that is,

f (n) := 1

2π

∫ π

−πf (eiθ ) e−inθ dθ for all n ∈ Z .


If, as in Part I, Exercise 1.5.4, we let g(θ ) := f (e iθ ) for all θ ∈ [−π, π ],then, for every n ∈ Z \ {0}, repeated integrations by parts lead to the formula

f (n) = 1

(in) p1

2π

∫ π

−πg(p)(θ ) e−inθ dθ for all p ∈ Z+ .

This shows that | f (n)| is O(|n|−p) for any given p ∈ Z+. Since ‖T n‖ = 1 forall n ∈ Z, it follows that the infinite series

∞∑n=−∞

f (n) T n

converges in the operator norm of B(X ), and hence there is a well-definedcontinuous linear map � : C∞(C)→ B(X ) with �(T ) =∑ f (n)T n . (Thisargument works not just for invertible isometries, but also for any T ∈ B(X )that satisfies σ (T ) ⊆ T and has polynomial growth in the sense that there isa bound ‖T n‖ = O(|n|α) for some α ≥ 0, as |n| → ∞. In particular, everydoubly power-bounded operator is generalized scalar.) �

Incidentally, knowledge of the order of a generalized scalar operator T inB(X ) (which is defined to be the smallest k ∈ Z+ for which the estimatesinvolved in the above specification of continuity of � hold, for appropriatechoice of �, Q, and c) makes it possible to give a very precise description ofthe local spectral subspaces. The proof of that part of the next theorem is ratherinvolved, and we omit it here. It may be found in Laursen and Neumann (2000,Theorem 1.5.4).

Theorem 22.3.2 Let T ∈ B(X ) be a generalized scalar operator of orderk ∈ Z+. Then T is super-decomposable, and the representation

XT (F) = E T (F) =⋂

λ∈C\F(T − λ)p X

holds for every closed set F ⊆ C and every integer p for which p ≥ k + 3. Inparticular, T has no non-trivial divisible subspace. �

Finally, we may combine the observation of Example 22.3.1 with the fact(noted in Chapter 21) that a restriction of a decomposable operator to a closedinvariant subspace will have (β). The conclusion is that (β) holds for anyisometry. A simple proof comes from the following extension result, due toDouglas (1969).


Proposition 22.3.3 Let T ∈ B(X ) be an isometry. Then there is a Banachspace Y intowhich X may be isometrically embedded and an invertible isometryS ∈ B(Y ) which (via the embedding) extends T .

Proof (sketched) Define Y0 to be the vector space (pointwise operations) ofall sequences u = {u1, u2, . . . } in X for which um+1 = Tum for all sufficientlylarge m. Since T is an isometry, the norms ‖um‖ are, for each such sequence,eventually constant, so the definition p(u) = limm→∞ ‖um‖ provides us witha seminorm on Y0. If we quotient out N = {u ∈ Y0|p(u) = 0} and completethe quotient space Y0/N , we obtain a Banach space, say Y. It is easy to seethat the assignment x → {T nx}∞n=1/N yields an isometric embedding of Xinto Y .In Y0 we may define the map S0 by S0({um}∞m=1) = {Tum}∞m=1. Clearly, S0

induces an isometry S on Y. This isometry is invertible and extends T (Exercise22.4.5). �

Corollary 22.3.4 Every isometry has property (β). �

22.4 Exercises

In these exercises, let X be a Banach space, and consider an operator T inB(X ).

1. Show that, XT (F) = XT (F) for every closed set F ⊆ C if and only ifXT (∅) = {0}, and this happens precisely when T has SVEP.

2. Show that, for each λ ∈ C \ σsu(T ), there is a constant c > 0 for whichX = XT (C \ D(λ; c)). In particular, show that σ (T ) \ σsu(T ) ⊆ σp(T ), andσ (T ) = σsu(T ), if T has SVEP.

3. Show that σsu(T ) =⋃ {σT (x) : x ∈ X} .

4. Show that E T (F) = E T (F ∩ σ (T )) for all sets F ⊆ C.

5. Fill in the details of the proof of Proposition 22.3.3.


As already mentioned, a general source for virtually everything that is touchedon here is Laursen and Neumann (2000). A specific instance of this: as we sawin Example 22.3.1, if the operator T has powers (negative and positive) of poly-nomial growth, then T must be generalized scalar; in Laursen and Neumann(2000) there are more specific results along these lines: it turns out that, foroperators with spectrum in T, these two properties are essentially equivalent.


This result was originally due toColojoara and Foias, but Laursen andNeumann(2000, Theorem 1.5.12) has a simpler proof. There it is also shown that theseconditions are equivalent to the condition that σ (T ) ⊆ T provided that thefollowing norm estimate holds for the growth of the resolvent near the spectrum:there exist a constant d > 0 and an m ∈ N such that

‖(T − λ)−1‖ ≤ d |1− |λ||−m

for all λ ∈ C with |λ| �= 1.

23

Duality theory

In this chapter we shall explore the surprising relations between decomposabil-ity and properties (β) and (δ).

23.1 Duality between (β) and (δ)

Wehave already seen, in Theorem21.2.8, that T is decomposable if and only if ithas both (β) and (δ). But the main reason for emphasizing these two propertiesis not just that they together describe decomposability – surely (β), in particular,is too technical, and too non-intuitive, to gain fame just for that! Their mainconceptual raison d’etre is that they possess a remarkable dual relationship: anoperator will have one of them (either one!) precisely when its adjoint operatorhas the other one. This is the significant conclusion that is provided by theduality theory. (You should note that the adjoint operator is called the dualoperator in Part I.)The duality theory for operators that we are talking about here goes back to

Errett Bishop’s PhD thesis, published in Bishop (1959), where he developed aspectral theory for an arbitrary bounded linear operator on a reflexive Banachspace. Bishop called his development a duality theory, because the operatorand its adjoint are involved. In Bishop (1959) we can see, in more or less fullydeveloped form, much of what are now basic tools and concepts of the field,such as the glocal spectral subspaces, and conditions (β) and (δ).Therewas evena precursor of decomposability (which Bishop called duality theory of type 3).Theorem 21.2.8 (that T is decomposable if and only if it has both (β) and

(δ)) was of course not available to Bishop. But he did show that, if both T andT ′ satisfy (β), then T has a duality theory of type 3, that is, T is decomposable.This result was established without Bishop’s reflexivity condition by Frunza(1971). Bishop also showed that, if T ′ has (β), then T has (δ). This was a resultthat rather glaringly used reflexivity, because it showed that (T ′)′ has (δ).

226

23 Duality theory 227

The full duality theory owes its spectacular completion to Albrecht andEschmeier (1997). It contains the following results. They are not complicatedto state, but a lot of ingenuity lies behind their proofs. You will find reasonablyaccessible proofs in Laursen and Neumann (2000, Chapter 2).

Theorem 23.1.1 A continuous linear operator on a Banach space has (δ) ifand only if it is a quotient of a decomposable operator with respect to a closedinvariant subspace. �

Theorem 23.1.2 A continuous linear operator on a Banach space has (β) ifand only if it is a restriction of a decomposable operator to a closed invariantsubspace. �

Incidentally, the latter theorem explains why operators with (β) are oftencalled sub-decomposable operators.In both theorems, the if parts are easy enough. The approach to proving, say,

Theorem 23.1.2 may be explained in fairly non-technical terms as follows. Atits root is a pair of 3-space lemmas both of which may be combined into thefollowing statement. The proof for the (β) case is easy, the (δ) case is not.

Lemma 23.1.3 Consider a commutative diagram of continuous linear oper-ators between Banach spaces

0 −→ XJ−→ Y

Q−→ Z −→ 0

A |↓ B |↓ C |↓0 −→ X −→

JY −→

QZ −→ 0

with exact rows. If both A and C have (β) (or (δ)) then so does B. �

Now, through considerable technical effort, it may be established that for anarbitrary T there is a particular commutative diagram

0 −→ XJ−→ Y

Q−→ Z −→ 0

T |↓ B |↓ C |↓0 −→ X −→

JY −→

QZ −→ 0

with exact rows, in which the operator C is generalized scalar, in particular de-composable, and B, as a quotient of a generalized scalar operator (not displayedin this diagram), has (δ). Now, if we suppose that T has (β), then the 3-space


lemma implies that B will also have (β), and hence B is decomposable. Thisestablishes that T is the restriction of a decomposable operator.Although the next result would be an easy consequence of Theorem 23.1.5,

we state it separately, because in the actual development of the duality theoryit is used in the proof of Theorem 23.1.5 – and doing so makes it possible forus to sketch quite a bit of the arguments.

Theorem 23.1.4 An operator T ∈ B(X ) is decomposable if and only if theadjoint T ′ ∈ B(X ′) is decomposable. �

The proof may be found in Laursen and Neumann (2000, Section 2.5). Nottoo surprisingly it makes extensive use of annihilators and preannihilators oflocal spectral subspaces. The only if part begins by showing that, if T has theproperty that all its local spectral subspaces XT (F) are closed, for every closedF ⊆ C (this is called property (C) for T ), then X ′ = XT (C\U )⊥ + XT (C\V )⊥for every open cover {U, V } of C. This is followed by establishing that, ifT is decomposable, then X ′T ′ (F) = XT (C\F)⊥. Finally, this then implies thatT ′ is decomposable. To establish that decomposability of T ′ implies the sameproperty for T is somewhat more delicate, but the main step consists in showingthat, on the assumption that T ′ have SVEP, X ′T ′ (F) is norm-closed if and onlyif it is weak-∗ closed.

Theorem 23.1.5 A operator T ∈ B(X ) will have one of the properties (β) or(δ) if and only if its adjoint T ′ ∈ B(X ′) has the other one.

Proof We sketch how to show that, if T has (β), then T ′ has (δ). If T has(β), then, by Theorem 23.1.2, T has a decomposable extension, say S. ByTheorem 23.1.4, the adjoint operator S′ is also decomposable. Moreover, T ′

may be realized as a quotient of S′. Hence, by Theorem 23.1.1, T ′ has (δ). �

The hardest part of all this, by far, is the proof that, if T ′ has (β), then T has(δ). You may wish to consult Laursen and Neumann (2000, Chapter 2).

23.2 Exercises

1. Fill in the details of the sketch of the part of the proof of Theorem 23.1.5that is included above.

2. Use Theorems 23.1.2, 23.1.1, and 23.1.4 to show that, if T has (δ), then T ′

has (β). Then prove that, if T ′ has (δ), then T has (β).

23 Duality theory 229


The actual functional model is explained in some detail in the Appendix toPart IV, ‘The functional model’. A fairly elementary account, from which thisbackground material is culled, is in Laursen and Neumann (2000, Chapter 2).The complete duality theory was first described in Eschmeier’s Habilitations-schrift and in Albrecht and Eschmeier (1997).The properties (β) and (δ) have been used in some of the most general so-

lutions of the invariant-subspace problem. Here is a sample, from Eschmeierand Prunaru (1990): if T ∈ B(X ) has (β) or (δ) and if V ⊆ C is open andbounded, then if σ (T ) is dominating in V (a technical condition that is satis-fied, for instance, when σ (T ) has non-empty interior), then T has a non-trivialclosed invariant subspace. Moreover, if the smaller set consisting of the essen-tial spectrum of T is dominating in V (the essential spectrum is defined at thebeginning of the next chapter), then T has many closed invariant subspaces.These invariant-subspace results, and much more, are described in Part III ofthis book.

24

Preservation of spectra and index

It is well known that, if two linear operators on a finite-dimensional space aresimilar, then their spectra are equal, and that this equality extends to distin-guished parts of the spectra, such as the point spectra, etc. Here we considernatural extensions of this situation, where similarity is replaced by weakerconditions, and where concepts and techniques from local spectral theory areused.

24.1 An intertwining result

We will concentrate on proving the following result.

Theorem 24.1.1 Suppose that S ∈ B(Y ) and T ∈ B(X ) both have property(β) and that A ∈ B(X, Y ) and B ∈ B(Y, X ) are operators with dense rangesfor which SA = AT and T B = BS. Then σ (T ) = σ (S), σe(T ) = σe(S) andind(T − λ) = ind(S − λ) for all λ ∈ ρe(T ).

First we shall explain the terms. Recall that an operator T ∈ B(X ) is aFredholm operator (or just Fredholm) if its null space ker T is of finite di-mension and its range T X is of finite codimension in X ; the range is thenautomatically closed (Exercise 24.3.1). The essential spectrum σe(T ) of T isthe set of complex numbers λ for which T − λ is not Fredholm (often expressedthis way: λ is not a Fredholm point for T .) Fredholm theory is developed morefully in Part V.A relation such as SA = AT is described by saying that A intertwines the

pair (S, T ). If A is a bijection, then S and T are similar, and A is a similarity. Aswe mentioned, a similarity preserves spectra, and even the finer structure of thespectra (point spectrum, approximate point spectrum, Fredholm points, etc.).Here we shall see what happens if the conditions on the intertwining operator

230

24 Preservation of spectra and index 231

are weakened, for example to that of being a quasi-affinity, that is, an injectivemap with dense range, or even if both (S, T ) and (T, S) are intertwined byquasi-affinities; in this latter case the operators S and T are said to be quasi-similar.Note, incidentally, that the answer provided by Theorem 24.1.1 is madeon intertwining assumptions that are slightly weaker than quasi-similarity.The index of a Fredholm operator T ∈ B(X ) is

ind(T ) := dim ker T − dim(X/T X ) .

Here are a few basic results from classical Fredholm theory: an operatorT ∈ B(X ) is Fredholm if andonly if its adjoint operatorT ′ ∈ B(X ′) is Fredholm;in fact, since the dimension of the null space of one equals the codimension ofthe range of the other, we see that

ind(T ′) = − ind(T ) .

The product of two Fredholm operators is again Fredholm. In particular,every power of a Fredholm operator is Fredholm. The index theorem says thatthe index is a homomorphism, that is,

ind(T S) = ind(T )+ ind(S)

for all Fredholm operators T, S ∈ B(X ). Since invertible operators areFredholm operators of index zero, it follows that the Fredholm property aswell as the index are preserved under similarity.Does this hold for quasi-similarities also?

Example 24.1.2 (Fialkow (1977), here reproduced from Laursen andNeumann 2000, Example 3.7.12). Let T be the unilateral right shift on 2(N),and let S be the unilateral weighted right shift on 2(N) with weight sequence(1/(n + 1))n∈N.Then T has (β), σ (T ) = D, and σe(T ) = T,while S is compactwith σ (S) = σe(S) = {0}. Thus σe(T ) and σe(S) are disjoint. However, the pair(S, T ) is intertwined by a quasi-affinity, namely the multiplication operator Aon 2(N) given by

Ax := (xn/n!)n∈N for all x = (xn)n∈N ∈ 2(N).

It is easy to see that A is injective, has dense range, and that SA = A T . �

Even when (β) or (δ) is assumed, very little is true about inclusions for theessential spectra of one-sidedly intertwined operators,


Example 24.1.3 Let H p(D) be the classical Hardy space of all complex-valued analytic functions f defined on the open unit disc D for which

sup

{∫ π

−π| f (re iθ )| p dθ : 0 ≤ r < 1

}<∞ .

Observe that, for any choice of p, q such that 1 < p < q <∞, the canonicalembedding A of H q (D) into H p(D) is injective and it has dense range (thepolynomials on D are dense in H p(D)). For arbitrary 1 < p <∞, the Cesarooperator Cp on the Hardy space H p(D) is defined by

(Cp f )(λ) := 1λ

∫ λ

0

f (ζ )

1− ζ dζ for all f ∈ H p(D) and λ ∈ D .

In Miller, Miller and Smith (1998) it is shown that Cp has (β), that

σ (Cp) = D(p/2; p/2),

and that σe(Cp) = ∂D(p/2; p/2). The intertwining Cp A = ACq is clear, sohere we have one-sided intertwining, but no nice containment for the essentialspectra; in fact, we have σe(Cp) ∩ σe(Cq ) = {0}. �

The proof of Theorem 24.1.1 is relatively elementary, but lengthy. It beginsby using an observation about restrictions of Fredholm operators to certaininvariant subspaces.

Proposition 24.1.4 Suppose that T ∈ B(X ), and that Z ⊆ X is a T -invariantclosed linear subspace of finite codimension in X. Then T is a Fredholm op-erator on X if and only if T | Z is a Fredholm operator on Z . Moreover, if thetwo operators are both Fredholm, they have the same index.

Proof Pick a projection P ∈ B(X ) with range Z , and observe that the space(I − P)X = ker P is finite-dimensional. Since T (I − P) is of finite-rank,and T equals T P + T (I − P), it follows that T and T P are Fredholmsimultaneously, and that, in this case, ind(T ) = ind(T P). Moreover, since aneasy calculation will reveal that we have ker(T P) = ker(T | Z )⊕ ker P and(T P)(X ) = (T | Z )(Z ),we conclude that T P and T | Z also are Fredholm si-multaneously and that, when this happens, they will have the same index. �

Another fundamental fact of Fredholm theory, the punctured disc theorem,says that, if T is Fredholm, then there is a positive number c with the propertythat, for each λ ∈ D(0; c), the operator T − λ is Fredholm, and the quanti-ties dim ker(T − λ) and dim(X/(T − λ)X ) are constant on the punctured discD(0; c) \ {0}.Moreover,dim ker(T − λ) ≤ dim ker T and dim(X/(T − λ)X ) ≤ dim(X/T X )


for all λ ∈ D(0; c). The jump of T is then the non-negative integer

dim ker T − limλ→0dim ker(T − λ) = dim(X/T X )− lim

λ→0dim(X/(T − λ)X ) ,

and the zero-jump theorem says that, for a Fredholm operator T with positivejump, there exists an invariant closed linear subspace of finite codimension onwhich the restriction of T is Fredholm, with jump zero. Versions more generalthan we need here may be found in Kato (1958) and West (1990). See alsoPart V.The zero-jump theorem that we will use, Proposition 24.1.6, has a simple

local-spectral-theory-based proof. We make one observation first.

Lemma 24.1.5 Every surjective operator with complemented kernel is rightinvertible.

Proof If T ∈ B(X ) is surjective and if there is a closed subspace Y for whichX = ker T ⊕ Y, then, for every x ∈ X, there is a unique element y ∈ Y suchthat T y = x . It is clear that T |Y : Y → X is a bijection, so it has an inverse, sayW ∈ B(X, Y ).Wemay regardW as a bounded linear operator on X.ObviouslyTW = IX . �

The converse of the above lemma happens to be true as well, but we shallnot need this.

Proposition 24.1.6 Given a Fredholm operator T ∈ B(X ), there are numbersn ∈ Z+ and c > 0 for which the dimension of ker(T − λ) | T nX is constant forall λ ∈ D(0; c). This constant value is dim ker(T − λ) for all λ ∈ D(0; c) \ {0}.If T also has SV E P, then this constant value is 0.

Proof If we add the assumption that T be surjective, this is quite easy: byLemma 24.1.5 there is an operator S ∈ B(X ) for which T S = I.The invertiblesform an open subset of B(X ), so the operator (T − λ)S is invertible for all λ insome open discD(0; c); in fact, by standard arguments (see Part I, Theorem1.4.2(iii)), this holds for c = 1/ ‖S‖ . By shrinking the radius c > 0, if necessary,we may suppose that D(0; c) ⊆ ρe(T ). Thus T − λ is a surjective Fredholmoperator for all λ ∈ D(0; c). Because the index is locally constant, it followsthat dim ker(T − λ) = dim ker T for every λ ∈ D(0; c).Now we drop the surjectivity assumption on T . Recall the generalized range

T ∞X :=⋂{

T nX : n ∈ N}.


It can be proved that R := T | T ∞X is surjective and Fredholm (Exercise24.3.2). So the first part shows that there is a c > 0 such that dim ker(R − λ) isconstant for all λ ∈ D(0; c). If we choose an n ∈ Z+ so that

ker T ∩ T nX = ker T ∩ T ∞X ,then we see that the constant dim ker(R − λ) is dim ker T | T nX. However, forevery non-zero λ ∈ C, ker(T − λ) ⊆ T ∞X , and therefore

ker(T − λ) = ker(T | T nX − λ) = ker(R − λ) .This shows that the constant value is dim ker(T − λ) for all λ ∈ D(0; c) \ {0}.Finally, if T also has SVEP, then its restriction R is a surjective operator with

SVEP. Thus ker R = {0} (Exercise 24.3.3. Exercise before Proposition 22.1.3:if ker R is non-trivial then surjectivity of R allows the construction of a powerseries with sum f, say, for which (R − µ) f (µ) = 0 in a neighbourhood of 0)and the last assertion follows. �

The principal tool in the rest of the proof is the next lemma. For an open setU ⊆ C and a W ∈ B(X, Y ), recall from Chapter 21 the notation

W ' : H (U, X )→ H (U, Y )

for the operator given by (W ' f )(λ) = W ( f (λ)) for all f ∈ H (U, X ) andλ ∈ U.

Lemma 24.1.7 Suppose that T ∈ B(X ) is Fredholm, and that the operatorC ∈ B(X ) commutes with T and has dense range. Then there is an open discV := D(0; r ) for which C induces a surjection

C ♦ : H (V, X )/TV H (V, X )→ H (V, X )/TV H (V, X )

by the assignment

C ♦( f + TV H (V, X )) := C ' f + TV H (V, X ) for all f ∈ H (V, X ).Additionally, if T is also injective, then r > 0 may be chosen so that C ♦ isbijective.

Proof Evidently, since C T = T C, the definition of C ♦ makes sense. Therange T X is of finte codimension, so there is a finite-dimensional subspace Fof X for which X = T X ⊕ F. Using this splitting, we observe that

CX = C T X + CF = TCX + CF ⊆ T X + CF.The space T X + CF is closed and C has dense range, so X = T X + CF. Inparticular, the dimension of CF is at least as big as that of F, that is, C | F isinjective and X = T X ⊕ CF. Similarly, X = T X ⊕ C nF for every n ∈ N.


If we now define an analytic operator function η on C taking its values inB(X × F, X ) by

η(λ)(x, u) := (T − λ)x + Cu for all λ ∈ C and (x, u) ∈ X × F,then we have shown that η(0) is surjective. Moreover, since T X ∩ C F = {0},the space ker η(0) = ker T × {0} is finite-dimensional. It follows, by Lemma24.1.5, that η(0) has a continuous linear right inverse, so that η(0)S = IX forsome S ∈ B(X, X × F). Continuity of η then implies that invertibility holdsfor η(λ)S for all λ near 0, that is, there is an r > 0 such that η(λ)S is invertiblefor every λ ∈ V := D(0; r ). Thus, given any f ∈ H (V, X ), the assignment

( f1(λ), f2(λ)) := S (η(λ)S)−1 f (λ) for all λ ∈ Vyields analytic functions f1 ∈ H (V, X ) and f2 ∈ H (V, F) such that we havef = TV f1 + C ' f2. The surjectivity of C ♦ is now obvious.Next we add the assumption that T is also injective, and we define

ξ (λ)(x, u) := (T − λ)x + u for all λ ∈ C and (x, u) ∈ X × F.The function ξ : C → B(X × F, X ) is analytic, and ξ (0) is bijective. Hence,shrinking the radius of V = D(0; r ), if necessary, we may suppose that, forevery λ in V, the operator ξ (λ) is both right and left invertible, that is bijective.This implies that the operator T − λ is injective and Fredholm, and that(T − λ)X + F = X for all λ ∈ V . Since, by continuity of the index,dim(X/(T − λ)X ) = − ind(T − λ) = − ind(T ) = dim F for all λ ∈ V,we conclude that, in fact, X = (T − λ)X ⊕ F for all λ ∈ V, and the sameargument as before then shows that (T − λ)X ∩ C nF = {0} for every n ∈ N.

Now, if f, g ∈ H (V, X ) are functions for which C ' f = TV g, then, writing, asabove, f = TV f1 + C ' f2 for suitable f1 ∈ H (V, X ) and f2 ∈ H (V, F),we seethat (C 2) ' f2 ∈ TV H (V, X ), and consequently f2 = 0, so that f ∈ TV H (V, X ).This shows that C ♦ is injective, and therefore bijective. �

Proof of Theorem 24.1.1 We begin by establishing the equality of the twospectra. For this we need two useful observations:

1. Let T ∈ B(X ) be arbitrary. For each closed F for which F ∩ σ (T ) = ∅,we have XT (F) = {0}. This follows from Liouville’s classic theorem: infact, if x = (T − λ) f (λ) for all λ ∈ C \ F, where f is analytic, then in aneighbourhood of F we must have f (λ) = (T − λ)−1x . This implies that fmay be extended analytically (by means of the expression (T − λ)−1x) toall ofC, and since this yields a bounded entire function, f must be constant.The only possible constant is 0.


2. If T ∈ B(X ), S ∈ B(Y ), and if A ∈ B(X, Y ) intertwines (S, T ), thenσS(Ax) ⊆ σT (x) for every x ∈ X. This is immediate from the fact that, if theequation x = (T − λ) f (λ) holds, with f analytic, in some open subset ofC, then Ax = (S − λ)A f (λ) in that same subset, and A f is analytic. By thesame argument we conclude that, if F ⊆ C is an arbitrary closed set, thenAXT (F) ⊆ YS(F).

Now suppose that T and S are intertwined by A, as in Observation 2, andsuppose that A is injective (this part of the argument will soon be applied to theadjoint operators, thus the injectivity assumption, which is dual to the densityassumption of Theorem 24.1.1). If λ ∈ σp(T ), then

{0} �= ker(T − λ) ⊆ XT ({λ})(the last inclusion is Exercise 24.3.4), and hence {0} �= AXT ({λ}) ⊆ YS({λ}).But then, byObservation 1, λ ∈ σ (S).Wehave shown that σp(T ) ⊆ σ (S). Sincewe have σ (T ) = σp(T ) ∪ σsu(T ), we next consider points of the surjectivityspectrum σsu(T ).Continuing for a little while in our dual mode, we impose the assumption

that T has (δ). LetU be an open neighbourhood of σ (S) and pick another openset V with closure V disjoint from σ (S) such that C = U ∪ V . By Observation1, YS(V ) = {0}, and consequently, by the injectivity of A, also XT (V ) = {0}.By using (δ), we see that

X = XT (U )+ XT (V ) ,

and hence X = XT (U ). An immediate consequence of this is that σsu(T ) ⊆ U .Since the set U is an arbitrary neighbourhood of σ (S), we may conclude thatσsu(T ) ⊆ σ (S). All in all σ (T ) ⊆ σ (S), if T has (δ) and A is injective.Now we position ourselves in the setting of Theorem 24.1.1, and suppose

that S and T have (β) and that the intertwining maps A and B have denseranges. Then S′ and T ′ have (δ), they are intertwined by B ′ and A′, and thesetwo maps are both injective. It follows that σ (T ′) = σ (S′), and consequentlyσ (T ) = σ (S).To establish the claim about the essential spectra and the index, we must

show that if T is Fredholm, then so is S, and ind(T ) ≤ ind(S).If we replace X by T nX, where n is specified in Proposition 24.1.6, and T

by T | T nX, then this restriction also has (β).Moreover, by Proposition 24.1.4,T | T nX is a Fredholm operator with the same index as T .What will change in the passage from S to S | S nY ? First of all, S | S nY

inherits (β) from S.Moreover, obviously A | T nX maps into S nY , densely, andB | S nY maps into T nX, also densely. The originally assumed intertwining


relationships hold for the four restrictions. Since T ′n is Fredholm andA′ ker S ′n ⊆ ker T ′n, it follows from the injectivity of the adjoint A ′ thatker S ∗n is finite-dimensional. The space S nY is the pre-annihilator of ker S ′n,hence it is of finite codimension (this is a consequence of the annihilator the-orem; see, for example, Laursen and Neumann (2000, Theorem A.1.8)). Thus,by Proposition 24.1.4, S is Fredholm precisely when S | S nY is Fredholm.Moreover, when they are, S and S | S nY have the same index.Consequently, we may continue the argument with T | T nX. Because T has

SVEP, this means that, by Proposition 24.1.6, we may invoke the additionalassumption that T is injective.We shall do this, but continue using the shortersymbols T, S, A, B, rather than those of their restrictions.The operator BA commuteswith T, and has dense range.Consequently, since

T is injective, Lemma 24.1.7 tells us that there is an open disc V := D(0; r ) forwhich

(BA)♦ : H (V, X )/TV H (V, X )→ H (V, X )/TV H (V, X ) ,

defined, as before, by (BA)♦( f + TV H (V, X )) := (BA) # f + TV H (V, X ) forall f ∈ H (V, X ), is a bijection.Since T and S have (β),we know that the ranges TV H (V, X ) and SV H (V, Y )

are closed (cf. the remark immediately after Definition 21.2.1). Hence

H (V, X )/TV H (V, X ) and H (V, Y )/SV H (V, Y )

are Frechet spaces in the natural quotient space topology. By the open mappingtheorem, the continuous bijection (BA)♦ is a topological linear isomorphism.The canonically defined map

A♦ : H (V, X )/TV H (V, X )→ H (V, Y )/SV H (V, Y )

is obviously continuous, and we now show that A♦ is a bijection. Since(BA)♦ = B ♦A♦, and (BA)♦ is bijective, A♦ is certainly injective. Moreover,as a map from the range of A♦ to H (V, X )/TV H (V, X ), B ♦ is a continuousbijection. Since the linear span of functions of the form λ �→ f (λ)y, V → Y (asf ranges through H (V ) and y through Y ) is dense in H (V, Y ), and A has denserange, so does A♦. But this range is closed because (BA)♦ is a topologicalisomorphism (so that the range of A♦ is the pre-image with respect to B ♦ ofH (V, X )/TV H (V, X )), and therefore A♦ is surjective. It then follows that also

B ♦ : H (V, Y )/SV H (V, Y )→ H (V, X )/TV H (V, X )

is bijective.


Since A♦ is surjective, there is, for every g ∈ H (V, Y ), a function f inH (V, X ) for which

g − A # f ∈ SV H (V, Y ) .

In particular, if we evaluate this expression at 0, and use the fact that g isarbitrary, we see that, for every y ∈ Y, there are elements x ∈ X and z ∈ Y sothat y = S z + Ax .ThusY = S Y + AX. If F , as before, is a finite-dimensionalsubspace of X for which X = T X ⊕ F, then

AX ⊆ A T X + AF = SAX + AF ⊆ S Y + AF,

and therefore Y = S Y + AF. This shows that S Y is of finite codimension inY. In fact, dim(Y/S Y ) ≤ dim(AF) ≤ dim F.Moreover, if y ∈ ker S and we view y as an analytic function on V, then its

class y + SV H (V, Y ) satisfies B ♦(y + SV H (V, Y )) = 0; this holds becauseT B y = BS y = 0, and T is assumed to be injective. We have already estab-lished that B ♦ is injective, so it follows that y ∈ SV H (V, Y ), and therefore

y ∈ YS({0}) ∩ YS(C \ V ) = YS(∅) = {0},

because S has SVEP. This shows that S is injective, and hence establishes thatS is Fredholm. Moreover, it follows that

ind(S) = − dim(Y/SY ) ≥ − dim F = ind(T ),

and therefore ind(T ) ≤ ind(S). By symmetry, we are done. �

There is an analogous result for operators with (δ). It says that if S ∈ B(Y )and T ∈ B(X ) both have (δ), and if A ∈ B(X, Y ) and B ∈ B(Y, X ) are injectiveoperators for which SA = AT and T B = BS, then σ (T ) = σ (S), σe(T ) =σe(S) and

ind(T − λ) = ind(S − λ) for all λ ∈ ρe(T ) .

This is proved by relying heavily on the natural dualities; but you should beaware that, although the adjoint of an injective operator always has weak-∗dense range, this range is not necessarily norm-dense. Hence the (δ) result isnot an immediate consequence of Theorem 23.1.5 via the duality between theproperties (β) and (δ). The details may be found in Laursen and Neumann(2000, Section 3.7).


24.2 Automatic continuity

Automatic continuity theory for homomorphisms and derivations is discussedin Part I, Chapter 5. Just to give an impression of the kind of automatic con-tinuity results that may be established on the basis of local spectral theory, Iquote one theorem. There is not sufficient space here to dwell on how it isproved, but it and others like it may be found in Laursen and Neumann (2000,Chapter 5). However, significant in the arguments is the simple observation thatthe algebraic spectral subspaces, discussed immediately after Example 22.2.5,have the property (analogous to what is true for both the local and the glocalspectral subspaces and continuous�, in the notation of this next theorem) that�ET (F) ⊆ ES(F), when S� = �T .

Theorem 24.2.1 Suppose that T ∈ B(X ) is decomposable, while S ∈ B(Y )is generalized scalar. Suppose also that� : X → Y is a linear transformationfor which S� = �T . Then there is a finite set � ⊂ C such that �|XT (F) iscontinuous for every closed F ⊆ C which is disjoint from �. Moreover, thereis a non-zero polynomial p, all of whose roots are eigenvalues of S, such thatthe composition p(S)� : X → Y is continuous. In particular, � is continuousif S has no eigenvalues. �

The finite set � is called the singularity set. The condition on T may berelaxed to that of assuming (δ), while that on S could be replaced by super-decomposable and ES(∅) = {0}, and even more general conditions. For otherresults on automatic continuity, see Part I, Chapter 5.

24.3 Exercises

1. Show that, if the range T X of the operator T is of finite codimension in X,then the range is automatically closed.

2. Referring to the proof of Theorem 24.1.6, assume that T is Fredholm. Provethat R := T | T ∞X is surjective and Fredholm.

3. (cf. the exercise before Proposition 22.1.3) Show that, if R is a surjective,non-injective operator, then there is a power series with sum f, say, for which(R − µ) f (µ) = 0 in a neighbourhood of 0.

4. Prove the inclusion ker(T − λ) ⊆ XT ({λ}) for an arbitrary operator T .


The issue of spectral overlaps has received a lot of attention. For general Banachspaces, the classical Lumer andRosenblum (1959) result says that two operators


(S, T ) intertwined by a non-zero intertwinermust have overlapping spectra, andthis result was refined by Davis and Rosenthal (1974) to σsu(T ) ∩ σap(S) �= ∅.Both of these results are provable, in even greater generality, by local spectralmethods. Laursen and Neumann (2000, Chapter 3) has details, and work on thismay also be found in Laursen and Neumann (1994): it is not even necessary tosuppose that SA = AT . There are ways of saying how close parts of the twospectra must be in terms of the commutator SA − AT .Fialkow (1977) also showed that, in contrast to Example 24.1.2, the essential

spectra of arbitrary quasi-similar operators on a separable Hilbert space alwayshave non-empty intersection. Moreover, Herrero (1988) showed that, for everypair of quasi-similar operators T ∈ B(X ) and S ∈ B(Y ) on the Banach spacesX and Y, each connected component of σe(T ) touches σe(S), and vice versa.This result now has an attractive sheaf-theoretic proof (see Eschmeier 2000).In this chapter we presented a proof of the (β) part of Theorem 24.1.1.

This proof is quite elementary compared with previous ones. Theorem 24.1.1has a background that may be traced to a problem from Clary (1975): havingestablished the equality of the spectra for quasi-similar hyponormal opera-tors (hyponormal T means that the commutator T ∗T − T T ∗ ≥ 0), Clary askedwhether this extends to the essential spectra. This problem received consider-able attention. The final solution, due to Putinar (1992) and, independently, toYang (1993), revealed that the assumption of hyponormal is not really needed:all that matters is that hyponormal operators have (β). But generally, of course,Hilbert space operators have their own theory.

25

Multipliers on commutative Banach algebras

25.1 Multipliers

In this chapter I want to give an impression of what sort of information becomesavailable via local spectral theory when it is applied to a particular class of op-erators, namely the multipliers on a commutative Banach algebra. Throughoutthis chapter, the letter A will denote a commutative, complex Banach algebra.

Definition 25.1.1 A linear map T : A→ A is a multiplier if aT (b) = T (a)bfor all a, b ∈ A. The set of multipliers on A is denoted by M(A).

The most obvious example, given A, is the multiplication operator La in-duced by a fixed element a ∈ A, that is, the operator La(b) := ab for all b ∈ A.If A has a unit e (A is unital) then everymultiplier T is amultiplication operator.In this case T = LT (e) (Exercise 25.3.2).If the map a→ La : A→ B(A) is injective (faithful), then A is said to be

a faithful algebra. Every unital algebra is faithful, as is every semisimple, andalso every semiprime algebra – the latter term means, in our commutative case,that the algebra contains no non-zero nilpotent elements.

Example 25.1.2 Let A := C0() be the Banach algebra of all continuouscomplex-valued functions vanishing at ∞ on the locally compact Hausdorffspace , and let f ∈ Cb() be a bounded continuous function on . ThenT := L f (notation self-explanatory by now) is a multiplier. Conversely, it canbe shown that a multiplier gives rise to a bounded continuous function on with respect to which the multiplier acts by pointwise multiplication (Exercise25.3.4). �

The group algebras are perhaps the most important class of examples.

241


Example 25.1.3 LetG be a locally compact abelian group, withHaarmeasurem, and let L1(G) denote the group algebra of G, as described in Parts I and II.Also let M(G) denote the measure algebra, that is, the algebra of all bounded,regular, complex-valuedmeasures onG.Equippedwith the total variation norm,given by ‖µ‖ := |µ| (G), M(G) is a commutative unital Banach algebra (pointmass at the unit e ∈ G is the identity) and since the convolution of an elementf ∈ L1(G) and a measure µ ∈ M(G),

( f � µ)(s) =∫Gf (st−1) dµ(t) ,

is an element in L1(G), we see that every µ ∈ M(G) defines a multiplier onL1(G). It is a classical result of Wendel (1952) and of Helson (1953) (seeTheorem 9.1.11) that all multipliers on L1(G) arise this way. This shows thatM(L1(G)) = M(G). �

The next result (the proof of which is left to the reader, Exercise 25.3.5)claims, inter alia, that M(A) is a full subalgebra of B(A). This means that, ifT ∈ M(A) is invertible as an operator on A, that is, invertible in B(A), thenT−1 ∈ M(A). You should also note that the definition of a multiplier doesnot require the map to be a priori continuous, so the proposition contains anautomatic continuity result.

Proposition 25.1.4 Let A be faithful. Then the multiplier algebra M(A) is aclosed, commutative, unital, full subalgebra of B(A) which contains A as anideal. �

Here is our first explicit allusion to local spectral theory.

Proposition 25.1.5 Let A be semiprime. Then every multiplier T ∈ M(A) hasSVEP.

Proof We observe, to begin with, that ker T ∩ T A = {0}: if u ∈ ker T ∩ T A,then Tu = 0 and u = T v for some v ∈ A; hence (T v)2 = (T 2v) v = 0 v = 0,and therefore u = T v = 0, since A contains no non-zero nilpotent elements.Now, on an open subset U of the complex plane we consider an analytic

function f : U → A such that (T − λ) f (λ) = 0 for all λ ∈ U. For each λ ∈ U,it is clear that f (λ) ∈ ker(T − λ); moreover f (λ) ∈ (T − λ)A, by Lemma 22.1,and so

f (λ) ∈ ker(T − λ) ∩ (T − λ)A = {0} ,since T − λ is also a multiplier. Thus f (λ) = 0 for all λ ∈ U. This proves thatT has SVEP. �

25 Multipliers on commutative Banach algebras 243

We next proceed to study the extent to which decomposability is transferredby algebra homomorphisms. This development has some of the same flavouras Theorem 22.2.4. We suppose that A is faithful and that ϕ : A→ B(X ) is analgebra homomorphism. This turns X into a left A-module, with module actiondefined by a · x := ϕ(a)x for all a ∈ A and x ∈ X. Although the followingfacts may be established without continuity assumptions about ϕ, the simplerproofs presented here cover only this case.As our first step we extend the homomorphism ϕ from A to its multiplier

algebra M(A). For this to go through, we shall impose two conditions. The firstcondition is one of non-degeneracy.

Definition 25.1.6 The left A-module X is non-degenerate if, for each non-zerox ∈ X, there exists an element a ∈ A for which a · x is non-zero.

We also say that the homomorphism ϕ is non-degenerate. Note that whenthe algebra A is viewed as a module over itself (with module action given bythe algebra multiplication), non-degeneracy is the same as faithfulness.In the non-degenerate case, there is at most one algebra homomorphism

� : M(A)→ B(X )

that extends ϕ, in the sense that�(La) = ϕ(a) for all a ∈ A. Indeed, if�1 and�2 are two such extensions, then

ϕ(a) (�1(S)x −�2(S)x) = ϕ(Sa)x − ϕ(Sa)x = 0for all S ∈ M(A), x ∈ X, and a ∈ A,

and therefore �1 = �2 if ϕ is non-degenerate. Non-degenerateness will also,with much the same argument, establish that the extension � of ϕ maps theidentity operator on A to the identity operator on X.We shall also assume about the module action that AX = X. Here, for arbi-

trary subsets B ⊆ A and Y ⊆ X, we define

BY = lin{b · y; b ∈ B and y ∈ Y } ,

where lin denotes the linear span. Evidently, the condition that AX = X isfulfilled whenever

⋃ {ranϕ(a) : a ∈ A} = X.If A has an identity element e, and if ϕ(e) = IX , then X is clearly non-

degenerate and satisfies AX = X. Actually, these two conditions are fulfilledin lots of situations: for instance, if the homomorphism ϕ is continuous, and ifA has a bounded approximate identity (eλ)λ∈� for which ϕ(eλ)x → x for each


x ∈ X, then the left A-module X is obviously non-degenerate, and the Cohenfactorization theorem implies that AX = X.

Lemma 25.1.7 Let A be faithful, and consider an algebra homomorphismϕ : A→ B(X ) which provides us with the module action a · x := ϕ(a)x forall a ∈ A and x ∈ X. Suppose that X is non-degenerate and that AX = X. Ifx ∈ Xis represented as x =∑n

k=1 ϕ(ak)xk with ak ∈ A and xk ∈ X for eachk = 1, . . . , n, and if we define

�(S)x :=n∑k=1

ϕ(Sak)xk

for all S ∈ M(A), then � : M(A)→ B(X ) is a unital algebra homomorphismthat extends ϕ, in the sense that �(La) = ϕ(a) for every a ∈ A. The extension� is continuous on M(A) if and only if ϕ is continuous on A.

Proof Let S ∈ M(A). First we must make sure that the above definition of�(S) : X → X makes sense: since AX = X, every element x ∈ X has a repre-sentation x =∑n

k=1 ϕ(ak)xk , so if∑ n

k=1 ϕ(ak)xk = 0, then, for every a ∈ A,we see that

ϕ(a)

(n∑k=1

ϕ(Sak)xk

)= ϕ(Sa)

(n∑k=1

ϕ(ak)xk

)= 0 ,

because ϕ is a homomorphism and S is a multiplier. The module X is non-degenerate, so this implies that

∑ nk=1 ϕ(Sak)xk = 0.

The linearity of the operator �(S) is immediate, and, by the closed graphtheorem, �(S) is continuous: if un → 0 in A and �(S)un → v in X, thenclearly, for each a ∈ A, ϕ(Sa)un → 0 and ϕ(a)�(S)un → ϕ(a)v as n→∞ .

But a computation similar to the one above shows that

ϕ(a)�(S)un = ϕ(Sa)un for all n ∈ N .

Thus ϕ(a)v = 0 for each a ∈ A, and therefore v = 0 by non-degeneracy.It is straightforward to check that � is a unital algebra homomorphism that

extends ϕ, and it is clear that ϕ is continuous whenever � is. On the otherhand, if ϕ is continuous on A, then, for every x ∈ X, the defining expressionfor�(S)x shows that the map S �→ �(S)x is continuous on M(A). This, by theprinciple of uniform boundedness, is enough to establish continuity of �. �

The following theorem is a key result for the preservation of decomposabilityunder homomorphisms. I have chosen to show only a special case; there is a


rather more elaborate version of it, avoiding continuity assumptions on ϕ, inLaursen and Neumann (2000, Lemma 4.2.4).

Theorem 25.1.8 Let A have a bounded approximate identity (eλ)λ∈� andϕ : A→ B(X ) be a continuous algebra homomorphism for which ϕ(eλ)x → xfor each x ∈ X. If S ∈ M(A) is amultiplier with (δ) and SVEP, then the operatorT := �(S) ∈ B(X ) is decomposable, and

XT (F) = {x ∈ X : ϕ(a)x = 0 for all a ∈ A for which σS(a) ∩ F =∅} ,

for all closed sets F ⊆ C.

Proof We shall establish the decomposability of T by checking the verydefinition, and show that the spaces ZT (F) := {x ∈ X : ϕ(a)x = 0for all a ∈ A for which σS(a) ∩ F = ∅} give us the claimed spectral sub-spaces.Right away we note that, for every closed set F ⊆ C, ZT (F) is a closed

linear subspace of X with the property that �(R) ZT (F) ⊆ ZT (F) for eachR ∈ M(A) . In particular, ZT (F) is T -invariant.We first show that X = ZT (U )+ ZT (V ) for every open cover {U, V } of C.

ByCohen’s factorization theorem, AX = X, and wemay represent an arbitraryx ∈ X as x = ϕ(a0)x0 with a0 ∈ A and x0 ∈ X. Since S has (δ), there exist ele-ments u0, v0 ∈ A for which σS(u0) ⊆ U , σS(v0) ⊆ V , and a0 = u0 + v0. Thusx = ϕ(u0)x0 + ϕ(v0)x0.We claim that ϕ(u0)x0 belongs to ZT (U ): consider anarbitrary a ∈ A with the property that σS(a) ∩U = ∅. Since

σS(au0) ⊆ σS(a) ∩ σS(u0) ⊆ σS(a) ∩U = ∅,

au0 = 0, because S has SVEP. This implies that ϕ(a)ϕ(u0)x0 = 0, and hencewe have u ∈ ZT (U ). The same argument shows that v ∈ ZT (V ). We haveproved that

X = ZT (U )+ ZT (V ) .

We now claim that σ (T | ZT (F)) ⊆ F for every closed set F ⊆ C. To es-tablish this, fix an arbitrary open neighbourhood U of F, and observe thatAZT (F) = ZT (F), again byCohen factorization.Hence, given any x ∈ ZT (F),we have a representation of the form x = ϕ(a)z with a ∈ A and z ∈ ZT (F).Since S has (δ) and C = U ∪ (C \ F), we may write a = u + v with u, v ∈ Achosen so that σS(u) ⊆ U and σS(v) ∩ F = ∅.Because z is in ZT (F),we knowthat ϕ(v)z = 0, and hence x = ϕ(u)z. Since S has SVEP, and σS(u) ⊆ U, thereis an analytic function f : C \U → A for which (S − λ) f (λ) = u for every


λ ∈ C \U . Now apply the homomorphism �; this gives(T − λ)ϕ( f (λ)) = ϕ(u),

and therefore (T − λ)yλ = x, where we set yλ := ϕ( f (λ))z ∈ ZT (F) for allλ ∈ C \U .Consequently, for each λ ∈ C \U , the operator T − λmaps ZT (F)onto itself.Additionally, it does it injectively. To see this, let

λ ∈ C \U and z ∈ ZT (F) ∩ ker(T − λ) .Since X is non-degenerate, it suffices to show that ϕ(a)z = 0 for each a ∈ A.Clearly x := ϕ(a)z belongs to ZT (F). Hence the preceding argument suppliesan analytic function f : C \U → A for which (T − λ)ϕ( f (λ))z = x . Thus

x = ϕ( f (λ))(T − λ)z = 0 ,proving that (T − λ) | ZT (F) is injective. It follows that σ (T | ZT (F)) ⊆ U forevery open neighbourhood U of F, and therefore that σ (T | ZT (F)) ⊆ F.We conclude that T is decomposable, and that ZT (F) ⊆ XT (F) for every

closed set F ⊆ C.

The opposite inclusion ZT (F) ⊇ XT (F) follows easily from the continuityof ϕ. In fact, given any x ∈ XT (F) and a ∈ A for which σS(a) ∩ F = ∅, weobtain from the continuity of ϕ that ρS(a) ⊆ ρT (ϕ(a)x), and hence that

σT (ϕ(a)x) ⊆ σS(a) ∩ σT (x) ⊆ σS(a) ∩ F = ∅ .Since T has SVEP, this implies that ϕ(a)x = 0, and consequently x ∈ ZT (F).

�

Here is a pleasing observation which is immediate from what we have saidso far.

Corollary 25.1.9 Let A have a bounded approximate identity. Then a multi-plier on A is decomposable if and only if it has (δ) and SVEP. In particular, if Ais also semiprime, a multiplier on A is decomposable precisely when it has (δ).

Proof Simply let ϕ : A→ B(A) be the left regular representation given byϕ(a) := La for all a ∈ A. Then ϕ is a continuous algebra homomorphism, andthe extension of ϕ to the multiplier algebra M(A), as in Lemma 25.1.7, isthe identity map on M(A). By Theorem 25.1.8, every multiplier on A with(δ) and SVEP is decomposable. The last statement is immediate from this inconjunction with Proposition 25.1.5. �


25.2 The hull-kernel topology

The secondmain result of this chapter is Theorem 25.2.2. To appreciate all of it,it is necessary to realize that, in addition to theGelfand topology, there is anothertopology on the character space �A of A. And it is also worth knowing thatthe condition of continuity of the Gelfand transform of a given algebra elementwith respect to this other topology, which is the main assumption of Theorem25.2.2, is really a decomposability condition in disguise: in Proposition 25.2.5it is established that, if A is faithful and if T ∈ M(A) is a multiplier with (δ),then the restriction T |�A is hk-continuous. This of course applies in particularto a multiplication operator.The topology that we are referring to is introduced in the next proposition,

stated here without proof; it is called the hull-kernel topology, or hk-topology,of �A. This topology is defined via specification of the closure of any givensubset of the character space. This involves the following steps, already givenas Exercise 3.6.1 in Part I. For a set S ⊆ �A, the kernel of S is the ideal of Adefined as

k(S) := {a ∈ A : ϕ(a) = 0 for all ϕ ∈ S} ,

and, symmetrically, for an ideal J ⊆ A, the hull of J in �A is

h(J ) := {ϕ ∈ �A : ϕ(a) = 0 for all a ∈ J } .

For an arbitrary subset S of �A, we then define the hk-closure of S by

hk(S) := h(k(S)) = {ϕ ∈ �A : a(ϕ) = 0 for all a ∈ A with a ≡ 0 on S} .

Proposition 25.2.1 Let A be a commutative complex algebra. The sets of theform hk(S), where S ranges over the subsets of the character space�A, are theclosed sets of a topology, the hk-topology of�A. This topology has the followingproperties:

(i) for any S ⊆ �A, the closure of S in the hk-topology is precisely hk(S);(ii) the hk-topology is coarser than (or equal to) the Gelfand topology. �

As noted in Part I, these two topologies coincide exactly when A is regular(regular: whenever F is a Gelfand closed subset of �A and ϕ ∈ �A\F thenthere is a ∈ A for which a vanishes on F , but not at ϕ).With these basic facts listed, we can state the theorem.

Theorem 25.2.2 Suppose that A is semisimple and that a ∈ A is an elementfor which a is hk-continuous on �A. Then, for an arbitrary homomorphism


ϕ : A→ B(X ), the operator ϕ(a) ∈ B(X ) is super-decomposable on X. If,additionally, ϕ is non-degenerate, then

Xϕ(a)(F) ={x ∈ X : ϕ(u)x = 0 for every u ∈ A for which a (supp u) ∩ F = ∅} .

Note that one consequence of this is that, if a ∈ A is an element for whicha is hk-continuous on �A, then a defines a super-decomposable multiplicationoperator on A. The converse is also true, as we shall see in Proposition 25.2.5.The proof of Theorem 25.2.2 comes after the next two lemmas. Both of these

are classical; see Laursen and Neumann (2000, Section 4.3).

Lemma 25.2.3 The following statements are true for every commutativeBanach algebra A.(i) If S is a compact hull in �A, and if S ⊆ U1 ∪ · · · ∪Un for finitely many

hk-open sets U1, . . . ,Un ⊆ �A, then there are elements u1, . . . , un ∈ A suchthat u1 + · · · + un ≡ 1 on S and uk ≡ 0 on �A \Uk for k = 1, . . . , n.(ii) If S1 and S2 are two disjoint hulls in �A, and if S1 is compact in the

Gelfand topology, then there is an element u ∈ A for which u ≡ 1 on S1 andu ≡ 0 on S2.(iii) If S ⊆ �A is a hull such that there exist v ∈ A and δ > 0 for which

| v | ≥ δ on S, then there is an element w ∈ A for which vw ≡ 1 on S. �

Lemma 25.2.4 Suppose that A is semisimple and has an identity elemente, and suppose that ϕ : A→ B(X ) is a homomorphism such that ϕ(e) = I.Moreover, let a ∈ A be an element for which a is hk-continuous on �A. Then,for every open cover {U, V } of C, there exists an element r ∈ A for whichr ≡ 0 on a −1(C \U ), and for which r ≡ 1 on a −1(C \ V ), and is such thatσ (T | RX ) ⊆ U and σ (T | (I − R)X ) ⊆ V, where R, T ∈ B(X ) denote thecommuting operators given by R := ϕ(r ) and T := ϕ(a).

Proof Start with the open cover {U, V } of C. Choose open sets G, H ⊆ C forwhich G ⊆ U, H ⊆ V, and G ∪ H = C. Since the complements C \ G andC \ H are closed and disjoint, hk-continuity of a implies that their preimagesa −1(C \ G) and a −1(C \ H ) are hulls in�A, of course also disjoint. Moreover,since A has an identity, both sets are compact in the Gelfand topology of thecompact space �A. Consequently, by Lemma 25.2.3 (ii), there is r ∈ A forwhich r ≡ 0 on a −1(C \ G) and r ≡ 1 on a −1(C \ H ). Evidently, r satisfies


the first two conditions of this lemma, and the operator R := ϕ(r ) commuteswith T := ϕ(a).To show that σ (T | RX ) ⊆ U, look at an arbitrary λ ∈ C \U. The distance δ

from λ to G is strictly positive. Furthermore, |a − λe | ≥ δ on the hull a −1(G).Hence, by Lemma 25.2.3 (iii), there is a uλ ∈ A for which

(a − λ)uλ ≡ 1 on a −1(G) .

Since r ≡ 0 on a −1(C \ G), the identity (a − λ)uλ r ≡ r holds everywhereon the character space �A. Thus (a − λ)uλr = r, by semisimplicity of A.Now apply the homomorphism ϕ to the last identity. If Uλ := ϕ(uλ) ∈ B(X ),then

(T − λ)Uλ(Rx) = Uλ(T − λ)(Rx) = Rx for all x ∈ X ,

and therefore (T − λ)Uλ = Uλ(T − λ) = I on RX . This space isUλ-invariant,so λ ∈ ρ(T | RX ). This proves that σ (T | RX ) ⊆ U. A similar argument willshow that σ (T | (I − R)X ) ⊆ V . �

Proof of Theorem 25.2.2 First extend the homomorphism ϕ to the unitizationA# = A ⊕ C by defining

�(u + λe) := ϕ(u)+ λ I for all u ∈ A and λ ∈ C .

Clearly,� : A#→ B(X ) is a homomorphism with�(e) = I and�(u) = ϕ(u)for all u ∈ A. It may be shown that a remains hk-continuous on�A# , so Lemma25.2.4 establishes the super-decomposability of T .To prove the formula for the local spectral subspaces of T when ϕ is assumed

non-degenerate, we consider, for each closed set F ⊆ C, the space

ZT (F) := {x ∈ X : ϕ(u)x = 0 for all u ∈ A for which a (supp u) ∩ F = ∅} .

This is the space which is claimed to equal XT (F).Note that ZT (F) consists ofall x ∈ X which satisfy ϕ(u)x = 0 for all u ∈ A for which u ≡ 0 on a −1(U )for some open neighbourhood U of F, depending on u.The equality XT (F) = ZT (F) is established in several steps. The first step

is to show that ZT (F) ∩ ZT (G) = {0} for arbitrary closed and disjoint F,G inC. No number can belong to both F and G, so we may as well suppose that0 /∈ F. Then we may select open sets U, V ⊆ C such that F ⊆ U, G ⊆ V,0 /∈ U , and U ∩ V = ∅. By the hk-continuity of a on �A, the sets a −1(U ) and


a −1(V ) are disjoint hulls in �A. Moreover, if δ denotes the positive distancefrom 0 to U , then |a| ≥ δ on a −1(U ). This shows that a −1(U ) is Gelfandcompact in �A. By Lemma 25.2.3 (ii), there is an element u ∈ A for which

u ≡ 1 on a −1(U ) and u ≡ 0 on a −1(V ) .

Hence, for arbitrary c ∈ A, we have that c − cu ≡ 0 on a −1(U ) and cu ≡ 0on a −1(V ). For every x ∈ ZT (F) ∩ ZT (G), we get both that ϕ(c − cu)x = 0,and ϕ(cu)x = 0, and consequently that ϕ(c)x = 0 for all c ∈ A. The homo-morphism ϕ is non-degenerate, so it follows that x = 0. Thus we have as ourfirst step that the spaces ZT (F) and ZT (G) have trivial intersection.The next step consists in showing that XT (F) ⊆ ZT (F) for every closed set

F ⊆ C. To this end, consider an arbitrary u ∈ A for which u ≡ 0 on a −1(U )for some open neighbourhood U of F, and choose an open set V ⊆ C sothat F ∩ V = ∅ and U ∪ V = C. We now apply Lemma 25.2.4 to the opencover {U, V } of C and to the unitization of A and ϕ. Let r ∈ A# denote thecorresponding element provided byLemma25.2.4. Since r ≡ 0 on a −1(C \U ),wehave that ur ≡ 0on�A, andhence, by semisimplicity,ur = 0.Furthermore,with R := ϕ(r ) ∈ B(X ), σ (T | (I − R)X ) ⊆ V, and so (I − R)X ⊆ XT (V ).Because R T = T R, this implies that

(I − R)XT (F) ⊆ XT (F) ∩ XT (V ) = XT (F ∩ V ) = XT (∅) = {0} .

For each x ∈ XT (F), we conclude that (I − R)x = 0, hence x = Rx, andtherefore ϕ(u)x = ϕ(u)Rx = ϕ(ur )x = 0. Thus x ∈ ZT (F), and this showsthat XT (F) ⊆ ZT (F).It remains to establish that ZT (F) ⊆ XT (F), for every closed set F ⊆ C. For

this it is enough to prove that ZT (F) ⊆ XT (U ), for every openneighbourhoodUof F.As in the preceding paragraph, givenU , wemay choose an openV ⊆ C forwhich F ∩ V = ∅ andU ∪ V = C, and put R := �(r ) ∈ B(X ), where r ∈ A#denotes the element provided by Lemma 25.2.4, again applied to the open cover{U, V } ofC and to the unitization of A and ϕ. The two spectral inclusions fromLemma 25.2.4 imply that R X ⊆ XT (U ) and (I − R) X ⊆ XT (V ). Moreover,it is clear that the space ZT (F) is invariant under R. Gathering the results ofthe preceding two paragraphs, we obtain

(I − R) ZT (F) ⊆ ZT (F) ∩ XT (V ) ⊆ ZT (F) ∩ ZT (V ) = {0} ,

and consequently ZT (F) = R ZT (F) ⊆ XT (U ). This completes the proof ofthe inclusion ZT (F) ⊆ XT (F). �


I want to conclude this chapter by stating a converse to Theorem 25.2.2. As aconsequence of this, and because (as pointed out in connection with Proposition25.2.1) there is a close connection between equality of the two topologies on�A and regularity of A, we obtain a nice result that expresses this connectionin terms of decomposability.

Proposition 25.2.5 Suppose that A is faithful and that T ∈ M(A) is a multi-plier with (δ). Then the restriction T |�A is hk-continuous.

Proof Given any closed F ⊆ C. Our task is to show that

E := {ϕ ∈ �A : T (ϕ) ∈ F}is hk-closed.So consider any ψ ∈ �A \ E, and choose an x ∈ A so thatψ(x) = 1. Since

λ := T (ψ) does not belong to F, there exists an open cover {U, V } of C forwhich U ⊆ C \ {λ} and V ⊆ C \ F. By (δ), we may decompose x = y + z,with y ∈ AT (U ) and z ∈ AT (V ). Since λ /∈ U and y ∈ AT (U ), we know thaty = (T − λ)u for some u ∈ A, and therefore

ψ(y) = (T (ψ)− λ)ψ(u) = 0 .

Thus ψ(z) = ψ(x) = 1. On the other hand, for any ϕ ∈ E, clearly µ := T (ϕ)belongs to F, and hence not to V . Since z ∈ AT (V ), z = (T − µ)v for suitablev ∈ A, and therefore

ϕ(z) = (T (ϕ)− µ)ϕ(v) = 0 .

Since ϕ(z) = 0 for all ϕ ∈ E and ψ(z) = 1, we see that ψ /∈ hk(E). Thus theset E = hk(E) is closed, and we have established the hk-continuity of T |�A.

�

Corollary 25.2.6 Suppose that A is semisimple. Then A is regular if and onlyif, for every a ∈ A, the multiplication operator La is decomposable.

Proof If A is regular then, by the remark immediately after Proposition 25.2.1,a is hk-continuous for every a ∈ A.Conversely, if every La is decomposable then Proposition 25.2.5 implies that

a is hk-continuous for every a ∈ A. But the Gelfand topology is, by definition,the coarsest topology on�A for which all these functions are continuous. Con-sequently, the hk-topology coincides with the Gelfand topology on �A. By theremark after Proposition 25.2.1, this establishes the regularity of A. �


25.3 Exercises

1. Show that a multiplier on a faithful commutative Banach algebra is contin-uous.

2. Show that if A is unital then every multiplier T is a multiplication operator,by showing that in this case T = LT (e).

3. Show that, if A has approximate units, that is, for every a ∈ A and everyε > 0 there is u ∈ A such that ‖a − ua‖ < ε, then A is faithful.

4. Let f be a bounded continuous function on . Show that T := L f is amultiplier on C0(). Establish also that the converse is true: every multi-plier is given by pointwise multiplication by a bounded continuous functionon .

5. Let A be faithful. Prove that the multiplier algebra M(A) is a closed, com-mutative, unital, full subalgebra of B(A) and contains A as an ideal.

6. Show that hkM(A)(�A) = h({0}) = �M(A) , hence that �A is hk-dense in�M(A).

7. Show that if C is a commutative Banach algebra, if c ∈ C, if c is hk-continuous on �C , and if B is any commutative algebra that contains Cas a subalgebra, then c is hk-continuous on �B .


Concerning examples of non-degeneratemodules, it is easily seen that X is non-degenerate and satisfies AX = X whenever the left A-module X is irreducible;see Proposition 24.4 of Bonsall and Duncan (1973).The best results about decomposability of multipliers are obtained when

certain vanishing-at-infinity conditions are imposed.

Definition 25.4.1 Suppose that A is semisimple and has multiplier algebraM(A), and character space �A (respectively, �M(A)). Then

M0(A) := {T ∈ M(A) : T |�A vanishes at infinity on �A ⊆ �M(A)}

and

M00(A) := {T ∈ M(A) : T vanishes on �M(A)\�A} .

In the simple case of A = C(), where is a locally compact Hausdorffspace, it is easy to picture these two ideals. Evidently, we always have

A ⊆ M00(A) ⊆ M0(A) ⊆ M(A) ,


and in general all these containments are strict; notably, this holds whenA = L1(G) is the group algebra for a locally compact, non-discrete, abeliangroup.The next result, and its proof, may be found in Laursen and Neumann (2000,

Theorem 4.5.4).

Theorem 25.4.2 Let A be semisimple. For T ∈ M0(A), the following areequivalent:

(a) T is super-decomposable on A;(b) T is decomposable on A;(c) T has (δ);(d) T ∈ M00(A) and T |�A is hk-continuous on �A;(e) T is hk-continuous on �M(A);(f) multiplication by T is decomposable on M(A).

In the case where A is also regular, T ∈ M0(A) satisfies all these conditionsexactly when T ∈ M00(A). �

Appendix to Part IV: The functional model

The following material forms a key background to our subject; a fuller accountof it is given in Laursen and Neumann (2000, Chapter 2).We first review some notions from the theory of scalar-valued functions and

distributions in two variables. Fuller expositions of this may also be found inRudin (1991) and Treves (1967).As before, we identify C and R2, write z = x + i y = (x, y) for z ∈ C and

x, y ∈ R, and let ∂x and ∂y denote the usual operators of partial differentiationwith respect to x and y, acting on functions of the real variables x and y. InChapter 21 we touched on C∞(C), in particular the notation

D α := ∂ α1x ∂ α2y , |α| := α1 + α2 and α! := α1!α2!

for every pair α = (α1, α2) ∈ (Z+)2.We then consider, for a compact subset Kof C, the space of smooth functions

D(K ) := { f ∈ C∞(C) : supp f ⊆ K } ,

topologized by the submultiplicative seminorms given by

‖ f ‖m,K :=∑|α|≤m

1

α!sup {|D α f (z)| : z ∈ K }

for all f ∈ C∞(C) and m ∈ Z+. Thus the convergence of a sequence ( fn)n∈N

in D(K ) with respect to this topology is that of uniform convergence of allsequences of derivatives (D α fn)n∈N, for every α ∈ (Z+)2. It is a standard factthat D(K ) is a Frechet space.For an open subset U of C, we choose a sequence of compact sets

(Kn)n∈N such that Kn ⊂ int Kn+1 for all n ∈ N and U =⋃ {Kn : n ∈ N} ,and let D(U ) :=⋃∞n=1D(Kn). This space we topologize as the locally convex

254


inductive limit of the spaces D(Kn), which means that a set in D(U ) is open ifand only if its intersection with every D(Kn) is open (this definition does notdepend on the choice of the sequence (Kn)n∈N). Moreover, a linear map fromD(U ) into an arbitrary locally convex topological vector space is continuous ifand only if its restriction to D(K ) is continuous for every compact subset K ofU. Topologized this way, the space D(U ) is the space of test functions on U.The space of continuous linear functionals onD(U ), customarily denoted by

D ′(U ), is the space of distributions on U.Adistributionϕ ∈ D ′(U ) is said to vanish on anopen subset V ofU ifϕ( f ) =

0 for all f ∈ D(V ). There is a largest open set V ⊆ U on which ϕ vanishes,and the complement of this set with respect toU is called the support of ϕ, andwill be denoted by suppϕ. Evidently, suppϕ is a relatively closed subset of U.If g : U → C is locally integrable with respect to the restriction of two-

dimensional Lebesgue measure µ, then g induces a distribution ϕ ∈ D ′(U ) viathe formula

ϕ( f ) :=∫Ug f dµ for all f ∈ D(U ) .

Since two locally integrable functions induce the same distribution exactlywhen they coincide almost everywhere with respect to µ, this formula leads toa canonical embedding of the space L 1loc(U ) intoD ′(U ),where L 1loc(U ) consistsof all equivalence classes modulo µ of the locally integrable functions onU. Inparticular, each element of L 1loc(U ) has a support in the sense of distributions.It also follows that each of the spacesC(U ) and L p(U ) (1 ≤ p <∞) may be

regarded as a linear subspace of D ′(U ). Here of course L p(U ) is the classicalBanach space of all pth power integrable complex-valued functions onU (withtwo functions identified when they coincide almost everywhere with respect toLebesgue measure). The fact that all pth power integrable functions on U arelocally integrable is a consequence of Holder’s inequality. For the space L∞(U )of all µ-essentially bounded measurable complex-valued functions on U, it isimmediate that L∞(U ) is contained in L 1loc(U ), and hence in D ′(U ).We equip D ′(U ) with the strong topology, that is, the topology of uniform

convergence on the bounded subsets of D(U ).It is immediate that, for every α ∈ (Z+)2, the differential operator D α defines

a continuous linear map on D(K ) for each compact subset K of U, and henceon D(U ). To extend this differential operator from D(U ) to the larger spaceD ′(U ), one uses the notion of a signed adjoint: for each ϕ ∈ D ′(U ), let

(D αϕ) ( f ) := (−1) |α| ϕ (D α f ) for all f ∈ D(U ) .

It is obvious that D αϕ ∈ D ′(U ).


Given a function g ∈ C m(U ) for some m ∈ N , integration by parts leads to∫U(D αg) f dµ = (−1) |α|

∫Ug D α f dµ

for all f ∈ D(U ) and α ∈ (Z+)2 with |α| ≤ m. It follows that, for |α| ≤ m, thedistribution induced by the function D αg ∈ C(U ) coincides with the distribu-tional derivative D αϕ of the distribution ϕ corresponding to g. This justifiesthe factor (−1) |α| in the definition of the distributional derivative, and showsthat differentiation in the sense of distributions extends the usual notion ofdifferentiation for continuously differentiable functions.An important tool in the theory of partial differential equations is the class of

Sobolev spaces. FollowingTreves (1967), the classical Sobolev space Hm(U ) oforder m ∈ Z+ consists of all functions f ∈ L 2(U ) for which the distributionalderivative D α f belongs to L 2(U ) for every α ∈ (Z+)2 with |α| ≤ m.Of course,here the space L 2(U ) is considered as a linear subspace of D ′(U ).Weshall need avariant of this,more convenient for the studyof the ∂-operator.

As usual, let

∂ := (∂x + i ∂y)/2 and ∂ := (∂x − i ∂y)/2.

As we have just explained, these differential operators (and their powers) arecanonically defined on the spaceD ′(U ), and hence act, in the sense of distribu-tions, on functions in L 2(U ).Be aware, however, that, for arbitrary f ∈ L 2(U ),the distributions ∂ f and ∂ f need not be implemented by functions in L 2(U ).For every m ∈ Z+, our Sobolev-like space of order m is defined as

W m(U ) := { f ∈ L 2(U ) : ∂ k f ∈ L 2(U ) for k = 0, . . . ,m}.

The proof of the following resultmay be found in Laursen andNeumann (2000).

Proposition A.1 For each open set U ⊆ C and each m ∈ Z+, the spaceW m(U ) is a Hilbert space when equipped with the inner product given by

[ f, g] :=m∑k=0

∫U∂kf ∂

kg dµ

for all f, g ∈ W m(U ). The corresponding norm on W m(U ) satisfies

‖ f ‖ =(

m∑k=0

∥∥∥∂ k f ∥∥∥22

)1/2


for all f ∈ W m(U ),where ‖ · ‖2 denotes the standard normof theHilbert spaceL 2(U ). �

The connection between distributional and old-fashioned differentiation isdescribed very well by the Sobolev embedding theorem.

Proposition A.2 The inclusion W m+2(U ) ⊆ Cm(U ) holds for every open setU ⊆ C and every m ∈ Z+. �

We proceed now to the vector-valued situation, and let L 2(U, X ) denote thespace of all X -valued measurable functions on U (here X is a given complexBanach space andU an open subset ofC) which are Bochner square-integrablewith respect to two-dimensional Lebesgue measureµ onU.With respect to thenorm given by

‖ f ‖2 :=(∫

U‖ f (ζ )‖2 dµ(ζ )

)1/2for all f ∈ L 2(U, X ) ,

L 2(U, X ) is a Banach space. Concerning integration of functions with values inaBanach space, itmay be sufficient here to indicatewhat is involved by referringto a classical result due to Pettis (Dunford and Schwartz 1958, Theorem 3.6.11),which shows that an X -valued function f on U is measurable (with respect toµ) if and only if it is µ-essentially separably valued in X and has µ-measurablecomposition ϕ ◦ f with every continuous linear functional ϕ on X.For every m ∈ Z+, we define the vector-valued Sobolev-like space

W m(U, X ) := { f ∈ L 2(U, X ) : ∂ k f ∈ L 2(U, X ) for k = 0, . . . ,m} ,

where the differential operator ∂ := (∂x + ∂y)/2 is of course taken distribution-ally. A result like Proposition A.1 will then show that W m(U, X ) is a Banachspace when endowed with the obvious norm.If X is a Hilbert space, then so are the two spaces L 2(U, X ) and W m(U, X )

with respect to the canonical inner product.Next we mention a vector analogue of the Sobolev embedding theorem,

sufficient for our purposes. This one relates the Sobolev-like spaces to the spaceC∞(U, X ) of all infinitely differentiable X -valued functions on U. This spaceis a Frechet space with respect to the topology of locally uniform convergencefor the functions and all their derivatives.We letD(U, X ) denote the space of all functions inC∞(U, X ) with compact

support inU.One convenient thing about the spaceC∞(U, X ) is that a functionf : U → X belongs to C∞(U, X ) if and only if ϕ ◦ f ∈ C∞(U ) for every


ϕ ∈ X ′ (this follows from the nuclearity of the Frechet space C∞(U ) and isincluded in Theoreme 2.13 of Grothendieck (1955)).

Proposition A.3 For every open set U ⊆ C and every Banach space X, wehave

∞⋂m=0

W m(U, X ) ⊆ C∞(U, X ) .

Sketch of proof We refer the interested reader to Laursen and Neumann(2000, Proposition 2.3.6). The argument is a beautiful piece of functional anal-ysis: it makes use of the Pettis result mentioned before. It also relies on theKrein–Smulian theorem: a linear functional ψ on X ′ is weak-∗ continuouson X ′ exactly when the restriction ψ |(X ′)[r ] is weak-∗ continuous for eachr > 0. Additional classic tools employed are Lebesgue’s dominated conver-gence theorem, the result of Grothendieck (1955) mentioned above, and theHahn–Banach theorem. �

Proposition A.3 may be used to show that the vector-valued Bergman space

A 2(U, X ) := H (U, X ) ∩ L 2(U, X )

consists of all f ∈ L 2(U, X ) for which ∂ f = 0 in the distributional sense (ex-ercise). Consequently, A 2(U, X ) is a closed linear subspace of W m(U, X ) forevery m ∈ Z+. Moreover, it is easily seen that the Banach space A 2(U, X ) iscontinuously embedded in the Frechet space H (U, X ). As in the scalar-valuedsetting, it is clear that the ∂-operator acts as a continuous linear map fromW m(U, X ) into W m−1(U, X ).From now on, let T ∈ B(X ) be given, and suppose thatU is a bounded open

subset of C containing the spectrum σ (T ). We need the definition of the localanalytic functional calculus

�( f ) = 1

2π i

∫�

(λ− T )−1 f (λ) dλ for all f ∈ H (U, X ),

where � is a contour in U surrounding σ (T ). The map � : H (U, X )→ Xis continuous, and the Bergman space A 2(U, X ) is continuously embedded inH (U, X ), so that

B 2(U, X ) := { f ∈ A 2(U, X ) : �( f ) = 0}

is a closed linear subspace of A 2(U, X ). The null space of the natural surjectionfrom

W m(U, X )/B 2(U, X )


onto W m(U, X )/A 2(U, X ) may be identified with A 2(U, X )/B 2(U, X ). Thuswe obtain a short exact sequence

0→ A2(U, X )/B 2(U, X )→ Wm(U, X )/B 2(U, X )

→ Wm(U, X )/A2(U, X )→ 0

of continuous linear operators between Banach spaces.Here the surjectivity of the local analytic functional calculus allows us to

identify the quotient A 2(U, X )/B 2(U, X ) with the given Banach spaceX. Moreover, the ∂-operator induces an identification of the spaces W m

(U, X )/A 2(U, X ) and W m−1(U, X ). Hence the preceding exact sequence maybe rephrased as

0→ X → W m(U, X )/B 2(U, X )→ W m−1(U, X )→ 0 .

This explains the meaning of the spaces appearing in this next theorem.

Theorem A.4 Let T ∈ B(X ) be an arbitrary operator, let U be a boundedopen subset ofC such that U contains the spectrum σ (T ), and let m ∈ N. Thenthe assignments

� J (x) = 1⊗ x + B 2(U, X ) for all x ∈ X,� Q( f + B 2(U, X )) = ∂ f for all f ∈ W m(U, X ),� S( f + B 2(U, X )) = Z f + B 2(U, X ) for all f ∈ W m(U, X ),� R (g) = Zg for all g ∈ W m−1(U, X )

yield a commutative diagram with exact rows of continuous linear operatorsbetween Banach spaces:

0 −→ XJ−→ W m(U, X )/B 2(U, X )

Q−→ W m−1(U, X ) −→ 0

T |↓ S |↓ R |↓0 −→ X −→

JW m(U, X )/B 2(U, X ) −→

QW m−1(U, X ) −→ 0 .

Moreover, R is generalized scalar with σ (R) = U , while S is a quotient of ageneralized scalar operator with σ (S) = U. �

Since the statement of Theorem A.4 includes the specific forms of the mapsinvolved, carrying out the proof of this result is more a matter of tenacitythan of ingenuity, as may be seen by looking at Laursen and Neumann (2000,Theorem 2.4.3). I have elected to leave out the details, and instead concentrateon our main uses for this result.


It is interesting that the above theorem establishes a certain kind of universal-ity of generalized scalar operators: every bounded linear operator on a Banachspace is a restriction of a quotient of a generalized scalar operator. However,our main concern is expressed in the next result, which is now simple. Its prooffollows by applying the 3-space lemma, Lemma 23.1.3, for (β).

TheoremA.5 A continuous linear operator on a Banach space has (β) if andonly if it is similar to the restriction of a decomposable operator to one of itsclosed invariant subspaces. Moreover, if T is an operator with (β), then theoperator S, in the functional model of Theorem A.4 for T, is a decomposableextension of T . �

To get a handle on how to establish the remaining duality property we haveto mention a second functional model.

Theorem A.6 Let T ∈ B(X ) be an arbitrary operator on a Banach space X,let m ∈ N be given, and suppose that U is a bounded open subset of C suchthat U contains σ (T ). Let

Y := {V ∈ B(W m(U ), X ) : V ( f ) = f (T )V (1) for all f ∈ A 2(U )} ,

where the operator f (T ) ∈ B(X ) is given by the analytic functional calculusfor T . Then the assignments:

� J (C) := C ◦ ∂;� R (C) := C ◦ MZ for all C ∈ B(W m−1(U ), X );� Q (V ) := V (1);� S (V ) := V ◦ MZ for all V ∈ Y

yield a commutative diagram with exact rows of continuous linear operatorsbetween Banach spaces:

0→ B(W m−1(U ), X )J→ Y

Q→ X → 0

R |↓ S |↓ T |↓0→ B(W m−1(U ), X )→

JY →

QX → 0 .

Moreover, R is generalized scalar, and S is subscalar. �

The 3-space lemma for (δ), also contained in Lemma 23.1.3, is used inproving the ‘only if’ part.


Finally, here are the detailed duality statements. Evidently, it is an immediateconsequence of the fact (Theorem 21.2.8) that an operator is decomposable ifand only if it has both (β) and (δ), combinedwith TheoremA.8, that an operatorand its adjoint are simultaneously decomposable. However, the proof of A.8proceeds by showing this latter fact first, so we state it separately. That will alsoserve to emphasize its importance.

Theorem A.7 An operator T ∈ B(X ) is decomposable if and only if the ad-joint T ′ ∈ B(X ′) is decomposable. �

Theorem A.8 For every operator T ∈ B(X ), the following assertions hold.(i) If T has (β), then T ′ has (δ).(ii) If T has (δ), then T ′ has (β).(iii) If T

′has (δ), then T has (β).

(iv) If T′has (β), then T has (δ).

Proof We sketch the proof of (i). If T has (β), then T has a decomposableextension, say S. By Theorem 23.1.4, the adjoint operator S′ is also decom-posable. Moreover, T ′ may be realized as a quotient of S′. Hence, by Theorem23.1.1, T ′ has (δ). Exercise: fill in the details.(ii) and (iii): exercises.We leave out entirely the details of how to establish part (iv) because this

is so much harder than the rest. This result is one of J. Eschmeier’s greatachievements. It can be done via a tensor version of the functional model ofTheorem A.4. The details are omitted here, except for the diagram itself. Forfull details, see Laursen and Neumann (2000). �

For an arbitrary operator T ∈ B(X ) on a Banach space X, the followingdiagram is commutative and has exact rows:

0→ W m−1(U )′ ⊗ε X∂′ ⊗ ε I→ ⊥ker &

Q→ X → 0

M ′Z ⊗ε I

|↓ S |↓ T |↓0→ W m−1(U )′ ⊗ε X →

∂′ ⊗ ε I

⊥ker & →Q

X → 0,

where S := (M ′Z ⊗ε I ) |⊥ker & is a restriction of a generalized scalar operator,

and the operator M ′Z ⊗ε I on the left-hand side is generalized scalar.


Theorem A.9 The dual of the above diagram may be identified with the fol-lowing commutative diagram with exact rows:

0 −→ X ′J−→ (W m(U )⊗π X ′

)/ ker &

[∂ ⊗π I ′]−→ W m−1(U )⊗π X ′ −→ 0

T′ |↓

(MZ ⊗π I ′

)/ ker &

|↓ MZ ⊗π I ′ |↓0 −→ X ′ −→

J

(W m(U )⊗π X ′

)/ ker & −→

[∂ ⊗π I ′]W m−1(U )⊗π X ′ −→ 0 .

Finally, for every λ ∈ C and every sequence of vectors (xn)n∈N in X for which(T − λ)xn converges to 0 as n→∞, there exist yn ∈⊥ker & such thatQ yn = xn for all n ∈ N and (S − λ) yn → 0 as n→∞. In particular, wehave σap(T ) ⊆ σap(S).

�

Here

& : A2(U )⊗π X

′ → X ′

denotes the local analytic functional calculus of the adjoint T ′ ∈ B(X ′) and

J : X ′ → (W m(U )⊗π X′) / ker &

is the corresponding continuous linear map given by

J (ϕ) := 1⊗ ϕ + ker & for all ϕ ∈ X ′.

Morever, the information in the last part of the theorem may be used in a proofof Eschmeier–Prunaru’s result on the existence of closed invariant subspacesfor operators with (β) and thick spectrum. There is more on this in Chapter 18.As a consequence we have Theorem 23.1.5, that is, every operator T ∈ B(X )

has one of the properties (β) or (δ) if and only if its adjoint operator T ′ ∈ B(X ′)has the other one.

References

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Part V

Single-valued extension property and Fredholmtheory

PIETRO AIENAUniversita degli Studi, Palermo, Italy

26

Semi-regular operators

Let T be a bounded operator on a Banach space X . It has been observed inChapter 21 that, if the spectrum σ (T ) is totally disconnected, then T is decom-posable. In particular, every compact operator K is decomposable, becauseσ (K ) is either finite or a sequence which converges to 0. The classical Riesz–Schauder theory for compact operators shows that, for every λ �= 0, the spaceλI − T has a finite-dimensional kernel and a finite-codimensional range, thatis, λI − T is a Fredholm operator, see Part I, Theorem 2.2.5 or Heuser (1982).Similarly, the class of normal operators on a Hilbert space consists of decom-posable operators (see Part IV, Chapter 21), and hasmany remarkable propertiesfrom the point of view of Fredholm theory. For instance, for normal operatorssome of the spectra originating from Fredholm theory coincide, as in the caseof compact operators.A natural question is to what extent the results which hold for compact and

normal operators in Fredholm theory may be extended to decomposable oper-ators. In this part of the book we shall give some answers to this question, andestablish important connections betweenFredholm theory and the single-valuedextension property, a property which plays a leading role in the investigationof decomposable operators.

26.1 Definitions

Among the various concepts of regularity originating fromFredholm theory, theconcept of semi-regularity, which will be introduced in this chapter, seems tobe the most appropriate to investigate some important aspects of local spectral

H.G. Dales, P. Aiena, J. Eschmeier, K.B. Laursen, and G. A. Willis, Introduction to BanachAlgebras, Operators, and Harmonic Analysis. Published by Cambridge University Press.c© Cambridge University Press 2003.

267

268 Part V SVEP and Fredholm theory, Pietro Aiena

theory. The concept of semi-regularity originates in the classical Kato’s treat-ment (1958) of perturbation theory of Fredholm operators.To introduce the class of semi-regular operators, we need first to establish

some connections between the kernels and the ranges of the iterates T n of anoperator T on a vector space X .

Proposition 26.1.1 For a linear operator T on a vector space X, the followingstatements are equivalent:

(i) ker T ⊆ Tm(X ) for each m ∈ N;(ii) ker T n ⊆ T (X ) for each n ∈ N;(iii) ker T n ⊂ Tm(X ) for each n ∈ N and each m ∈ N;(iv) ker T n = Tm(ker Tm+n) for each n ∈ N and each m ∈ N.

Proof (i)⇒(ii) We proceed by induction. The case n = 1 is obvious from theassumption (i). Assume that (ii) holds for n = k, and let x ∈ ker T k+1. Then

T kx ∈ ker T ⊆ T k+1(X ) ,by assumption (i). Hence there exists y ∈ X such that T kx = T k+1y. It is obvi-ous that T k(x − T y) = 0. Hence, if z := x − T y, then z ∈ ker T k ⊆ T (X ) bythe inductive assumption. Therefore x = z + T y ∈ T (X ), so that the inclusionker T k+1 ⊆ T (X ) is proved.(ii)⇒(iii) We proceed by induction onm. The casem = 1 is true by assump-

tion.Assume that (iii) is valid form = k and eachn ∈ N, that is, ker T n ⊆ T k(X )for everyn ∈ N. Let x ∈ ker T n ,wheren ∈ N is arbitrary. Since ker T n ⊆ T (X ),we have x = T y for some y ∈ X . From 0 = T nx = T n+1y, we obtain thaty ∈ ker T n+1 ⊆ T k(X ), and this implies that x = T y ∈ T k+1(X ). Hence (iii) isproved.

(iii)⇒(iv) Clearly, if x ∈ Tm(ker T n+m), then T nx = 0, so thatTm(ker T n+m) ⊆ ker T n.

To prove the opposite inclusion, let x ∈ ker T n for a fixed n ∈ N. If m ∈ N,then, by assumption (iii), there exists y ∈ X such that x = Tm y. From theequality 0 = T nx = T n+m y, we conclude that y ∈ ker T n+m ; so (iv) is proved.

(iv)⇒ (i) It suffices to consider the equality (iv) in the case where n = 1. Infact, in this case, for each m ∈ N, we have

ker T = Tm(ker Tm+1) ⊆ Tm(X ). �

As in Part I, Example 1.2(vii), B(X ) denotes the Banach algebra of allbounded linear operators on a Banach space X .

26 Semi-regular operators 269

Definition 26.1.2 Given a Banach space X, a bounded operator T ∈ B(X ) issemi-regular if T (X ) is closed and if T satisfies one of the equivalent conditionsof Proposition 26.1.1.

It is not difficult to show that T is semi-regular if and only if the dual operatorT ′ is semi-regular.Given a bounded operator T ∈ B(X, Y ), where X , Y are Banach spaces, the

property of T (X ) being closed may be characterized by means of the followingquantity associated with T .

Definition 26.1.3 If T ∈ B(X, Y ), where X and Y are Banach spaces, theminimal modulus of T is

γ (T ) := infx /∈ker T

‖T x‖dist(x, ker T )

,

with the convention that γ (T ) = ∞ if T = 0.

A classical result from perturbation operator theory (which is easy to seedirectly) establishes that γ (T ) > 0 if and only if T (X ) is closed. Moreover, ifT ′ is the dual of T , then γ (T ) = γ (T ′) for every T ∈ B(X ); see Kato (1966) orGoldberg (1966) for details. Of course, this implies that T (X ) is closed if andonly if T ′(X ′) is closed.The next result gives a useful condition which ensures that T (X ) is closed.

Proposition 26.1.4 Let T ∈ B(X ), where X is a Banach space, and supposethat there exists a closed subspace Y of X such that T (X ) ∩ Y = {0} andT (X )⊕ Y is closed. Then T (X ) is also closed.

Proof Consider the product space X × Y under the norm

‖(x, y)‖ := ‖x‖ + ‖y‖ (x ∈ X, y ∈ Y ) .

Then X × Y is aBanach space, and the continuousmap S : X × Y → X definedby S(x, y) := T x + y has range S(X × Y ) = T (X )⊕ Y , which is closed byassumption. From this, it follows that the minimal modulus

γ (S) := inf(x,y)/∈kerS

‖S(x, y)‖dist((x, y), ker S)

> 0 .


Moreover, ker S = ker T × {0}, so dist((x, 0), ker S) = dist(x, ker T ), andhence

‖T x‖ = ‖S(x, 0)‖ ≥ γ (S) dist((x, 0), ker S)) = γ (S) dist(x, ker T ) .

This implies that γ (T ) ≥ γ (S), and therefore T has closed range. �

Recall that every subspaceM of a vector space X admits at least one algebraiccomplement N , that is, X = M + N and M ∩ N = {0}. The codimension of asubspace M of X is the dimension of each algebraic complement N of M . Animmediate consequence of Proposition 26.1.4 is that, if Y is a finite-dimensionalsubspace of X and T (X )+ Y is closed, then T (X ) is closed. In particular, itfollows from Proposition 26.1.4 that, if codim T (X ) <∞, then T (X ) is closed.The following T -invariant subspace for a linear operator T on a vector space

has been introduced by Saphar (1964).

Definition 26.1.5 Let T be a linear operator on a vector space X. The al-gebraic core C(T ) is defined to be the greatest subspace M of X for whichT (M) = M.

Evidently, C(T ) coincides with the algebraic spectral subspace ET (C \ {0})defined in Chapter 22.It is not difficult to show that C(T ) may be characterized in terms of se-

quences. In fact, a recursive argument proves that x ∈ C(T ) if and only if thereexists a sequence (un)n∈Z+ ⊂ X such that x = u0 and Tun+1 = un (n ∈ Z+).

The subspaceC(T ) is defined in purely algebraic terms. The second subspacethat we shall consider is, in a certain sense, the analytic counterpart of C(T ).

Definition 26.1.6 Let X be a Banach space. The analytical core of T ∈ B(X )is the set K (T ) of all x ∈ X for which there is a sequence (un) and a δ > 0 suchthat:

(i) x = u0 and Tun+1 = un (n ∈ Z+);(ii) ‖un‖ ≤ δn‖x‖ (n ∈ Z+).

It is easily seen from definition that K (T ) is a linear subspace of X con-tained in C(T ) and that T (K (T )) = K (T ). We shall see in Theorem 27.2.2that K (T ) = XT (C \ {0}), where XT denotes the local analytic subspace, asdefined in Definition 22.1.1. Observe that in general neither K (T ) nor C(T ) is


closed. The next result shows that, under the assumption that C(T ) is closed,these two subspaces coincide.

Proposition 26.1.7 Let T ∈ B(X ), where X is a Banach space, and let F bea closed subspace of X such that T (F) = F. Then F ⊆ K (T ). In particular, ifC(T ) is closed, then C(T ) = K (T ).

Proof Let T0 : F → F denote the restriction of T to F . By assumption, F is aBanach space and T (F) = F , and so, by the openmapping theorem, T0 is open.Thismeans that there is a constant δ > 0with the property that, for every x ∈ F ,there exists u ∈ F such that Tu = x and ‖u‖ ≤ δ‖x‖. Let x ∈ F be arbitrarilygiven, let u0 = x , and consider an element u1 ∈ F such that Tu1 = u0 and‖u1‖ ≤ δ‖u0‖. By repeating this procedure, we can find an element un ∈ F forevery n ∈ Z+ such that Tun+1 = un and ‖un‖ ≤ δ‖un−1‖. The last inequalitygives the estimate ‖un‖ ≤ δn‖u0‖ = δn‖x‖ for n ∈ Z+. Thus x ∈ K (T ), andhence F ⊆ K (T ).The last assertion is clear. �

Definition 26.1.8 Let T be a linear operator on a vector space X. The gen-eralized range and generalized kernel of T are

T∞(X ) :=⋂n∈N

T n(X ) and N∞(T ) :=⋃n∈N

ker T n .

A simple inductive argument shows that the inclusion C(T ) ⊆ T n(X ) holdsfor all n ∈ N. From this it follows that C(T ) ⊆ T∞(X ). By Proposition 26.1.1,we easily obtain that T ∈ B(X ) is semi-regular if and only if T (X ) is closedand N∞(T ) ⊆ T∞(X ).The next lemma shows that under certain conditions the algebraic core and

the generalized range of an operator coincide.

Lemma 26.1.9 Let T be a linear operator on a vector space X. Suppose thatthe equality

ker T ∩ Tm(X ) = ker T ∩ Tm+k(X ) (k ∈ Z+) , (26.1.1)

holds for some m ∈ N. Then C(T ) = T∞(X ).

Proof We need only to prove the inclusion T∞(X ) ⊆ C(T ), that is,T (T∞(X )) = T∞(X ). The inclusion T (T∞(X )) ⊆ T∞(X ) is obvious for everylinear operator, so we need only to prove the opposite inclusion.


Suppose that (26.1.1) holds for m ∈ N, and consider D := ker T ∩ Tm(X ).Clearly, D = ker T ∩ T∞(X ). Now, let y be an arbitrary element of T∞(X ).Since y ∈ T n(X ) (n ∈ N), there exists xk ∈ X such that y = Tm+k xk . We nowset

zk := Tmx1 − Tm+k−1xk (k ∈ N) .

Then zk ∈ Tm(X ), and from the fact that T zk = Tm+1x1 − Tm+k xk = 0 weobtain that zk ∈ ker T . Hence zk ∈ D. The inclusion

D = ker T ∩ Tm+k(X ) ⊆ ker T ∩ Tm+k−1(X )

now entails that zk ∈ Tm+k−1(X ). This implies that

Tmx1 = zk + Tm+k−1xk ∈ Tm+k−1(X ) (k ∈ N) ,

and therefore Tmx1 ∈ T∞(X ). From the equalities T (Tmx1) = Tm+1x1 = y,we finally conclude that y ∈ T (T∞(X )), and therefore T∞(X ) ⊆ T (T∞(X )),as required. �

Proposition 26.1.10 Let T be a linear operator on a vector space X. Supposethat one of the following conditions holds:

(i) dim ker T <∞;(ii) codim T (X ) <∞;(iii) ker T ⊆ T n(X ) (n ∈ N).

Then C(T ) = T∞(X ).

Proof (i) It is evident that, if ker T is finite dimensional, then there ex-ists a positive integer m ∈ N such that the equality (26.1.1) holds. HenceLemma 26.1.9 applies.(ii) Assume that X = F ⊕ T (X ) with dim F <∞. Clearly, if

Dn := ker T ∩ T n(X ),

then Dn ⊇ Dn+1 for all n ∈ N. Suppose that there exist k distinct subspaces Dn .There is no loss of generality if we assume that Dj �= Dj+1 for j = 1, 2, . . . , k.Then for each one of these j , there exists w j ∈ X such that Tjw j ∈ Dj

and T jw j /∈ Dj+1. By means of the decomposition X = F ⊕ T (X ), we canfind u j ∈ F and v j ∈ T (X ) such that w j = u j + v j . We claim that the set{u1, . . . , uk} is linearly independent.


To see this, let us suppose that∑k

j=1 λ j u j = 0. Then∑k

j=1 λ jw j =∑kj=1 λ j v j and hence, since T

kw1 = · · · = T kwk−1 = 0, we have

T k(

k∑j=1

λ jw j

)= λkT

kwk = T k(

k∑j=1

λ j v j

)∈ T k(T (X )) = T k+1(X ) .

From T kwk ∈ ker T , we obtain λkT kwk ∈ Dk+1 and, since T kwk /∈ Dk+1, thisis possible only if λk = 0. Analogously we have λk−1 = . . . = λ1 = 0, and con-sequently the set {u1, . . . , uk} is linearly independent. From this, we concludethat k ≤ dim F . But then, for a sufficiently large m, we have

ker T ∩ Tm(X ) = ker T ∩ Tm+ j (X ) ( j ∈ Z+) .

So we are again in the situation of Lemma 26.1.9, and so (ii) implies thatC(T ) = T∞(X ).(iii) Obviously, if ker T ⊆ T n(X ) for all n ∈ N, then

ker T ∩ T n(X ) = ker T ∩ T n+k(X ) = ker T (k ∈ Z+) .

Hence also in this case we can apply Lemma 26.1.9. �

Lemma26.1.11 Suppose that T ∈ B(X ), where X is a Banach space, and thatT has closed range T (X ). For each (not necessarily closed) subspace Y ⊆ X,T (Y ) is closed whenever Y + ker T is closed.

Proof Let x be the equivalence class x + ker T in the quotient space X/ker T ,and denote by T : X/ker T → X the canonical injection which is defined byT (x) := T x , where x ∈ x . By assumption, T (X ) is closed, so that T has abounded inverse T

−1: T (X )→ X/ker T . Let Y := {y : y ∈ Y }. It is evident

that T (Y ) = T (Y ) is the inverse image of Y under the continuous map T−1.Thus T (Y ) is closed whenever Y is closed.It remains to show that Y is closed whenever Y + ker T is closed. Let (xn) be

a sequence of Y which converges to x ∈ X/ker T . Then there exists a sequence(xn) with xn ∈ xn such that dist(xn − x, ker T ) converges to zero, and so thereexists a sequence (zn) ⊂ ker T with xn − x − zn → 0 as n→∞. Evidently,the sequence (xn − zn) ⊂ Y + ker T converges to x , and since, by assumption,Y + ker T is closed, we conclude that x ∈ Y + ker T . This implies x ∈ Y , andhence Y is closed. �


Proposition 26.1.12 Let X be a Banach space, and let T ∈ B(X ) be semi-regular. Then:

(i) the subspace T n(X ) is closed for each n ∈ N;(ii) C(T ) is closed and C(T ) = K (T ) = T∞(X ).

Proof (i) We proceed by induction. The case n = 1 is obvious because, byassumption, Y := T (X ) is closed. Assume that T n(X ) is closed for some n ∈ N.Since T is semi-regular, ker T n ⊆ T (X ) = Y . From this, it follows that the sumY + ker T n = Y is closed, and hence, by Lemma 26.1.11, T n(Y ) = T n+1(X )is closed.(ii) The operator T is semi-regular, and so, by Proposition 26.1.10,

C(T ) = T∞(X ) =∞⋂n=1

T n(X ) ,

and therefore C(T ) is closed by part (i). From Proposition 26.1.7, we nowconclude that C(T ) = K (T ). �

26.2 The Kato spectrum

Among the many concepts dealt with in Kato’s extensive treatment of perturba-tion theory (Kato 1958), there is a very important part of the spectrum definedas follows.

Definition 26.2.1 Let X be a Banach space, and let T ∈ B(X ). The Katoresolvent set of T is

ρK (T ) := {λ ∈ C : λI − T is semi-regular}.

The Kato spectrum of T is the set σK (T ) := C \ ρK (T ) .

It is clear that σK (T ) ⊆ σ (T ) and ρ(T ) ⊆ ρK (T ). Later we shall prove thatσK (T ) is a non-empty, compact subset of C.Recall that an operator T ∈ B(X ) is bounded below if T is injective and has

closed range. This is equivalent to saying that there exists K > 0 such that‖T x‖ ≥ K‖x‖ for all x ∈ X . Clearly, if T is bounded below or surjective, thenT is semi-regular. A standard result of duality theory shows that T is boundedbelow (respectively, surjective) if and only if T ′ is surjective (respectively,bounded below).


Lemma 26.2.2 Let X be a Banach space, and let T ∈ B(X ). Then T isbounded below (respectively, surjective) if and only if λI − T is bounded below(respectively, surjective) for every |λ| < γ (T ).

Proof Suppose T bounded below. Since T (X ) is closed, γ (T ) > 0 and

γ (T ) · dist(x, ker T ) = γ (T )‖x‖ ≤ ‖T x‖ (x ∈ X ) .

Take |λ| < γ (T ). Then

‖(λI − T )x‖ ≥ ‖T x‖ − |λ|‖x‖ ≥ (γ (T )− |λ|)‖x‖ ,

and so λI − T is bounded below.The case where T is surjective now follows easily by considering the dual

T ′ of T . �

Theorem 26.2.3 Let X be a Banach space, and let T ∈ B(X ) be semi-regular.Then λI − T is semi-regular for |λ| < γ (T ). Consequently, σK (T ) is closed.

Proof First we show that C(T ) ⊆ C(λI − T ) for all |λ| < γ (T ). Indeed, letT0 : C(T )→ C(T ) denote the restriction of T toC(T ). By Proposition 26.1.12,C(T ) is closed and T0 is surjective. Thus, by Lemma 26.2.2, the equalities

(λI − T0)(C(T )) = (λI − T )(C(T )) = C(T )

hold for all |λ| < γ (T0).On the other hand, T is semi-regular so, by Proposition 26.1.10, we have

ker T ⊆ T∞(X ) = C(T ). From this it easily follows that γ (T0) ≥ γ (T ) and,consequently, we see that (λI − T )(C(T )) = C(T ) for all |λ| < γ (T ). Notethat this last equality implies that

C(T ) ⊆ C(λI − T ) (λ ∈ D(0, γ (T )) . (26.2.1)

Moreover, for every λ �= 0, we have T (ker(λI − T )) = ker(λI − T ), and so,by Proposition 26.1.12 and Proposition 26.1.7, we have ker(λI − T ) ⊆ C(T )for λ �= 0. From the inclusion (26.2.1), we now obtain that the inclusions

ker(λI − T ) ⊆ C(λI − T ) ⊆ (λI − T )n(X ) (26.2.2)

hold for all |λ| < γ (T ), λ �= 0 and n ∈ N. This is still true for λ = 0 since T issemi-regular, so that (26.2.2) is valid for all |λ| < γ (T ).


To show that λI − T is semi-regular for all |λ| < γ (T ), it only remainsto prove that (λI − T )(X ) is closed for all |λ| < γ (T ). Observe that, as aconsequence of Lemma 26.2.2, we can limit ourselves to considering onlythe case where {0} �= C(T ) �= {X}. Indeed, if C(T ) = {0}, then we have kerT ⊆ C(T ) = {0}, and hence T is injective, while, if C(T ) = X , the operator Tis surjective.Let X := X/C(T ), and let T : X → X be the continuous quotient map,

defined by T (x) := T x where x ∈ x . The operator T is injective, since fromT x = T x = 0 we have T x ∈ C(T ) and this easily implies, since T is semi-regular, that x ∈ C(T ), which yields x = 0.Next we prove that T is bounded below. Evidently, we only need to prove

that T has closed range. To see this we show the inequality γ (T ) ≥ γ (T ). Infact, for x ∈ X and u ∈ C(T ), we have

‖x‖ = dist(x,C(T )) = dist(x − u,C(T ))≤ dist(x − u, ker T ) ≤ ‖T x − Tu‖/γ (T ) .

From the equality C(T ) = T (C(T )), we obtain ‖x‖ = ‖T x‖/γ (T ) and, con-sequently, γ (T ) ≥ γ (T ). Therefore T is bounded below. By Lemma 26.2.2,λI − T is bounded below for all |λ| < γ (T ) and a fortiori for all |λ| < γ (T ).Finally, to show that (λI − T )(X ) is closed for all |λ| < γ (T ), let us con-

sider a sequence (un) of (λI − T )(X ) which converges to x ∈ X . Clearly, thesequence (xn) converges to x and xn ∈ (λI − T )(X ). The last space is closedfor all |λ| < γ (T ), and hence x ∈ (λI − T )(X ). Let x = (λI − T )v and v ∈ v .Then

x − (λI − T )v ∈ C(T ) = (λI − T )(C(T )) (|λ| < γ (T )) ,

and so there exists u ∈ C(T ) such that x = (λI − T )(v + u), that is,x ∈ (λI − T )(X ), for each |λ| < γ (T ). Therefore (λI − T )(X ) is closed for|λ| < γ (T ), and consequently λI − T is semi-regular for |λ| < γ (T ), that is,ρK (T ) is an open subset of C. �

The set ρK (T ) is open, and therefore it can be decomposed into open,connected, pairwise disjoint, non-empty components. We want to prove thatC(λI − T ) is constant on each component of ρK (T ). To show this, we firstneed some preliminary results on gap theory.Let M , N be two closed linear subspaces of a Banach space X , and define

δ(M, N ) := sup{dist(u, N ) : u ∈ M, ‖u‖ = 1}.


Lemma 26.2.4 Let M and N be two closed subspaces of a Banach space X.For every x ∈ X and 0 < ε < 1, there exists x0 ∈ X such that x − x0 ∈ M and

dist(x0, N ) ≥ (1− ε)1− δ(M, N )1+ δ(M, N )‖x0‖ . (26.2.3)

Proof The case x ∈ M is obvious: it suffices to take x0 = 0. Hence as-sume that x /∈ M . Let X := X/M be the quotient space, and set x := x + M .Then ‖x‖ = infz∈X ‖z‖ > 0. We claim that there exists x0 ∈ X such that‖x0‖ = dist(x0,M) ≥ (1− ε)‖x0‖. Indeed, were this not so, then

‖x‖ = ‖z‖ < (1− ε)‖z‖for every z ∈ x and, consequently, ‖x‖ ≤ (1− ε) infz∈x ‖z‖ = (1− ε)‖x‖ .This is impossible because ‖x‖ > 0.Let µ := dist(x0, N ) = infu∈N ‖x0 − u‖. We know that there exists y ∈ N

such that ‖x0 − y‖ ≤ µ+ ε‖x0‖. From this we obtain‖y‖ ≤ (1+ ε)‖x0‖ + µ .

On the other hand we have dist(y,M) = δ(N ,M) · ‖y‖, and so(1− ε)‖x0‖ ≤ dist(x0,M) ≤ ‖x0 − y‖ + dist(y,M)

≤ µ+ ε‖x0‖ + δ(N ,M) · ‖y‖≤ µ+ ε‖x0‖ + δ(N ,M)[(1+ ε)‖x0‖ + µ] ,

and therefore

µ ≥(1− ε − δ(N ,M)1+ δ(N ,M) − ε

)‖x0‖ .

Since ε > 0 is arbitrary, this implies the inequality (26.2.3). �

Lemma 26.2.5 Let X be a Banach space, and let T ∈ B(X ) be semi-regular.Then

γ (λI − T ) ≥ γ (T )− 3|λ| (λ ∈ C) . (26.2.4)

Proof Of course, for every T ∈ B(X ) and |λ| ≥ γ (T ), we haveγ (λI − T ) ≥ 0 ≥ γ (T )− 3|λ| ,

hence we need only to prove the inequality (26.2.4) in the case where|λ| < γ (T ).


If C(T ) = {0}, from the assumption that ker T ⊆ T∞(X ) = C(T ), we con-clude that ker T = {0}, that is, T is bounded below. From an inspection of theproof ofLemma26.2.2,weobtain thatγ (λI − T ) ≥ γ (T )− |λ| ≥ γ (T )− 3|λ|for all |λ| < γ (T ). Also the case whereC(T ) = X is trivial, because in this caseT is surjective and hence T ′ is bounded below, so that

γ (λI − T ) = γ (λI ′ − T ′) ≥ γ (T ′)− 3|λ| = γ (T )− 3|λ| .It remains to prove the inequality (26.2.4) in the case where C(T ) �= {0} and

C(T ) �= X . Suppose that |λ| < γ (T ), and let x ∈ C(T ) = T (C(T )). Then thereexists u ∈ C(T ) such that x = Tu, and hence

dist(u, ker T ) ≤ ‖Tu‖/γ (T ) = ‖x‖/γ (T ) .Let ε > 0 be arbitrary, and choose w ∈ ker T such that

‖u − w‖ ≤ [(1− ε)‖x‖/γ (T ).Let u1 := u − w , and µ := (1− ε)γ (T ). Clearly, u1 ∈ C(T ), Tu1 = x , and‖u1‖ ≤ µ−1‖x‖. Since u1 ∈ C(T ), by repeating the same procedure, we obtaina sequence (un), where u0 := x , such that un ∈ C(T ), Tun+1 = un , and ‖un‖ ≤µ−n‖x‖.Let us consider the function f : D(0, µ)→ X defined by

f (λ) := u0 +∞∑n=1

λnun.

Obviously, f (0) = x and f (λ) ∈ ker (λI − T ) for all |λ| < µ. Moreover,

‖x − f (λ)‖ =∥∥∥∥∥ ∞∑n=1

λnun

∥∥∥∥∥ ≤ |λ|µ− |λ| .

From this we obtain

dist(x, ker (λI − T )) ≤ |λ|µ− |λ| ,

which implies that

δ(ker T, ker (λI − T )) ≤ |λ|µ− |λ| =

|λ|(1− ε)γ (T )− |λ| (|λ| < µ) .

Since ε > 0 is arbitrary, we then conclude that

δ(ker T, ker (λI − T )) ≤ |λ|γ (T )− |λ| ( |λ| < γ (T )) . (26.2.5)

Now, let δ := δ(ker T, ker (λI − T )). By Lemma 26.2.4, to the element u andε > 0 there corresponds an element v ∈ X such that z := u − v ∈ ker (λI − T )


and

dist(v, ker T ) ≥ 1− δ1+ δ (1− ε)‖v‖ .

From this it follows that

‖(λI − T )u‖ = ‖(λI − T )v‖ ≥ ‖T v‖ − |λ|‖v‖≥ γ (T ) · dist(v, ker T )− |λ|‖v‖

≥ γ (T )1− δ1+ δ (1− ε)‖v‖ − |λ|‖v‖ .

Then, by using the inequality (26.2.5), we obtain

‖(λI − T )u‖ ≥ [(1− ε)(γ (T )− 2|λ|)− |λ|]‖v‖≥ [(1− ε)(γ (T )− 2|λ|)− |λ|]‖u − z‖≥ [(1− ε)(γ (T )− 2|λ|)− |λ|] · dist(u, ker (λI − T )) .

From the last inequality, it easily follows that

γ (λI − T ) ≥ (1− ε)(γ (T )− 2|λ|)− |λ| ,and, since ε is arbitrary, we conclude that γ (λI − T ) ≥ γ (T )− 3|λ|. �

Theorem 26.2.6 Let T ∈ B(X ), where X is a Banach space, and considera component of ρK (T ). If λ0 ∈ , then C(λI − T ) = C(λ0 I − T ) for allλ ∈ , that is, the subspaces C(λI − T ) = K (λI − T ) are constant on .

Proof Observe first that, by the first part of the proof of Theorem 26.2.3, wehave C(T ) ⊆ C(δ I − T ) for every |δ| < γ (T ). Now take |δ| < γ (T )/4 anddefine S = δ I − T . From Lemma 26.2.5, we obtain

γ (S) = γ (δ I − T ) ≥ γ (T )− 3|δ| > |δ| ,and hence, again by the observation above,

C(S) = C(δ I − T ) ⊆ C(δ I − (δ I − T )) = C(T ) .This shows that C(δ I − T ) = C(T ) for δ sufficiently small.Assume now that λ,µ ∈ . Then λI − T = (λ− µ)I − (T − µI ), and the

previous argument shows that, if we take λ, µ sufficiently close to each other,then

C(λI − T ) = C((λ− µ)I − (T − µI )) = C(µI − T ) .A compactness argument shows that we have C(λI − T ) = C(µI − T ) for allλ, µ ∈ . �


In the next definition we introduce another important T -invariant subspacefor a bounded operator T .

Definition 26.2.7 Let T ∈ B(X ), where X is a Banach space. The quasi-nilpotent part of T is the set

H0(T ) := {x ∈ X : limn→∞‖T

nx‖1/n = 0} .

As in Definition 2.1.2, the operator T ∈ B(X ) is said to be quasi-nilpotentif its spectral radius ν(T ) := inf{‖T n‖1/n} = limn→∞ ‖T n‖1/n is zero.Clearly, H0(T ) is a linear subspace of X , generally not closed.

Lemma 26.2.8 Let X be a Banach space, and let T ∈ B(X ) be semi-regular.Then γ (T n) ≥ γ (T )n (n ∈ N).

Proof We proceed by induction. The case n = 1 is trivial. Assume inductivelythat γ (T n) ≥ γ (T )n . For every element x ∈ X and u ∈ ker T n+1, we have

dist(x, ker T n+1) = dist(x − u, ker T n+1) ≤ dist(x − u, ker T ) .

Since T is semi-regular, Lemma 26.1.1 implies that ker T = T n(ker T n+1).Therefore

dist(T nx, ker T ) = dist(T nx, T n(ker T n+1)) = infu∈ker T n+1

‖T n(x − u)‖

≥ γ (T n) · infu∈ker T n+1

dist(x − u, ker T n)

≥ γ (T n) dist(x, ker T n+1) .

From this we obtain

‖T n+1x‖≥ γ (T ) dist(T nx, ker T )≥ γ (T ) γ (T n) · dist(x, ker T n+1) .

Consequently, γ (T n+1) ≥ γ (T )γ (T )n = γ (T )n+1, so the proof is com-plete. �

Let M⊥ denote the annihilator of M ⊆ X , and ⊥N the pre-annihilator ofN ⊆ X ′.

Proposition 26.2.9 For every bounded operator T ∈ B(X ), where X is aBanach space, we have H0(T ) ⊆ ⊥K (T ′) and K (T ) ⊆ ⊥H0(T ′). Moreover, if Tis semi-regular, then H0(T ) = N∞(T ) = ⊥K (T ′) and K (T ) = ⊥H0(T ′) .


Proof Consider an element u ∈ H0(T ) and f ∈ K (T ′). Then, according to thedefinition of K (T ′), there exists δ > 0 and a sequence (gn) ⊂ X ′ such that g0 =f, T ′gn+1 = gn , and ‖gn‖ ≤ δn‖ f ‖ for every n ∈ Z+. These equalities entailthat f = (T ′)ngn , and hence that f (u) = (T ′)ngn(u) = gn(T nu), for n ∈ Z+.From this it follows that | f (u)| ≤ ‖T nu‖‖gn‖ for n ∈ Z+, and therefore

| f (u)| ≤ δn‖ f ‖‖T nu‖ (n ∈ Z+) . (26.2.6)

Now, from u ∈ H0(T ), we obtain that limn→∞ ‖T nu‖1/n = 0, and hence,by taking the nth root in (26.2.6), we conclude that f (u) = 0. Therefore,H0(T ) ⊆ ⊥K (T ′).The inclusion K (T ) ⊆ ⊥H0(T ′) is proved in a similar way.Now, suppose that T is semi-regular. The inclusion N∞(T ) ⊆ H0(T ) is

obvious for every operator. To show the reverse inclusion, let suppose thatx /∈ N∞(T ) and take

δ := dist(x,N∞(T )) = dist(x,

∞⋃n=1ker T n

).

Obviously, δ > 0 and, by Lemma 26.2.8, we have

‖T nx‖ ≥ γ (T n) dist(x, ker T n)) ≥ γ (T n) δ ≥ γ (T )n δ ,so that limn→∞ ‖T nx‖1/n ≥ γ (T ) > 0, and hence x /∈ H0(T ). This shows thereverse inclusion, so that we have H0(T ) = N∞(T ).Finally, we show the equality H0(T ) = ⊥K (T ′). From the first part, it is

enough to show the inclusion ⊥K (T ′) ⊆ H0(T ). For T ∈ B(X ) and n ∈ N, wehave ker T n ⊆ H0(T ), and hence

H0(T )⊥ ⊆ ker T n⊥ = (T ′)n(X ′) ,

where the last equality holds since T ′ is semi-regular and therefore, by Propo-sition 26.1.12(i), (T ′)n(X ′) is closed for n ∈ N. This easily implies that

H0(T )⊥ = H0(T )

⊥ ⊆ (T ′)∞(X ′) = K (T ′) ,

again by Proposition 26.1.12. Consequently, ⊥K (T ′) ⊆ H0(T ) .The equality K (T ) =⊥ H0(T ′) is proved in a similar way. �

Theorem 26.2.10 Let T ∈ B(X ), where X is a Banach space, and let ⊂ C

be a connected component of ρK (T ). If λ0 ∈ , then

H0(λI − T ) = H0(λ0 I − T ) (λ ∈ ) .


Proof We know that ρK (T ) = ρK (T ′). By Theorem 26.2.6, it follows thatK (λI ′ − T ′) = K (λ0 I ′ − T ′) for all λ ∈ . From Proposition 26.2.9, we thenobtain that

H0(λI − T ) = ⊥K (λI ′ − T ′) = ⊥K (λ0 I ′ − T ′) = H0(λ0 I − T ) ,

for all λ ∈ . �

Theorem 26.2.11 Let T ∈ B(X ), and denote by� a connected component ofσ (T ). Then the topological boundary ∂� is contained in σK (T ). In particular,if X �= {0}, the Kato spectrum σK (T ) is a non-empty, compact subset of C

containing ∂σ (T ).

Proof Let λ0 ∈ ∂�, and suppose that λ0 ∈ ρK (T ). Let denote the componentof ρK (T ) containing λ0. The set is open, so there exists a neighbourhood Uof λ0 contained in and, since λ0 ∈ ∂σ (T ), U also contains points of ρ(T ).Hence ∩ ρ(T ) �= ∅.Consider a point λ1 ∈ ∩ ρ(T ). Clearly, N∞(λ1 I − T ) = {0}, and hence,

by Theorem 26.2.10 and Proposition 26.2.9, we have

H0(λI − T ) = H0(λ1 I − T ) = N∞(λ1 I − T ) = {0} (λ ∈ ) .

This shows that ker(λI − T ) = {0} for every λ ∈ , so that λI − T is injective.On the other hand, from Theorem 26.2.6, we know that

K (λI − T ) = K (λ1 I − T ) = X (λ ∈ ),

so that λI − T is onto for every λ ∈ . In particular, since λ0 ∈ , weconclude that λ0 ∈ ρ(T ), a contradiction. Hence the first assertion is proved.The second assertion is clear: σK (T ) is compact by Theorem 26.2.3 and

∂σ (T ) ⊆ σK (T ) by the first part. �

26.3 Exercises

1. Show that the equality (λI + T )(N∞(T )) = N∞(T ) holds for all λ �= 0.Show that N∞(λI + T ) ⊆ (µI + T )∞(X ) for λ �= µ.

2. Show that T ∈ B(X ) is semi-regular if and only if T ′ is semi-regular. Hint:show first that (ker T n)⊥ = (T ′)n(X ′) and ⊥(ker (T ′)n) = T n(X ) for everyn ∈ N.


3. Show that x ∈ C(T ) if and only if there exists a sequence (un)n∈Z+ ⊂ X suchthat x = u0 and Tun+1 = un . Check that, for each semi-regular operator, theabsorbency property x ∈ C(T )⇔ T x ∈ C(T ).

4. Show that T is quasi-nilpotent if and only if H0(T ) = X . Hint: for everyx ∈ X and λ �= 0, the series∑∞

n=0 Tnx/λn+1 converges to some y for which

(λI − T )x = y.5. Let T be a multiplier of a semisimple Banach algebra A, see Chapter 25.Show that H0(T ) = ker T . Hint: show first that, if x ∈ H0(T ), then T x be-longs to the radical rad A, for every a ∈ A. See Part I, §2.2 for definitionand basic properties.

6. Show that the operator T ∈ B(X ) is semi-regular if and only if the mappingλ �→ γ (λI − T ) is continuous at 0. Hint: use Lemma 26.2.5.

7. Show that T ∈ B(X ) is semi-regular if and only if there exists a closedsubspace M of X such that T (M) = M and the operator T : X/M → X/M ,induced by T , is injecive and has closed range.

8. Show that, if T, S ∈ B(X ) commute and if T S is semi-regular, then both Tand S are semi-regular. Find an example which shows that the product of twosemi-regular operators, also commuting semi-regular operators, need not besemi-regular. Hint: let H be a Hilbert space with an orthonormal basis (ei, j )where i, j are integers for which i j ≤ 0. Let T, S ∈ B(H ) be defined by theassignments:

T ei, j :={0 if i = 0, j > 0 ,ei+1, j otherwise ,

and Sei, j :={0 if j = 0, i > 0 ,ei, j+1 otherwise .

Show that T and S are semi-regular and that T S is not semi-regular.9. Check that the set of all semi-regular operators need not be an openset of B(X ). Hint: let H be a Hilbert space with an orthonormal basis(ei, j ) where i, j are integers and i ≥ 1. Let T ∈ B(H ) be defined by theassignment:

T ei, j :={ei, j+1 if j �= 0 ,0 if j = 0 .

Show first that T is semi-regular.Now, let ε > 0 be arbitrarily given and define S ∈ B(H ) by

Sei, j :={(ε/ i)ei,0 if j = 0 ,0 if j �= 0 .

Note first that ‖S‖ = ε. Show that (T + S)(H ) is not closed and hence T + Sis not semi-regular; see Muller (1994).



The concept of semi-regularity of an operator T ∈ B(X ), where X is a Banachspace, arose from the treatment of perturbation theory due to Kato (1958), evenif originally these operators were not given this name. Subsequently, this classof operators has received a lot of attention from several other authors (see, forinstance, Mbekhta 1987, 1990; Mbekhta and Ouahab 1994; Schmoeger 1990;Muller 1994). The Kato spectrum, also known as the Apostol spectrum in theliterature, was first introduced by Apostol (1984) for Hilbert space operatorsand was defined as the set of all complex λ such that either λI − T is not closedor λ is a discontinuity point for the function λ→ (λI − T )−1. Subsequently,the spectrum σK (T ) has been studied by different authors in the more generalframework of Banach spaces.The local constancy of the analytic core on the components of the Kato

resolvent has been established by several authors (see, for instance, Forster1966; O Searcoid and West 1989), but the methods adopted in this notes areinspired by the paper of Mbekhta and Ouahab (1994). A different approach tothis result may be found in Section 3.7 of Laursen and Neumann (2000). Thesection on the quasi-nilpotent part of an operator is modelled after Mbekhta(1990).The concept of algebraic core of an operator has been introduced by Saphar

(1964), while the analytical core has been introduced by Vrbova (1973) and,subsequently, studied byMbekhta (1990),Mbekhta andOuahab (1994). Lemma26.1.9 and Proposition 26.1.10 are taken from Aiena and Monsalve (2000).Proposition 26.1.12 is due to Schmoeger (1990), while the subsequent part,except Lemma 26.2.4 due to Kato (1958), can be found in Mbekhta (1990).Theorem 26.2.11 is due to Schmoeger (1990), who also showed that a spectralmapping theorem holds for σK (T ).

27

The single-valued extension property

The basic role of the single-valued extension property (SVEP) arises in thespectral decomposition theory, since every decomposable operator T enjoysthis property, as does its dual T ′. Indeed, in part IV, Chapter 21 it hasbeen shown that the decomposability of an operator may be viewed as theunion of two properties, the so-called Bishop’s property (β) and the prop-erty (δ). Property (β) for T implies the SVEP for T (see part IV, Chapter21) and, as observed in part IV, Chapter 23, properties (β) and (δ) have acomplete duality, so that, if T has (δ), then the dual T ′ has (β) and thereforeSVEP.The main goal of this chapter is to investigate in detail a localized version of

SVEP. First we shall show that local spectral theory provides a suitable framefor some characterizations of the analytical core and of the quasi-nilpotent part.Then we shall use these characterizations to describe the localized SVEP bymeans of a variety of conditions that involve the analytical core and the quasi-nilpotent part of an operator, aswell as the generalized range and the generalizedkernel.

27.1 The SVEP at a point

To explain the role of SVEP in local spectral theory we begin with some pre-liminary and well-known facts from operator theory.The resolvent function R(λ, T ) := (λI − T )−1 of T ∈ B(X ), where X is a

complex Banach space, is an analytic operator-valued function defined on theresolvent set ρ(T ). Setting fx (λ) := R(λ, T )x for each x ∈ X , we obtain avector-valued analytic function fx : ρ(T )→ X which satisfies

(λI − T ) fx (λ) = x (λ ∈ ρ(T )) . (27.1.1)

However, it is possible to find analytic solutions to the equation

285

286 Part V SVEP Fredholm theory, Pietro Aiena

(λI − T ) fx (λ) = x for some (sometimes even for all) values of λ that arein the spectrum of T , as the following example shows.Let T ∈ B(X ) be a bounded operator on a Banach space X such that the spec-

trumσ (T ) has a non-empty spectral subsetσ �= σ (T ), that is,σ andσ (T ) \ σ areclosed. From the functional calculus, if P := P(σ, T ) denotes the spectral pro-jection of T associated with σ , we know that P(X ) is a closed T -invariant sub-space and σ (T | P(X )) = σ , so the restriction (λI − T ) | P(X ) is invertible forall λ /∈ σ (see part I, Theorem 4.3.1). Let x ∈ P(X ). Then the equation (27.1.1)has the analytic solution fx (λ) := ((λI − T ) | P(X ))−1x for all λ ∈ C \ σ .These considerations lead in a natural way to the concepts of local resolvent

ρT (x) at x of a bounded operator T on a Banach space X . This is defined as theset of all λ ∈ C for which there is an open discD(λ, ε) and an analytic functionf : D(λ, ε)→ X such that the equation

(µI − T ) f (µ) = x (µ ∈ D(λ, ε))

holds. See Part IV, Definition 22.1.1. The local spectrum σT (x) at x is de-fined σT (x) := C \ ρT (x). Recall that, if λ ∈ ρT (x) and f : D(λ, ε)→ X is ananalytic function on D(λ, ε) which satisfies the equation (µI − T ) f (µ) = x(µ ∈ D(λ, ε)), then σT ( f (µ)) = σT (x) for all µ ∈ D(λ); for this, see Part IV,Lemma 22.1.4.The single-valued extension property has been defined in Part IV, Chapter 21.

In this chapter we shall consider a localized version of this property.Let X be a complex Banach space, and let T ∈ B(X ). The operator T has

the single-valued extension property at λ0 ∈ C, abbreviated ‘T has SVEP atλ0’, if, for every open disc D(λ0) centred at λ0, the only analytic functionf : D(λ0)→ X for which the equation (λI − T ) f (λ) = 0 holds is the constantfunction f ≡ 0. Obviously, if T has SVEP for every λ ∈ C, then T has SVEP.

Remark 27.1.1(i) If x ∈ X and T has SVEP at λ0 ∈ ρT (x), then there exist an open disc

D(λ0, ε) and a unique analytic function f : D(λ0)→ X satisfying the equa-tion (λI − T ) f (λ) = x for all λ ∈ D(λ0, ε). Consequently, SVEP impliesthe existence of a maximal analytic extension f of

R(λ, T )x := (λI − T )−1xto the set ρT (x) for every x ∈ X . It is evident that this function identicallysatisfies the equation

(λI − T ) f (λ) = x (λ ∈ ρT (x)) ,and that f (λ) = (λI − T )−1x for every λ ∈ ρ(T ).

27 The single-valued extension property 287

(ii) Trivially, a bounded operator T ∈ B(X ) has SVEP at every point of the re-solvent ρ(T ). Moreover, from the identity theorem for analytic functions,it easily follows that T ∈ B(X ) has SVEP at every point of the bound-ary ∂σ (T ) of the spectrum. In particular, every operator having discretespectrum has SVEP.

27.2 Local analytic subspace

For every subset of C, let XT () denote the local analytical subspace of Tassociated with , as in Definition 22.1.1, so that

XT () := {x ∈ X : σT (x) ⊆ } .

The next result gives a precise description of the analytical core K (T ) bymeans of the local spectrum σT (x).

Theorem 27.2.1 Let T ∈ B(X ) be a bounded operator on a Banach space X.Then

K (T ) = XT (C \ {0}) = {x ∈ X : 0 /∈ σT (x)} .

Proof Let x ∈ K (T ) and, according to the definition of K (T ), let δ > 0 and(un) ⊂ X a sequence for which x = u0, Tun+1 = un , ‖un‖ ≤ δn‖x‖ for everyn ∈ Z+. The function f : D(0, 1/δ)→ X , where D(0, 1/δ) is the open disccentred at 0 and with radius 1/δ, defined by f (λ) := −∑∞

n=1 λn−1un, is an

analytic vector-valued function which satisfies the equation (λI − T ) f (λ) = xfor everyλ ∈ D(0, 1/δ). Thismeans that 0 ∈ ρT (x), and henceσT (x) ⊆ C \ {0}.Conversely, if σT (x) ⊆ C \ {0}, then 0 ∈ ρT (x), and hence there is an open

disc D(0, ε) and an analytic function f : D(0, ε)→ X such that

(λI − T ) f (λ) = x (λ ∈ D(0, ε)) . (27.2.1)

Since f is analytic on D(0, ε), there exists a sequence (un) ⊂ X such that

f (λ) := −∞∑n=1

λn−1un (λ ∈ D(0, ε)) . (27.2.2)

Obviously, f (0) = −u1 and, taking λ = 0 in (27.2.1), we obtain the equationTu1 = −T ( f (0)) = x .


On the other hand, we have

x = (λI − T ) f (λ) = Tu1 + λ(Tu2 − u1)+ λ2(Tu3 − u2)+ · · · (λ ∈ D(0, ε)) .

From x = Tu1 we obtain Tun+1 = un for n ∈ N. Letting u0 = x , the sequence(un) satisfies the condition (1) of the definitionof K (T ), so it remains to prove thecondition ‖u − n‖ ≤ δn‖x‖ for a suitable δ > 0 and all n ∈ Z+. Takeµ > 1/ε.Then there exists c > 0 such that

‖un‖ ≤ c µn−1 (n ∈ N) . (27.2.3)

From (27.2.1) and (27.2.2) we easily obtain Tu1 = u0 and Tun = un−1 forevery n = 2, 3, . . . . Obviously, if x = 0, then x ∈ K (T ). Suppose that x �= 0.From the estimates (27.2.3), it then follows that ‖un‖ ≤ ‖x‖(µ+ c/‖x‖)n, andhence x ∈ K (T ). �

A certain variant of the local spectral subspaces which is better suited foroperatorswithout SVEP is given, for every closed subsetofC, by the subspaceXT () of all x ∈ X such that there is an analytic function f : C \→ X suchthat (λI − T ) f (λ) = x for all λ ∈ C \. Clearly, XT () ⊆ XT () for everyclosed subset ⊆ C. The set XT () is called the glocal spectral subspace ofT associated with in Definition 21.2.3.Note that XT () = XT () for every closed ⊆ C if and only if T has SVEP,

and this happens if and only if, for every x �= 0, the local spectrum σT (x) isnon-empty, that is, XT (∅) = {0}. See Proposition 22.1.2.

Theorem 27.2.2 For every operator T ∈ B(X ) on a Banach space X andevery ε ≥ 0, we have

XT (D(0, ε)) = {x ∈ X : lim supn→∞

‖T nx‖1/n ≤ ε} . (27.2.4)

In particular, H0(T ) = XT ({0}) and, if T has SVEP, then H0(T ) = XT ({0}).

Proof Let x ∈ X such that δ := lim supn→∞ ‖T nx‖1/n ≤ ε. The series

f (λ) :=∞∑n=1

λ−nT n−1x (λ ∈ C \ D(0, ε))

converges locally uniformly, so it defines an X -valued function on the open setC \ D(0, ε). Evidently, (λI − T ) f (λ) = x for all λ ∈ C \ D(0, ε). Thus we seethat x ∈ XT

(D(0, ε)

).


Conversely, let us assume that x ∈ XT (D (0, ε)), and let us consider an an-alytic function f : C \ D(0, ε)→ X such that (λI − T ) f (λ) = x holds for allλ ∈ C \ D(0, ε). For |λ| > max {ε, ‖T ‖}, we obtain

f (λ) = (λI − T )−1x =∞∑n

λ−nT n−1x ,

and therefore f (λ)→ 0 as |λ| → ∞. The analytic function g : D(0, 1/ε)→ Xdefined by

g(µ) :={f (1/λ) if 0 �= µ ∈ D (0, 1/ε) ,0 if µ = 0,

satisfies the equality

g(µ) =∞∑n=1

µnT n−1x (|µ| < 1/max{ε, ‖T ‖}) . (27.2.5)

Since g is analytic on D(0, 1/ε), it follows, exactly as in the scalar settingfrom Cauchy’s integral formula, that the equality (27.2.5) holds even for allµ ∈ D(0, 1/ε). This shows that the radius of convergence of the power seriesrepresenting g(µ) is greater than 1/ε. The standard formula for the radius ofconvergence of a vector-valued power series then implies that δ < ε. Thereforethe equality (27.2.4) holds.The equality H0(T ) = XT ({0}) is clear, taking ε = 0 in (27.2.4). Finally,

if T has SVEP, then XT () = XT () for every closed ⊆ C, in particularXT ({0}) = XT ({0}). �

We know that T does not have SVEP if and only if there exists an element0 �= x ∈ X such that σT (x) = ∅. The next result shows a localized version ofthis fact.

Theorem 27.2.3 Suppose that T ∈ B(X ), where X is a Banach space. Thenthe following conditions are equivalent:

(i) T has SVEP at λ0;(ii) ker (λ0 I − T ) ∩ XT (∅) = {0};(iii) ker(λ0 I − T ) ∩ K (λ0 I − T ) = {0} .

Proof (i)⇔ (ii)We can suppose that λ0 = 0. Assume that, for some x ∈ ker T ,we have σT (x) = ∅. Then 0 ∈ ρT (x), so there is an open disc D(0, ε) and ananalytic function f : D(0, ε)→ X such that (λI − T ) f (λ) = x (λ ∈ D(0, ε)).


Then

T ((λI − T ) f (λ)) = (λI − T )T ( f (λ)) = T x = 0 (λ ∈ D(0, ε)) .

Since T has SVEP at 0, then T f (λ) = 0, and therefore T ( f (0)) = x = 0.Conversely, suppose that, for every 0 �= x ∈ ker T , we haveσT (x) �= ∅. There

is an analytic function f : D(0, ε)→ X such that

(λI − T ) f (λ) = 0 (λ ∈ D(0, ε)).

Then f (λ) =∑∞n=0 λ

nun for a suitable sequence (un) ⊂ X . Clearly,

Tu0 = T ( f (0)) = 0,so that u0 ∈ ker T . Moreover, from the equalities

σT ( f (λ)) = σT (0) = ∅ (λ ∈ D(0, ε)),

we obtain σT ( f (0)) = σT (u0) = ∅, and therefore, from the assumption, weconclude that u0 = 0. For all 0 �= λ ∈ D(0, ε), we have

0 = (λI − T ) f (λ) = (λI − T )∞∑n=1

λnun = λ(λI − T )∞∑n=1

λnun+1 ,

and therefore 0 = (λI − T )(∑∞n=0 λ

nun+1) for every 0 �= λ ∈ D(0, ε). By con-tinuity this is still true for every λ ∈ D(0, ε). At this point, by using the sameargument as in the first part of the proof, it is possible to show that u1 = 0 and,by iterating this procedure, we conclude that u2 = u3 = · · · = 0. This showsthat f ≡ 0 on D(0, ε), and therefore T has SVEP at 0.To show the equivalence (ii)⇔ (iii), it suffices to prove the equality

ker T ∩ K (T ) = ker T ∩ XT (∅) .From Theorem 27.2.1 we have

ker T ∩ K (T ) = ker T ∩ XT (C \ {0}) ⊆ XT ({0}) ∩ XT (C \ {0}) = XT (∅) ,so that ker T ∩ K (T ) = ker T ∩ K (T ) ∩ XT (∅) = ker T ∩ XT (∅), as required.

�

Let σp(T ) denote the point spectrum of T ∈ B(X ), that is ,

σp(T ) := {λ ∈ C : λ is an eigenvalue of T } .

Corollary 27.2.4 Let T ∈ B(X ), where X is a Banach space. Then T doesnot have SVEP if and only if there exists λ0 ∈ σp(T ) and a correspondingeigenvector x0 (�= 0) such that σT (x0) = ∅. In such a case, T does not haveSVEP at λ0. �


It is clear from the definition of SVEP, that, if the set of eigenvalues of operatorT has empty interior, then T has this property. Consequently, all operatorsT ∈ B(X ) with real spectrum have SVEP. But not every bounded operatorenjoys SVEP. In fact, from Theorem 27.2.3, T has SVEP at 0 precisely whenker T ∩ K (X ) = {0}, so that every non-injective, surjective operator does nothave SVEP at 0.

Corollary 27.2.5 Let T ∈ B(X ), where X is a Banach space, and let λ0 ∈ C.

(i) If either

K (λ0 I − T ) ∩ H0(λ0 I − T ) = {0} orN∞(λ0 I − T ) ∩ (λ0 I − T )∞(X ) = {0},

then T has SVEP at λ0.(ii) T has SVEP⇔ K (λI − T ) ∩ H0(λI − T ) = {0} for every λ ∈ C.

Proof Part (i) is an obvious consequence of Theorem 27.2.3, since both con-ditions imply that ker (λ0 I − T ) ∩ K (λ0 I − T ) = {0}.Suppose that T has SVEP and x ∈ K (λI − T ) ∩ H0(λI − T ). By Theorem

27.2.1, we have σλI−T (x) ⊆ {0} and hence σT (x) ⊆ {λ}. On the other hand,we also have, by Theorem 27.2.2, 0 /∈ σλI−T (x), and hence λ /∈ σT (x). Conse-quently, σT (x) is empty and therefore, since T has SVEP, x = 0. The reverseimplication is clear from (i). �

Example 27.2.6 This example, based on the theory of weighted shifts, showsthat SVEP at a point λ0 does not necessarily imply that

H0(λ0 I − T ) ∩ K (λ0 I − T ) = {0}.Let β := (βn)n∈Z be the sequence of real numbers defined by

βn :={1+ |n| if n < 0 ,e−n

2if n ≥ 0 .

Let X := L2(β) denote the Hilbert space of all formal Laurent series∑∞n=−∞ anz

n for which∑∞

n=−∞ |αn|2βn2 <∞, endowed with the canonicalnorm ‖ · ‖β . Let us consider the bilateral weighted right shift T defined by

T

( ∞∑n=−∞

anzn

):=

∞∑n=−∞

anzn+1 ,

or equivalently, T zn = zn+1 (n ∈ Z). The operator T is bounded on L2(β) and

‖T ‖ = sup{βn+1/βn : n ∈ Z} = 1 .


Clearly T is injective, and thus has SVEP at 0. The following argument showsthat H0(T ) ∩ K (T ) �= {0}. Since‖zn‖β = βn (n ∈ Z), limn→∞ ‖zn−1‖β1/n = 0and limn→∞ ‖z−n−1‖β1/n = 1. By the formula for the radius of convergence ofa power series, we then conclude that the two series f (λ) :=∑∞

n=1 λ−nzn−1

and g(λ) := −∑∞n=1 λ

nz−n−1 converge in L2(β) for all |λ| > 0 and |λ| < 1,respectively. Clearly, f is analytic on C \ {0}, and

(λI − T ) f (λ) = −∞∑n=1

λ−nzn −∞∑n=1

λ1−nzn−1 = 1 (λ ∈ C \ {0}) ,

while g is analytic on the open unit disc D and satisfies

(λI − T )g(λ) =∞∑n=0

λnz−n −∞∑n=0

λ1+nz−n−1 = 1 (λ ∈ D) .

This means that 1 ∈ XT ({0}) ∩ XT (C \ D) = H0(T ) ∩ K (T ). �

An immediate consequence of part (ii) of Corollary 27.2.5 is that, for a quasi-nilpotent operator T , we have K (T ) = {0}. In fact, T hasSVEPand H0(T ) = X ,so that

H0(T ) ∩ K (T ) = K (T ) = {0}.We shall use this fact in the proof of the following result.

Proposition 27.2.7 Suppose that a bounded operator T ∈ B(X ) on a Banachspace X either has a closed quasi-nilpotent part H (λ0 I − T ) or is such thatthat H0(λ0 − T ) ∩ K (λ0 I − T ) is closed. Then

H0(λ0 − T ) ∩ K (λ0 I − T ) = {0},and hence T has SVEP at λ0.

Proof Without loss of generality, we may consider λ0 = 0.Assume first that H0(T ) is closed. Let T denote the restriction of T to the

Banach space H0(T ). It is easily seen that H0(T ) ∩ K (T ) = K (T ). On theother hand, we have H0(T ) = H0(T ). Thus T is quasi-nilpotent, and henceK (T ) = {0}.Suppose now thatY := H0(T ) ∩ K (T ) is closed. Clearly,Y is invariant under

T , so we can consider the restriction S := T | Y . If y ∈ Y , then‖Sn y‖1/n = ‖T n y‖1/n → 0 as n→∞.

Thus y ∈ H0(S), and hence H0(S) = Y . This shows that S is quasi-nilpotent,and hence has SVEP. This implies that σS(y) = {0} for all 0 �= y ∈ Y .


On the other hand, we also have K (S) = Y . Indeed, take an element y ∈ Y .Then there is a sequence (yn) ⊂ X and a δ > 0 such that

y0 = y, T yn = yn−1 and ‖yn‖ ≤ δn‖y‖

for all n ∈ Z+. From y ∈ Y = H0(T ), we obtain that yn ∈ H0(T ) for eachn ∈ N. Moreover, since y ∈ K (T ) = XT ({0}), we easily obtain that yn ∈ K (T )(n ∈ N). Thus yn ∈ Y , and therefore y ∈ K (S). This shows that Y ⊆ K (S). Theopposite inclusion is clear since K (S) = K (T ) ∩ Y ⊆ Y . Hence Y = K (S).Finally, from Theorem 27.2.1 we have

H0(T ) ∩ K (T ) = Y = XS({0}) = {0}. �

Proposition 27.2.8 Suppose that the sum H0(λ0 I − T )+ (λ0 I − T )(X ) isnorm-dense in X. Then T ′ has SVEP at λ0.

Proof Also here we suppose that λ0 = 0. From Proposition 26.2.9, we havethe inclusion K (T ′) ⊆ H0(T )⊥. Now a standard duality argument shows that

ker (T ′) ∩ K (T ′) ⊆ T (X )⊥ ∩ H0(T )⊥ = (T (X ) ∩ H0(T ))⊥ .

Finally, if H (T )+ T (X ) is norm-dense in X , then the last annihilator is zero.Thus ker T ′ ∩ K (T ′) = {0}, and hence, by Theorem 27.2.3, T ′ has SVEP at 0.

�

Corollary 27.2.9 Suppose for T ∈ B(X ), where X is a Banach space, eitherthat

H0(λ0 I − T )+ K (λ0 I − T ) or N∞(λ0 I − T )+ (λ0 I − T )∞(X )

is norm-dense in X. Then T ′ has SVEP at λ0. �

Let T ∈ B(X ), where X is a Banach space, and let f be an analytic functionon the open neighbourhoodU ofσ (T ). Let f (T ) ∈ B(X ) be the operator definedby the classical Riesz functional calculus, as explained in Part I, Chapter 4].We have the following remarks.

(i) If the operator T has SVEP, then f (T ) also has SVEP: see Theorem 3.3.6 ofLaursen andNeumann (2000). If f is non-constant on each of the connectedcomponents ofU , then T has SVEP if and only if f (T ) does: see Theorem3.3.9 of Laursen and Neumann (2000).


(ii) We have f (σT (x)) ⊆ σ f (T )(x) (x ∈ X ). If T has SVEP or if the func-tion f is non-constant on each of the connected components of U ,then f (σT (x)) = σ f (T )(x) (x ∈ X ): see Theorem 3.3.8 of Laursen andNeumann (2000).

For an arbitrary operator T ∈ B(X ) on a Banach space X , let

((T ) := {λ ∈ C : T does not have SVEP at λ} .

From the identity theorem for analytic functions it readily follows that ((T )is open and, consequently, is contained in the interior of the spectrum σ (T ).Clearly, ((T ) is empty precisely when T has SVEP.The next result shows that ((T ) behaves canonically under the Riesz func-

tional calculus.

Theorem 27.2.10 Let T ∈ B(X ), where X is a Banach space. Let f : U → C

be an analytic function on an open neighbourhoodU of σ (T ), and suppose thatf is non-constant on each of the connected components of U. Then f (T ) hasSVEP at λ ∈ C if and only if T has SVEP at every point µ ∈ σ (T ) for whichf (µ) = λ. Moreover, f (((T )) = ((( f (T )).

Proof Suppose first that f (T ) has SVEP at λ ∈ C. Let µ ∈ σ (T ) be such thatf (µ) = λ. In order to show that T has SVEP at µ, it suffices, by Theorem27.2.3, to show that

ker (µI − T ) ∩ XµI−T (∅) = {0} .

Let x ∈ ker (µI − T ) ∩ XµI−T (∅) be given. Since f (µ) = λ, there exists ananalytic function g on U such that λ− f = (µ− Z )g. The Riesz functionalcalculus preserves multiplication, and so

λI − f (T ) = (µI − T )g(T ) ,

and therefore x ∈ ker(λI − f (T )). Moreover, from σµI−T (x) = ∅, we obtainσT (x) = ∅, and hence σ f (T )(x) = f (σT (x)) = ∅. Consequently,

ker(µI − T ) ∩ XµI−T (∅) ⊆ ker(λI − f (T )) ∩ XλI− f (T )(∅) ,

and hence, again by Theorem 27.2.3, T has SVEP at µ.Conversely, let λ ∈ C and suppose that T has SVEP at every µ ∈ σ (T ) for

which f (µ) = λ. From the classical spectral mapping theorem, f (σ (T )) =σ ( f (T )), and so λ ∈ σ ( f (T )). By assumption, f is non-constant on each ofthe connected components of U , and thus the function f − λ has only finitely


many zeros in σ (T ), and these zeros are of finite multiplicity. Hence there existan analytic function g on U without zeros in σ (T ) and a polynomial p, ofthe form p(z) = (z − µ1) · · · (z − µn), with not necessarily distinct elementsµ1, . . . , µn ∈ σ (T ) such that the factorization f − λ = pg holds.Now, assume that x ∈ ker(λI − f (T )) ∩ XλI− f (T )(∅). In order to prove that

f (T ) has SVEP at λ, it suffices, again by Theorem 27.2.3, to show that x = 0.From the classical spectral mapping theorem, we know that g(T ) is invertible.Thus the equality f (T )− λI = g(T ) f (T ) implies that p(T )x = 0, and hence(µ1 I − T )y = 0, where y := q(T )x and q(z) := (z − µ2) · · · (z − µn). On theother hand, x ∈ XλI− f (T )(∅) and f is non-constant on each of the connectedcomponents of U . By Remark (ii) after Corollary 27.2.9, we have f (σT (x)) =σ f (T )(x) = ∅, and therefore, since T and q(T ) commute,

σT (y) = σT (q(T )x) ⊆ σT (x) = ∅ .

But T has SVEP at µ1, by assumption, so, again by Theorem 27.2.3, y = 0.Repetitions of this argument for µ2, . . . , µn then lead to the equality x = 0.Thus f (T ) has SVEP at λ.The last claim is obvious, being nothing else than a reformulation of the

equivalence proved above. �

Note that in the preceding result the additional condition on the analyticfunction is essential. In fact, if f is constant in U , then ((( f (T )) is certainlyempty, while f (((T )) is empty only when T has SVEP.An immediate consequenceofTheorem27.2.10 is that, in the characterization

of SVEP at a point λ0 ∈ C given in Theorem 27.2.3, the kernel ker(λ0 I − T )may be replaced by the generalized kernel N∞(λ0 I − T ).

Corollary 27.2.11 For every bounded operator T on a Banach space X thefollowing properties are equivalent:

(i) T has SVEP at λ0;(ii) T n has SVEP at λ0 for each n ∈ N;(iii) N∞(λ0 I − T ) ∩ XT (∅) = {0};(iv) N∞(λ0 I − T ) ∩ K (λ0 I − T ) = {0}.

Proof The equivalence (i)⇔ (ii) is obvious fromTheorem 27.2.10. Combiningthis equivalence with Theorem 27.2.3, we then obtain that T has SVEP at λ0if and only if ker(λ0 I − T )n ∩ XT (∅) = {0} for every n ∈ N. This argumentshows the equivalence (i) ⇔ (iii). The equivalence (i) ⇔ (iv) follows fromTheorem 27.2.3 in a similar way. �


27.3 Exercises

1. Show that T has SVEP at λ0 if and only if, for every 0 �= x ∈ ker (λ0 I − T ),we have σT (x) = {λ0}.

2. Show that a semi-regular operator T has SVEP at 0 if and only if T isinjective. Analogously, T ′ has SVEP at 0 if and only if T is surjective.

3. Show that if, for an operator T ∈ B(X ), one of the sums

H0(λ0 I′ − T ′)+ (λ0 I ′ − T ′)(X ′), H0(λ0 I

′ − T ′)+ K (λ0 I ′ − T ′) ,

or

N∞(λ0 I ′ − T ′)+ (λ0 I ′ − T ′)∞(X ′)

is weak-∗ dense in X ′, then T has SVEP at λ0.4. Let X := 2(N)⊕ 2(N)⊕ · · · , provided with the norm ‖ · ‖ specified by‖x‖ := (∑∞

n=1 ‖xn‖2)1/2 for x := (xn) ∈ X , and define

Tnei :={ei+1 if i = 1, . . . , n ,ei+1/(i − n) if i > n .

Let T := T1 ⊕ · · · ⊕ Tn ⊕ · · · . Show that T has SVEP, and that the quasi-nilpotent part H0(T ) is not closed. Hint: check that T n is quasi-nilpotent,and find an element x := (xn) ∈ X which shows that H0(T ) �= X . Showthat H0(T ) is norm-dense in X .

5. Find an example of operator forwhichN∞(T )+ T∞(X ) is not norm-densein X , while T ′ has SVEP. Hint: consider the Volterra operator defined inPart IV, Example 22.2.6 and Part I, Exercise 1.5.6.

6. Let be a component of ρK (T ). Show the following alternative: either Thas SVEP at every λ ∈ or T does not have SVEP at any λ ∈ . Hint:use Theorem 26.2.10.

7. Let K denote a compact subset of C with a non-empty interior, and letX := ∞(K ) denote the Banach space of all bounded, complex-valuedfunctions on K , as in Chapter 1, and consider the operator T : X → Xdefined by the assignment (T f )(λ) := λ f (λ) for f ∈ X and λ ∈ K . Showthat T has SVEP, while σp(T ) = K .

8. Let 1 ≤ p <∞, and denote by ω := (ωn) a bounded sequence of strictlypositive real numbers. The corresponding unilateral weighted right shiftoperator on the Banach space p(N) is the operator defined by:

T x =∞∑n=1

ωnxnen+1 (x = (xn) ∈ p(N)) .


Check that T has SVEP. Moreover, show that H0(T )+ T (X ) is norm-dense in p(N) if and only if limn→∞ sup(ω1 · · ·ωn)1/n = 0. Show that T ′has SVEP at 0 precisely when limn→∞ inf(ω1 · · ·ωn)1/n = 0.

9. Let T ∈ B(X ), where X is a Banach space. Given an element x ∈ X , thequantity νT (x) := lim supn→∞ ‖T nx‖1/n is called the local spectral radiusof T at x . Show that, if T has SVEP, then νT (x) = max{|λ| : λ ∈ σT (x)}.Further information on this local spectral radius may be found in Laursenand Neumann (2000).

10. Show that, if T is an isometric non-unitary operator on a Hilbert spaceH , then the adjoint T ′ does not have the SVEP at 0. See Example 1.7 ofColojoara and Foias (1968).


The single-valued extension property appeared first in Dunford (1952, 1954),and then received a systematic treatment in Dunford and Schwartz (1958, 1963,1971). This property also forms a preliminary topic of the book by Colojoaraand C. Foias (1968). The fundamental role of SVEP in local spectral theory ismore clearly explained in the recent monograph Laursen and Neumann (2000).The characterizations of the analytical core and the quasi-nilpotent part of

an operator given in Theorem 27.2.1 and Theorem 27.2.2 are due to Vrbova(1973) and Mbekhta (1987, 1990); see also Proposition 2.2 of Laursen (1992).The localizedSVEPat a point has been introduced byFinch (1975),who showedthat a surjective operator T has the SVEP at 0 precisely when it is injective.The characterization of SVEP at a single point λ0 given in Theorem 27.2.3 istaken from Aiena and Monsalve (2000).Except for Corollary 27.2.5, which is due to Mbekhta (1990), the source

of the results of the second section is essentially that of Aiena, Miller andNeumann (2001) and Aiena, Colasante and Gonzalez (2002). In particular,Example 27.2.6 and the local spectral mapping result of Theorem 27.2.10 aretaken fromAiena,Miller andNeumann (2001). In this paper the readermay findapplications to isometries, analytic Toeplitz operators, invertible compositionoperators on Hardy spaces, and weighted shifts.

28

SVEP for semi-Fredholm operators

In Chapter 27 we established many conditions which imply SVEP at a point.The main goal of this chapter is to show that all these implications becomeequivalences for an important class of operators, the class of all semi-Fredholmoperators. It will be also shown that, for these operators, SVEP at a pointλ0 ∈ C is equivalent to the finiteness of two important quantities, the ascent andthe descent of the operator λ0 I − T . These equivalences also provide usefulinformation on the fine structure of the spectrum. In particular, we shall showthat many spectra originating from Fredholm theory coincide whenever T orT ′ have SVEP.

28.1 Ascent, descent, and semi-Fredholm operators

Let us recall the definition of some classical quantities associated with an op-erator. Given a linear operator T on a vector space X , it is easy to see thatker T n ⊆ ker T n+1 and T n+1(X ) ⊆ T n(X ) for every n ∈ N.

Definition 28.1.1 The ascent of T is the smallest positive integer p = p(T ),whenever it exists, such that ker T p = ker T p+1. If such p does not exist, setp = ∞. Analogously, the descent of T is defined to be the smallest integerq = q(T ), whenever it exists, such that T q+1(X ) = T q (X ). If such q does notexist, set q = ∞. If both p(T ) and q(T ) are finite, then T has finite chains.Note that p(T ) = 0 means that T is injective and q(T ) = 0 means that T is

surjective. Clearly, T has finite ascent if and only ifN∞(T ) = ker T k for somek ∈ N, and, analogously, T has finite descent if and only if T∞(X ) = T k(X )for some k ∈ N.The relevant fact is that the finiteness of the ascent p(T ) may be expressed

by means of some intersection properties between kernels and ranges of theiterates of T .

298

28 SVEP for semi-Fredholm operators 299

Lemma 28.1.2 Let T be a linear operator on a vector space X, and letm ∈ Z+. Then

(i) p(T ) ≤ m <∞ if and only if, for every n ∈ N, T m(X ) ∩ ker T n = {0}.(ii) q(T ) ≤ m <∞ if and only if, for every n ∈ N, there exists a subspace

Yn ⊆ ker Tm such that X = Yn ⊕ T n(X ).

Proof (i) Suppose that p(T ) ≤ m <∞ and that n ∈ N. Let us consider anelement y ∈ Tm(X ) ∩ ker T n . Then there exists x ∈ X such that y = Tmx andT n y = 0. From this we obtain the equality Tm+nx = T n y = 0, and thereforewe have x ∈ ker Tm+n = ker Tm . Hence y = Tmx = 0.Conversely, suppose that Tm(X ) ∩ ker T n = {0} for some m ∈ N, and let

x ∈ ker Tm+1. Then Tmx ∈ ker T , and thereforeTmx ∈ Tm(X ) ∩ ker T ⊆ Tm(X ) ∩ ker T n = {0}.

Hence x ∈ ker Tm . We have shown that ker Tm+1 ⊆ ker Tm . Since the oppositeinclusion is satisfied for all operators, we conclude that ker Tm = ker Tm+1.(ii) Let q := q(T ) ≤ m <∞, and let Y be a complementary subspace to

T n(X ) in X . Let {x j : j ∈ J } be a basis of Y . For every element x j of thebasis there exists an element y j ∈ X such that T qx j = T q+n y j . This fol-lows because T q (Y ) ⊆ T q (X ) = T q+n(X ). Set z j := x j − T n y j . Then clearlyT qz j = T qx j − T q+n y j = 0. From this it follows that the linear subspace Yngenerated by the elements z j is contained in ker T q and a fortiori in ker Tm .From the decomposition X = Y ⊕ T n(X ), we obtain for every x ∈ X a repre-sentation of the form

x =∑j∈J

λ j x j + T n y =∑j∈J

λ j (z j + T n y j )+ T n y =∑j∈J

λ j z j + T nz ,

so that X = Yn + T n(X ). We show that this sum is direct. Indeed, choose anelement x ∈ Yn ∩ T n(X ). Then x =

∑j∈J µ j z j = T nv for some v ∈ X , and

therefore ∑j∈J

µ j x j =∑j∈J

µ j Tn y j + T nv ∈ T n(X ) .

From the decomposition X = Y ⊕ T n(X ), we then obtain that µ j = 0 for allj ∈ J , and hence that x = 0. Therefore Yn is a complement of T n(X ) containedin ker Tm .Conversely, if for n ∈ N the subspace T n(X ) has a complement Yn ⊆ ker Tm ,

then Tm(X ) = Tm(Yn)+ Tm+n(X ) = Tm+n(X ), and therefore q(T ) ≤ m. �

Theorem 28.1.3 If both p(T ) and q(T ) are finite, then p(T ) = q(T ).


Proof Set p := p(T ) and q := q(T ). Assume first that p ≤ q, so that theinclusion T q (X ) ⊆ T p(X ) holds. Obviously, wemay suppose that q > 0. Frompart (ii) of Lemma 28.1.2, we have X = ker T q + T q (X ), so every elementy := T p(x) ∈ T p(X ) admits the decomposition y = z + T qw ,with z ∈ ker T q .From z = T px − T qw ∈ T q (X ), we then obtain that z ∈ ker T q ∩ T q (X ), andthe last intersection is {0} by part (i) of Lemma 28.1.2. Therefore, we havey = T qw ∈ T q (X ), and this shows the equality T p(X ) = T q (X ), from whichwe obtain p ≥ q, so that p = q .Assume now that q ≤ p and p > 0, so that ker T q ⊆ ker T p. From part (ii)

of Lemma 28.1.2, we have X = ker T q + T p(X ), so that an arbitrary elementx of ker T p admits the representation x = u + T pv , with u ∈ ker T q . FromT px = T pu = 0, it then follows that T 2pv = 0, so that v ∈ ker T 2p = ker T p.Hence T pv = 0, and consequently x = u ∈ ker T q . This shows that we haveker T q = ker T p, and hence q ≥ p. Therefore p = q . �

If p(T ) <∞, then C(T ) = T∞(X ). This is a consequence of Lemma 26.1.9and part (i) of Lemma 28.1.2. The equality C(T ) = T∞(X ) is still true whenq := q(T ) <∞. Indeed, in this case, by definition,

ker T ∩ T q (X ) = ker T ∩ T q+k(X ) for k ∈ Z+,

so that Lemma 26.1.9 again applies.Two other important quantities associatedwith a linear operator T on a vector

space X are the nullity α(T ) := dim ker T and the defect β(T ) := codim T (X ).Let�(X ) denote the set of all linear operators on the vector space X forwhich

α(T ) and β(T ) are both finite. For every T ∈ �(X ), the index of T , defined byind T := α(T )− β(T ) ,

satisfies the basic index theorem:

ind (T S) = ind T + ind S ( T, S ∈ �(X )) .See Theorem 23.1 of Heuser (1982).In the next theoremwe establish the basic relationships between the quantities

α(T ), β(T ), p(T ), and q(T ).

Proposition 28.1.4 Let T be a linear operator on a vector space X. Then thefollowing properties hold.

(i) If p(T ) <∞ then α(T ) ≤ β(T ).(ii) If q(T ) <∞ then β(T ) ≤ α(T ).(iii) If p(T ) = q(T ) <∞, then α(T ) = β(T ) (possibly infinite).(iv) If α(T ) = β(T ) <∞, and, if one chain is finite, then p(T ) = q(T ).


Proof (i) Let p := p(T ) <∞. Obviously, if β(T ) = ∞, there is nothing toprove. Suppose that β(T ) <∞. It is easy to check that also β(T n) is finite. ByLemma 28.1.2(i), we have ker T ∩ T p(X ) = {0}, and this implies the inequalityα(T ) <∞. From the index theorem, we obtain

n · ind T = ind T n = α(T p)− β(T n) (n ≥ p) .

Now, suppose that q := q(T ) <∞. We see that, for each n ≥ max{p, q},the quantity n · ind T = α(T p)− β(T p) is constant, so that ind T = 0, that is,α(T ) = β(T ).Consider the other case, where q = ∞. Then β(T n)→ 0, as n→∞, so

n · ind T eventually becomes negative, and hence ind T < 0. Therefore in thiscase we have α(T ) < β(T ).(ii) Letq := q(T ) <∞. Also herewe can suppose thatα(T ) <∞, otherwise

there is nothing to prove. Consequently, as is easy to check, we also haveβ(T n) <∞ and, by Lemma 28.1.2(ii), X = Y ⊕ T (X ) with Y ⊆ ker T q . Fromthis, it follows that

β(T ) = dim Y ≤ α(T q ) <∞ .

With appropriate changes, the index argument used in the proof of (i) showsthat β(T ) = α(T ) if p(T ) <∞, and β(T ) < α(T ) if p(T ) = ∞.(iii) This is clear from (i) and (ii).(iv) This is an immediate consequence of the equality

α(T n)− β(T n) = ind T n = n · ind T = 0 ,

which is valid for every n ∈ N. �

Proposition 28.1.5 For a bounded operator T on a Banach space X, thefollowing implications hold:

(i) p(λ0 I − T ) <∞⇒ N∞(λ0 I − T ) ∩ (λ0 I − T )∞(X ) = {0} ⇒T has SVEP at λ0;

(ii) q(λ0 I − T ) <∞⇒ X = N∞(λ0 I − T )+ (λ0 I − T )∞(X )⇒T ′ has SVEP at λ0.

Proof (i) There is no loss of generality in supposing that λ0 = 0. We definep := p(T ) <∞. ThenN∞(T ) = ker T p and T∞(X ) ⊆ T p(X ). From Lemma28.1.2 (i), we then conclude that N∞(T ) ∩ T∞(X ) ⊆ ker T p ∩ T p(X ) = {0}.The second implication is a consequence of Theorem 27.2.3 because

ker T ∩ XT (∅) ⊆ N∞(T ) ∩ T∞(X ) = {0}.


(ii) Again we may suppose that λ0 = 0. Suppose that q := q(T ) <∞. ThenT∞(X ) = T q (X ) and X = T n(X )+ ker T q (n ∈ N) by Lemma 28.1.2(ii).From this it easily follows that X = N∞(T )+ T∞(X ).In order to show that the second implication of (ii) holds, we first note that, if

X = N∞(T )+ T∞(X ), thenN∞(T )⊥ ∩ T∞(X )⊥ = {0}. Now take an elementx ′ ∈ ker T ′ ∩ XT ′ (∅). Then

x ′ ∈ ker T ′ ⊆ ker (T ′)n = ker (T n)′ = T n(X )⊥ = T n(X )⊥ ,

for every n ∈ N and therefore x ′ ∈ T∞(X )⊥.On the other hand, from σT ′ (x ′) = ∅ we obtain, by Theorem 27.2.1, that

x ′ ∈ K (T ′) ⊆ (T ′)n(X ′) = (T n)′(X ′) ⊆ (ker T n)⊥

for every n ∈ N. From this, it easily follows that x ′ ∈ N∞(T )⊥, and this impliesthat x ′ ∈ N∞(T )⊥ ∩ T∞(X )⊥. Therefore x ′ = 0. From Theorem 27.2.3, wethen conclude that T ′ has SVEP at 0. �

The preceding result is quite useful to establish SVEP for several classes ofoperators. In fact, there are several operators which satisfy the condition thatp(λI − T ) <∞ for all λ ∈ C; see the Exercises.

28.2 Fredholm operators

Definition 28.2.1 A bounded operator T ∈ B(X ), where X is a Banach space,is upper semi-Fredholm if α(T ) <∞ and T (X ) is closed. The operator T inB(X ) is lower semi-Fredholm if β(T ) <∞. Denote by �+(X ) the class of allupper semi-Fredholm operators, and by �−(X ) the class of all lower semi-Fredholm operators. The class �(X ) of all Fredholm operators is defined by�(X ) := �+(X ) ∩�−(X ), while the class of all semi-Fredholm operators isdefined by �±(X ) := �+(X ) ∪�−(X ).

Note that, if β(T ) <∞, then T (X ) is closed (see Chapter 26). If T ∈ �±(X ),we can define the index of T as above. Obviously, the index of a semi-Fredholmoperator is an integer or ±∞.The three sets �+(X ), �−(X ), and �(X ) are semi-groups, in the sense that,

if T, S belong to one of these sets, then also the products T S and ST lie in thesame set. Moreover, these three sets are each open, T ∈ �+(X ) if and only ifT ′ ∈ �−(X ′), and dually, T ∈ �−(X ) if and only if T ′ ∈ �+(X ′); the readercan find these classical results on Fredholm theory in Caradus, Pfaffenbergerand Yood (1974).


Note that, if T ∈ �±(X ), then T n ∈ �±(X (n ∈ N), and this implies thatT∞(X ) is closed because the subspaces T n(X ) are closed for each n ∈ N.From Proposition 26.1.10, we then have T (T∞(X )) = T∞(X ), and hence it fol-lows from Proposition 26.1.7 that T∞(X ) ⊆ K (T ). But the opposite inclusionK (T ) ⊆ T∞(X ) is true for every operator T ∈ B(X ), so that K (T ) = T∞(X )for every T ∈ �±(X ).To explore SVEP at a point in the case of semi-Fredholm operators, we need

a decomposition property for these operators due to Kato (1958). We state thisclassical result without proving it (the proof of this decomposition is ratherinvolved; the interested reader will find a clear proof in West (1987)).

Theorem 28.2.2 Let T ∈ �±(X ), where X is a Banach space. Then thereexist closed T -invariant subspaces M, N of X such that X = M ⊕ N, T | Mis semi-regular, T | N is nilpotent, and N is finite-dimensional.

For an arbitrary bounded operator T ∈ B(X ), a pair (M, N ) of closed T -invariant subspaces is called a Kato decomposition for T if the decompo-sition observed in Theorem 28.2.2 for semi-Fredholm operators holds, thatis, X = M ⊕ N , T | M is semi-regular, T | N is nilpotent, and N is finite-dimensional. In the literature the operators which admit a Kato decompositionare called essentially semi-regular; see Muller (1994). A semi-regular operatoris obviously another example of an operator which admits a Kato decomp-osition. Note that not every semi-Fredholm operator is semi-regular; a semi-Fredholm T is semi-regular precisely when its jump j(T ) = 0; see Part IV,Chapter 24 for the definition and also West (1987).It is not difficult to show that, if (M, N ) is a Kato decomposition for T ,

then the pair (N⊥,M⊥) is a Kato decomposition for T ′. The proof of this factrequires methods of standard duality theory (Exercise).

Lemma 28.2.3 Suppose that (M, N ) is a Kato decomposition for T ∈ B(X ).Then:

(i) K (T ) = K (T | M) and K (T ) is closed;(ii) ker T | M = K (T ) ∩ ker T ;(iii) K (T ) = T∞(X ).

Proof (i) To prove the equality K (T ) = K (T | M), we need only to show theinclusion K (T ) ⊆ M . Let x ∈ K (T ) and, according to the definition of K (T ),let (un) be a sequence of X and δ > 0 be such that x = u0, Tun+1 = un , and

‖un‖ ≤ δn‖x‖ (n ∈ Z+).


Clearly, T nun = x (n ∈ N). From the decomposition X = M ⊕ N , we knowthat x = y + z, un = yn + zn , with y, yn ∈ M and z, zn ∈ N for every n ∈ N.Then x = T nun = T n yn + T nzn , hence, by the uniqueness of the decomposi-tion, y = T n yn and z = T nzn . Let P denote the projection of X onto N alongM . From the estimate

‖((T |N )P)n‖1/n ≤ ‖(T | N )n‖1/n‖Pn‖1/n = ‖(T | N )n‖1/n‖P‖ ,

we infer that (T |N )P is quasi-nilpotent, since, by assumption, T | N is nilpo-tent, and hence quasi-nilpotent. Therefore, if ε > 0, there is a positive integern0 such that ‖(T | N P)n‖1/n < ε (n > n0). Now we have

‖z‖ = ‖T nzn‖ = ‖T n Pvn‖ = ‖T n Pnvn‖‖(T P)nvn‖ ≤ εnδn‖x‖ (n > n0) .

Since ε is arbitrary, the last term of the previous inequality converges to 0, soz = 0 and hence x = y ∈ M .The final assertion is a consequence of Proposition 26.1.12 since T | M is

semi-regular.(ii) This equality is an immediate consequence of (i). Indeed, K (T ) ⊆ M and,

since T | M is semi-regular, from Proposition 26.1.12 and part (i) we obtainthat

ker T | M ⊆ (T | M)∞(M) = K (T | M) = K (T ) .

Hence

K (T ) ∩ ker T = K (T ) ∩ M ∩ ker T = K (T ) ∩ ker T | M = ker T | M .

(iii) If (T | N )d = 0, then we have

T n(X ) = T n(M)⊕ T n(N ) = T n(M)⊕ {0} (n ≥ d) ,

and this implies that T∞(X ) = (T | M)∞(M). The semi-regularity of T | Mthen yields that K (T | M) = (T | M)∞(M), and hence, by (i), we haveT∞(X ) = K (T | M) = K (T ). �

Remark 28.2.4 It should be noted that, if (M, N ) is a Kato decompositionfor λ0 I − T , then T has SVEP at λ0 if and only if T | M has SVEP at λ0. Infact, if T has SVEP at λ0, then T | M has SVEP at λ0 since the SVEP at a pointis inherited by the restrictions to closed invariant subspaces. On the other hand,if T | M has SVEP at λ0, the semi-regularity of (λ0 I − T ) | M entails that therestriction (λ0 I − T ) | M is injective, and therefore

ker(λ0 I − T ) | M = K (T ) ∩ ker(λ0 I − T ) = {0}.


Thus, by Theorem 27.2.3, T has SVEP at λ0. Note that this argument alsoshows T has SVEP at λ0 if and only if (λ0 I − T ) | M is injective. An obviousconsequence of this fact is that a semi-regular operator T has SVEP at λ0 if andonly if λ0 I − T is injective, or, equivalently, λ0 I − T is bounded below.

Lemma 28.2.5 Let T ∈ B(X ), and suppose that X is the direct sum of twoclosed subspaces M, N each invariant under T , forwhich T | M is semi-regular.Then T | M is surjective if and only if T ′ | N⊥ is injective.

Proof Suppose first that T (M) = M and x ′ ∈ ker T ′ | N⊥ = ker T ′ ∩ N⊥. Forevery m ∈ M , there exists m1 ∈ M such that Tm1 = m. We have

〈m, x ′〉 = 〈Tm1, x ′〉 = 〈m1, T ′x ′〉 = 0 ,

and hence x ′ ∈ M⊥ ∩ N⊥ = {0}.Conversely, suppose that T (M) ⊂ M and T (M) �= M . By assumption, T (M)

is closed since T | M is semi-regular, and hence, via theHahn–Banach theorem,there exists z′ ∈ X ′ such that z′ ∈ T (M)⊥ and z′ /∈ M⊥. Now, from the decom-position X ′ = N⊥ ⊕ M⊥, we have z′ = n′ + m ′ with n′ ∈ N⊥ and m ′ ∈ M⊥.For every m ∈ M , we obtain

〈m, T ′n′〉 = 〈Tm, n′〉 = 〈Tm, z′〉 − 〈Tm,m ′〉 = 0 .

Hence T ′n′ ∈ T ′(N⊥) ∩ M⊥ = N⊥ ∩ M⊥ = {0}, so that

0 �= n′ ∈ ker T ′ ∩ N⊥. �

Theorem 28.2.6 Suppose that λ0 I − T ∈ �±(X ). Then the implications (i)and (ii) of Proposition 28.1.5 are equivalences.

Proof To show the equivalence in (i) of Proposition 28.1.5, we need onlyprove that, if T has SVEP at λ0, then p(λ0 I − T ) <∞. Let (M, N ) be a Katodecomposition for λ0 I − T , and let k ∈ N be such that ((λ0 I − T ) | N )k = 0.Suppose that p := p(λ0 I − T ) = ∞. Then (λ0 I − T )|M is not injective; other-wise, if (λ0 I − T )|M were injective, for n ≥ k we would have

ker(λ0 I − T )n = ker(λ0 I − T ) | M)n = ker(λ0 I − T ) | N )n = {0} ,

and hence p(λ0 I − T ) <∞. Hence (λ0 I − T )|M is not injective, and thereforeT does not have the SVEP at λ0.Analogously, to show the equivalence in (ii) of Proposition 28.1.5, we need

only prove that, if T ′ has SVEP at λ0, then q(λ0 I − T ) <∞. Suppose that


q(λ0 I − T ) = ∞. Then λ0 I − T | M is not surjective; for otherwise, for everypositive integer n ≥ k, we would have

(λ0 I − T )n(X ) = (λ0 I − T )n(M)⊕ {0} = M,

and hence q(λ0 I − T ) <∞. Now, by Lemma 28.2.5, it follows thatλ0 I ′ − T ′ | N⊥ is not injective, and hence T ′ does not have SVEP at λ0. �

Recall that T is said to be a Riesz–Schauder operator if T is a Fredholmoperator having both ascent and descent finite. Note that, by Lemma 28.1.2, aRiesz–Schauder operator has index 0.The next result shows that the class of operators without SVEP is rather large,

since every Fredholm operator having index greater than 0 belongs to the class.

Corollary 28.2.7 If λ0 I − T ∈ �±(X ), where X is a Banach space, and if Thas SVEP at λ0, then λ0 I − T is upper semi-Fredholm and

ind(λ0 I − T ) ≤ 0.

Additionally, if also T ′ has SVEP at λ0, then λ0 I − T is a Riesz–Schauderoperator.

Proof The first assertion is clear since the condition p(λ0 I − T ) <∞ entailsthat α(λ0 I − T ) ≤ β(λ0 I − T ) by Proposition 28.1.4. The second assertionis also clear because, if both the operators T and T ′ have SVEP at λ0, thenp(λ0 I − T ) <∞ and q(λ0 I − T ) <∞. Consequently, p(λ0 I − T ) =q(λ0 I − T ) by Theorem 28.1.3, and therefore, by Proposition 28.1.4,

α(λ0 I − T ) = β(λ0 I − T ) <∞. �

It should be noted that the last assertion of Corollary 28.2.7 applies to everydecomposable operator T on a Banach space, that is, if T is semi-Fredholmand decomposable, then T is a Riesz–Schauder operator.

Proposition 28.2.8 Let λ0 I − T ∈ �±(X ) and suppose that (M, N ) is a Katodecomposition for λ0 I − T . Then the following properties are equivalent:(i) T has SVEP at λ0;(ii) H0(λ0 I − T ) = N;(iii) H0(λ0 I − T ) is finite-dimensional;(iv) H0(λ0 I − T ) is closed;(v) H0(λ0 I − T ) ∩ K (λ0 I − T ) = {0};


In this case, set p = p(λ0 I − T ). ThenH0(λ0 I − T ) = N∞(λ0 I − T ) = ker (λ0 I − T )p .

Proof There is no loss of generality if we suppose that λ0 = 0.Clearly, (ii)⇒ (iii)⇒ (iv), and, from Proposition 27.2.7, we know that the

implications (iv)⇒ (v)⇒ (i) hold, so we need only prove that (i)⇒ (ii).(i) ⇒ (ii) First note that, if the operator T admits a Kato decomposition

(M, N ), then H0(T ) = H0(T | M)+ H0(T | N ). The inclusion

H0(T ) ⊇ H0(T | M)+ H0(T | N )

is obvious. In order to show the opposite inclusion, let us consider an arbitraryelement x ∈ H0(T ), and let x = u + v , with u ∈ M and v ∈ N . Since T | N isquasi-nilpotent, we have N = H0(T | N ) ⊆ H0(T ). Consequently,

u = x − v ∈ H0(T ) ∩ M = H0(T | M),

and therefore H0(T ) ⊆ H0(T | M)+ H0(T | N ). HenceH0(T ) = H0(T | M)+ H0(T | N ) = H0(T | M)+ N .

Now, suppose that T has SVEP at 0. Then, as observed in Remark 28.2.4,T | M is injective and therefore, by Proposition 26.2.9,

H0(T | M) =∞⋃n=1ker (T | M)n = {0} .

This shows that H0(T ) = N .To prove the last assertion, observe first that the inclusions

ker T n ⊆ N∞(T ) ⊆ H0(T )

hold for every T ∈ B(X ) and n ∈ N. Let k ∈ N be such that (T |N )k = 0. ThenH0(T ) = N ⊆ ker T k , and hence H0(T ) = N∞(T ) = ker T k . From this, it fol-lows that p := p(T ) ≤ k, and therefore ker T k = ker T p. �

Proposition 28.2.9 Letλ0 I − T ∈ �±(X ), and suppose that (M, N ) is aKatodecomposition for T . Then the following assertions are equivalent:

(i) T ′ has SVEP at λ0;(ii) K (T ) = M;(iii) K (T ) is finite-codimensional;


(iv) X = H0(λ0 I − T )+ K (λ0 I − T );(v) H0(λ0 I − T )+ K (λ0 I − T ) is norm-dense in X.In this case, set q = q(λ0 I − T ). Then

K (λ0 I − T ) = T∞(λ0 I − T ) = (λ0 I − T )q (X ) .

Proof Also here we suppose that λ0 = 0.(i)⇒ (ii) We know that the pair (N⊥,M⊥) is a Kato decomposition for the

semi-Fredholm operator T ′, thus T ′ | N⊥ is semi-regular, T ′ | M⊥ is nilpotent,and further dimM⊥ = codimM <∞. From the assumption, T ′ has SVEP at 0,so the semi-regularity of T ′ | N⊥ entails that T ′ | N⊥ is injective, and hence, byLemma 28.2.5, T | M is surjective. This shows that M = K (T | M) = K (T ).The implication (ii)⇒ (iii) is obvious.(iii)⇒ (i) Since K (T ) = T∞(X ), the space T∞(X ) has finite codimension.

Because T∞(X ) ⊆ T q (X ) for every q ∈ N, we then conclude that q(T ) <∞,so T ′ has SVEP at 0 by Proposition 28.2.6.(i) ⇒ (iv) Assume that T ′ has SVEP at 0, or, equivalently, that q <∞,

where q = q(T ). Then K (T ) = T∞(X ) = T q (X ). Moreover, we see that X =ker T q + T n(X ) (n ∈ N) by Lemma 28.1.2(ii), and therefore it follows thatX = H0(λ0 I − T )+ K (λ0 I − T ).The implication (iv)⇒ (v) is obvious, while the implication (v)⇒ (i) has

been proved in Corollary 27.2.9.The last assertion is clear. �

Let σap(T ) be the approximate point spectrum of T and let σsu(T ) be the sur-jectivity spectrum of T ; for the definitions, see Part IV, Chapter 22. Obviously,these spectra are non-empty sets since they both contain σK (T ), and hence, byTheorem 26.2.11, they contain also the boundary of σ (T ). FromRemark 28.2.4it easily follows that, if T has SVEP, then σK (T ) = σap(T ). We have alreadyobserved that T has SVEP at every point which is not a limit point of σ (T ).What happens if we consider instead of σ (T ) some distinguished parts of thespectrum? A simple argument shows the following implications:

σap(T ) does not cluster at λ0 ⇒ T has SVEP at λ0 , (28.2.1)

and

σsu(T ) does not cluster at λ0 ⇒ T ′ has SVEP at λ0 . (28.2.2)

Indeed, if σap(T ) does not cluster at λ0, then there is an open disc D(λ0, δ)centred at λ0 such that λI − T is injective for every λ ∈ D(λ0, δ) with λ �= λ0.Let f : D (λ0, ε)→ X be an analytic function defined on another open discD(λ0, ε) centred at λ0 for which the equation (λI − T ) f (λ) = 0 holds for


every λ ∈ D(λ0, ε). Obviously, we may suppose that ε ≤ δ. It is clear thatf (λ) ∈ ker (λI − T ) = {0} for every λ ∈ D(λ0, ε) with λ �= λ0. Thus f (λ) = 0for every λ ∈ D(λ0, ε) with λ �= λ0. From the continuity of f at λ0, we concludethat f (λ0) = 0. Hence f ≡ 0 in D(λ0, ε), and therefore T has SVEP at λ0.The second implication is an immediate consequence of the equality

σsu(T ) = σap(T ′).Note that neither of the implications (28.2.1) and (28.2.2) may be reversed

in general. Indeed, if λ0 is a non-isolated boundary point of σ (T ), then σap(T )and σsu(T ) cluster at λ0, but, as observed before, T and T ′ have SVEP at λ0.An example of an operator T having the SVEP and such that every spectral

point is the limit of points ofσap(T )may be found among the unilateralweightedright shift operators. Indeed, there exist unilateral weighted right shift opera-tors T on p(N) which have SVEP and for which σ (T ) = σap(T ) = σsu(T ) andσ (T ) is a closed ball centred at 0 with radius r > 0; see Laursen and Neumann(2000).The next result shows that the implications (28.2.1) and (28.2.2) are actually

equivalences when λ0 I − T is semi-Fredholm.

Theorem 28.2.10 Suppose that λ0 I − T ∈ �±(X ). Then the following equiv-alences hold:

(i) T has SVEP at λ0⇔ σap(T ) does not cluster at λ0;(ii) T ′ has SVEP at λ0⇔ σsu(T ) does not cluster at λ0.

Proof (i) We need only to prove that, if T has SVEP at λ0, then σap(T ) doesnot cluster at λ0. Evidently, we may suppose that λ0 = 0. Assume that T hasSVEP at 0 and T ∈ �±(X ). Because�±(X ) is open there exists ε > 0 such thatλI − T is semi-Fredholm, and hence has closed range, for every |λ| < ε. Thenλ ∈ (D(0, ε) \ {0}) ∩ σap(T ) if and only if λ is an eigenvalue for T . Now it iseasy to see that ker (λI − T ) ⊆ T∞(X ) for every λ �= 0, so that every non-zeroeigenvalue of T belongs to the spectrum of the restriction T | T∞(X ).Finally, suppose that 0 is a cluster point of σap(T ). Let (λn) be a sequence

of non-zero eigenvalues which converges to 0. Then λn ∈ σ (T | T∞(X )) forevery n ∈ N, and hence 0 ∈ σ (T | T∞(X )) since the spectrum of an operatoris closed . But T has SVEP at 0, and thus T | M is injective. From part (ii) ofLemma 28.2.3, we have

{0} = ker T | M = ker T ∩ T∞(X ) ,

so the restriction T | T∞(X ) is injective.On the other hand, from the equality T (T∞(X )) = T∞(X ), we know that

T | T∞(X ) is surjective, so that 0 /∈ σ (T | T∞(X )), a contradiction.


(ii) The equivalence follows from the first part once we have observed thatalways σap(T ′) = σsu(T ). �

The result of Theorem 28.2.10 is quite useful for establishing the mem-bership of cluster points of some distinguished parts of the spectrum to thesemi-Fredholm spectrum σs f (T ) and to the essential spectrum σe(T ), whichare defined as follows:

σs f (T ) := {λ ∈ C : λI − T /∈ �±(X )}and

σe(T ) := {λ ∈ C : λI − T /∈ �(X )} .Clearly σs f (T ) ⊆ σe(T ). A first application is given in the following result,which improves a classical Putnam theorem about the non-isolated boundarypoints of the spectrum as a subset of the essential spectrum.

Corollary 28.2.11 For every operator T ∈ B(X ) on a Banach space X, everynon-isolated boundary point of σ (T ) belongs to σs f (T ), and in particular toσe(T ).

Proof If λ0 ∈ ∂σ (T ) is non-isolated in σ (T ), then σap(T ) clusters at λ0 because∂σ (T ) ⊆ σap(T ). But T has SVEP at every point of ∂σ (T ), and so, by Theorem28.2.10s, λ0 I − T /∈ �±(X ). �

Corollary 28.2.12 Suppose that T ∈ B(X ), where X is a Banach space, hasSVEP. Then all cluster points of σap(T ) belong to σs f (T ). Analogously, if T ′

has SVEP, then all cluster points of σsu(T ) belong to σs f (T ).

Proof Suppose that λ0 /∈ σs f (T ). Since T has SVEP, and in particular hasSVEP at λ0, then σap(T ) does not cluster at λ0, by Theorem 28.2.10.Analogously, suppose that λ0 /∈ σs f (T ). Since T ′ has SVEP atλ0, then σsu(T )

does not cluster at λ0, again by Theorem 28.2.10. �

All the results established above have interesting applications. Recall thatthe upper semi-Fredholm spectrum and lower semi-Fredholm spectrum aredefined by

σu f (T ) := {λ ∈ C : λI − T /∈ �+(X )} ,σl f (T ) := {λ ∈ C : λI − T /∈ �−(X )} ,

respectively. Clearly,

σs f (T ) ⊆ σu f (T ) ⊆ σap(T ) . (28.2.3)


In the next result we consider a situationwhich occurs in some concrete cases.

Proposition 28.2.13 Let T ∈ B(X ) be an operator for which σap(T ) = ∂σ (T )and every λ ∈ ∂σ (T ) is not isolated in σ (T ). Then

σs f (T ) = σu f (T ) = σap(T ) = σK (T ) .

Proof The operator T has SVEP at every point of the boundary as well as atevery point λwhich belongs to the remaining part of the spectrum since σap(T )does not cluster at this λ. Hence T has SVEP. By Corollary 28.2.12, we haveσap(T ) ⊆ σs f (T ), and hence, from the inclusions (28.2.3), we conclude thatσs f (T ) = σu f (T ) = σap(T ).Finally, σK (T )= σap(T ) because T has SVEP, so the proof is complete. �

Corollary 28.2.14 Suppose that σsu(T ) = ∂σ (T ) and that every λ ∈ ∂σ (T )is not isolated in σ (T ). Then

σs f (T ) = σl f (T ) = σsu(T ) = σK (T ) .

Proof From the assumption, we obtain σsu(T ) = σap(T ′) = ∂σ (T ) = ∂σ (T ′),and so we can apply Proposition 28.2.13. The statement then follows once wehave observed that σs f (T ) = σs f (T ′), that σl f (T ) = σu f (T ′), and finally thatσK (T ) = σK (T ′). �

Proposition 28.2.13 applies to arbitrary non-invertible isometries. In fact forthese operators we have σ (T ) = D(0, r (T )) and σap(T ) = ∂D(0, ν(T )), whereν(T ) is the spectral radius of T ; see Proposition 1.3.2 of Laursen and Neumann(2000).Proposition 28.2.13 also applies to theCesaro operator Cp defined in Part IV,

Example 24.1.3. As noted in T. L. Miller V. G. Miller, and Smith (1998), thespectrum of the operator Cp is the closed disc D(p/2, p/2) and

σe(Cp) = σap(Cp) = ∂D(p/2, p/2).

From Proposition 28.2.13, we also have

σs f (Cp) = σu f (Cp) = σK (Cp) = ∂D(p/2, p/2) .

Proposition 28.2.15 Let X be a Banach space, and suppose that λ0 I − T in�(X ) has index 0. Then the following statements are equivalent:

(i) T has SVEP at λ0;(ii) T ′ has SVEP at λ0;


(iii) p(λ0 I − T ) <∞;(iv) q(λ0 I − T ) <∞;(v) N∞(λ0 I − T ) ∩ (λ0 I − T )∞(X ) = {0};(vi) X = N∞(λ0 I − T )+ (λ0 I − T )∞(X );(vii) X = N∞(λ0 I − T )⊕ (λ0 I − T )∞(X );(viii) σ (T ) does not cluster at λ0;(ix) λ0 I − T is a Riesz–Schauder operator.

Proof If α(λ0 I − T ) = β(λ0 I − T ) <∞, then

p(λ0 I − T ) <∞⇔ q(λ0 I − T ) <∞.

The equivalence of the statements (i)–(ix) then easily follows by combining thepreceding results.

Two other important spectra originating from the theory of Fredholm op-erators are the Weyl spectrum and the Browder spectrum. The Weyl spectrumσw (T ) of T ∈ B(X ) is defined to be the complement of those complex λ ∈ C

such that λI − T ∈ �(X ) and ind(λI − T ) = 0. The Browder spectrum σb(T )of T is defined to be the complement of those complex λ ∈ C such that λI − Tis Riesz–Schauder. Obviously,

σs f (T ) ⊆ σe(T ) ⊆ σw (T ) ⊆ σb(T ) .

The next result extends to decomposable operators on Banach spaces someclassical results valid for normal operators on Hilbert spaces.

Theorem 28.2.16 Let X be a Banach space, and let T ∈ B(X ). Suppose thatT or T ′ has SVEP. Then σb(T ) = σw (T ). If both T and T ′ have SVEP, then

σs f (T ) = σe(T ) = σw (T ) = σb(T ) .

In particular, these equalities hold for every decomposable operator.

Proof Suppose first that T has SVEP. By Proposition 28.2.15, T is a Fredholmoperator having index 0 if and only if T is a Riesz–Schauder operator, and henceσb(T ) = σw (T ). The same argument works also in the case that T ′ has SVEPsince T and T ′ are simultaneously Weyl or Browder, so that σw (T ) = σw (T ′)and σb(T ) = σb(T ′). If both T and T ′ have SVEP then, from Corollary 28.2.7,we easily obtain that σs f (T ) = σe(T ) = σb(T ). �


28.3 Exercises

1. Let T be a bounded linear operator on a Banach space. Each of the two con-ditions ‘p(λ0 I − T ) <∞’ and ‘the space H0(λ0 I − T ) is closed’ impliesthat T has SVEP at 0. Prove that neither of these two conditions implies theother.Hint: consider the operator T defined in Exercise 27.3.4, and the opera-tor S : 2(N)→ 2(N) defined by Sx := (x2/2, . . . , xn/n, . . . ), where x =(xn) ∈ 2(N).

2. Show that every multiplier T on a semisimple Banach algebra satisfies thecondition that p(λI − T ) ≤ 1 (λ ∈ C).

3. A bounded operator T on a Banach space X is paranormal if

‖T x‖2 ≤ ‖T 2x‖‖x‖ (x ∈ X ) .

An operator T ∈ B(X ) on a Banach space X is totally paranormal if λI − Tis paranormal for every λ ∈ C. Show that every totally paranormal operatorhas SVEP.Hint: show that p(λI − T ) ≤ 1 (λ ∈ C). Alternatively, show thatH0(λI − T ) = ker (λI − T ) (λ ∈ C).

4. An operator T satisfies a polynomial growth condition if there exists K > 0and δ > 0 for which

‖exp(iλT )‖ ≤ K (1+ |λ|δ) (λ ∈ R) .

Let P(X ) denote the class of all operators which satisfy this condition. Thepolynomial growth condition may be reformulated as follows: T ∈ P(X ) ifand only if σ (T ) ⊆ R and there are constants K > 0 and δ > 0 such that

‖(λI − T )−1‖ ≤ K (1+ |Im λ|−δ)

for all λ ∈ C with Im λ �= 0; see Theorem 1.5.19 of Laursen and Neumann(2000). Show that p(λI − T ) <∞ (λ ∈ C). Hint: show that, if T ∈ P(X )and if c := Im λ > 0, then

(λI − T )−1 = −i∫ ∞0eiλt e−itT dt .

Assume then that λ = 0 and put m := [δ]+ 1, where δ is as above. Showthat p(T ) ≤ m for every λ ∈ C. See Barnes (1989) for details. Note thatP(X ) coincides with the class of all generalized scalar operators having realspectra; see Theorem 1.5.19 of Laursen and Neumann (2000).

5. Let λ0 be an isolated point of σ (T ), and denote by P0 the spectral projectionassociated with λ0, as in Part I, Chapter 4. Show that ker P0 = K (λ0 I − T )


and that P0(X ) = H0(λ0 I − T ). The proof of the second equality is notobvious; see Proposition 49.1 of Heuser (1982).

6. Let R and L denote the right and the left shift operator, respectively, on 2(N). Show that the operator T := L ⊕ R ∈ L( 2(N)× 2(N)) is a Fred-holm operator with ind T < 0 and that T does not have SVEP at 0.

7. Show that, if T ∈ B(X ) is essentially semi-regular, then there exists a discD(0, ε) for which λI − T is semi-regular for all λ ∈ D(0, ε) \ {0}. Hint: if(M, N ) is a Kato decomposition for T and T0 := T | T∞(X ), take

ε := min{γ (T | M), γ (T0)}.

8. Show that, if T ∈ B(X ) is essentially semi-regular, then there exists a finite-dimensional subspace F of X such that N∞(T ) ⊆ T∞(X )+ F . The con-verse is true if we assume that T (X ) is closed; see Muller (1994).

9. Suppose that T, S ∈ B(X ) are commuting operators for which T S is es-sentially semi-regular. Show that both T and S are essentially semi-regular.Hint: use the result of Exercise 8, and see the remark after Proposition 26.1.4.

10. Let T ∈ B(X ), where X is a Banach space, be essentially semi-regular.Then the operator T : X/T∞(X )→ X/T∞(X ) induced by T is uppersemi-Fredholm and also p(T ) <∞.

11. Let f be an analytic function defined on an open set containing σ (T ) suchthat f is non-constant on the connected components of ρ(T ). Show thatf (σap(T )) = σap( f (T )) and f (σsu(T )) = σsu( f (T )). Hint: show first thatσsu(T ) =

⋃x∈X σT (x), and then use remark (ii) before Theorem 27.2.10.

12. Let X be a Banach space, and take T ∈ B(X ). Show that

σb(p(T )) = p(σb(T ))

for every polynomial p.Hint: use Theorem 27.2.10. It should be noted thatthis equality holds if we replace p by an analytic function defined on anopen neighbourhood U of σ (T ); see Gramsch and Lay (1971).

13. Let X be a Banach space, and take T ∈ B(X ). Suppose that T is essentiallysemi-regular and quasi-nilpotent. Show that X is finite-dimensional and Tis nilpotent.


Most of the material of this chapter is a sample of results of Aiena andMonsalve (2000, 2001), Aiena, Colasante and Gonzalez (2001), and Aienaand Rosas (2003). In these papers the results have been established in the more


general situation ofKato type operators, that is, operators on a Banach space forwhich there exists a decomposition X = M ⊕ N , where M and N are closedT -invariant subspaces, such that T | M is semi-regular and T | N is nilpotent.Some of these results, in the special case of semi-Fredholm operators, may befound in Finch (1975) and Schmoeger (1995). Note that some of the equiva-lences of Proposition 28.2.6, Proposition 28.2.8, and Proposition 28.2.9 havebeen established in West (1987) by using methods of Fredholm theory, withoutinvolving local spectral theory.The condition that p(λI − T ) <∞ for all λ ∈ C has been investigated in

Laursen (1992) by different methods. In this paper the reader may find someinteresting open problems. Finally, results very close to those established in thecorollaries of Theorem 28.2.10 may be found in Section 3.7 of Laursen andNeumann (2000).

References

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Index of symbols

AE , 49(A1(r )), 184A(V ), 148A(X ), 5A(�), 25, 93, 94, 97A(D), 4, 27, 30, 146A := C0(), 241A ⊗ A, 54A+(D), 5, 31α(T ), 300

B(�), 95BM(T), 152B(E), 5B(E, F), 5Bn(A, E), 59B, 202B(G), 80β(T ), 300βX , 28(β), 106, 205, 226–228

(C2)∞, 77C(G), 82C(X ), Cb(X ), 4C(), 4, 27, 65, 66C (1)(I), 51C (n)(I), 5, 27, 55C∗(G), 91C ♦, 234Cm (), 216Cm (), 216C∞(C), 204C0(G), 82Cn , 76C∗r (G), 91C0·, 154, 188C·0, 154cσαβ , 101

C1(H ), 147Cϕ , 49

�, 81δ(M, N ), 276(δ), 106, 206, 226–228

E ⊗ F , 54E T (F), 220, 221εx , 23Eλ, 147

f ⊗ g, 62a f, fa , 82f , 25, 93, 222F2, 63, 76, 104, 107, 120F∞, 76, 127F(E), 17

GL(n,C), 76G, 93

H (U ), 7, 143H∞(U ), 8, 143, 146Hn(A, E), 58, 59, 65Hn(A, E), 65H0(T ), 280HYP (T ), 138, 139Hn(A, E), 60H1(A, X ), 116

InvA, 8Inv0A, 36I(θ ), 42I (F), 15ind T , 300

J, Jµ, 119, 123J(F), 15

319

320 Index of symbols

K (T ), 270, 287K(E), 17k((X )), 77

1(G), 6, 62, 63, 87 1(ω), 6, 14, 36, 46∞(S), 4L1(G), 11, 26, 27, 87, 108,

242L1(I), 10L 1(R), 7L∞(G), 83L10(G), 118, 123L p(G), 83L(E), 5, 49L(E, F), 5Ln(A, E), 58L(E), 49Lat, 138

µ, 95µJ , 119, 123(µ, m), µ ⊥ m, 150M(G), 66, 86, 242Ma(G), 86Mc(G), 86Md (G), 86Mn , 6

N 1(A, E), 52Nn(A, E), 59N 1(A, E), 52, 116N n(A, E), 59N∞(T ), 271νA(a), 13νT (x), 297

P(X ), 5p(T ), 298P1, 120�A , 20�(X ), 302�+(X ), 302�−(X ), 302�±(X ), 302

Qp , 77q(T ), 298Q(A), 13

R(K ), 171R(X ), 5Ra , 12R, 91ρK (T ), 274ρT (x), 212, 286ρA(a), 12

σA(a), 12, 27σb(T ), 312σe(A), 161, 310σH(T ), 189σK (T ), 274σp(T ), 290σT (x), 286, 312σw (T ), 312σap(T ), 213, 308σl f (T ), 310σs f (T ), 310σsu (T ), 213, 308σu f (T ), 310σre(A), 161σe(A), 161SL(n,R), 76SO(2), 77SO(n), 76S 2, 15S[2], 15S(T ), 39S(θ ), 41ST , 15S · T , 15

(T )′, 138T∞, 77T ∞X , 220T ', 206T∞(X ), 271Tr , 147TY , 207

Ux , 91

V N (G), 91V , 10

WT , 178W (T), 9

XT (), 287XT (F), 212, 221, 270X[1], 138x0 ⊗ λ0, 16X , 30XT (), 288XT (F), 205

Z , 4Z1(A, E), 51Zn(A, E), 59Zn(A, E), 59Z1(A, X ), 116Z( f ), 24Z1(A, E) , 52((T ), 294

Subject index

∗-algebra, symmetric, 106

adjoint, 26algebra, reflexive, 178amenable, 60, 63, 109, 114, 118weakly, 66

analytic, 182approximable operators, 17approximation by Lie groups, 79augmentation ideal, 118Aupetit, 40automatic continuity, 38, 121, 220, 239

Banachalgebra, 3bimodule, 116function algebras, 23left module, 50space, 3

bimodule, 48bounded approximate identity, 29, 87,

118Brennan, 167Brouwer, 139Brown, 166, 171Brown, Chevreau, and Pearcy, 160

C*-algebra, 26, 91Cauchy transform, 167character, 20, 25, 93coboundary, 59continuous, 59

cocycle, 59continuous, 59

Cohen factorization, 29, 55, 88cohomology, 58continuous cohomology group, 60

compact, 138operators, 17, 35, 137

compactly generated group, 80

connecting maps, 58continued bisection of the identity, 43continuity ideal, 42, 121, 128continuous cohomology group, 116contractionabsolutely continuous, 152

completely non-unitary, 149convolution, 7, 86, 87, 98, 242Conway and Olin, 181coordinate functional, 4operators, 98

decomposable, 98, 106derivation, 51, 116inner deriviation, 116

diagonal, 60virtual diagonal, 61

Dirichlet algebra, 179disc algebra, 4, 55, 57Ditkin, 103divisible subspace, 220dominating, 160, 190dual group, 25, 93dual module, 117duality theory, 226

eigenvalues, 137ergodic, 119Eschmeier and Prunaru, 171evaluation maps, 23

factors, 29Fejer’s theorem, 9fixed point property, 116formal power series in one variable, 6Følner’s condition, 113Fourier coefficients, 9, 222Fourier–Stieltjes algebra, 95Fourier transform, 25, 93, 94Frechet algebra, 7

321

322 Subject index

free group, 63, 76, 104, 107, 120two generators, 76, 110

functional calculus, 30, 179absolutely continuous, 187analytic, 203, 215continuous, 28H∞, 143map, 32non-analytic, 215, 216polynomial, 145

functional model, 254fundamental theorem of Banach algebras,

14

Gelfand–Mazur theorem, 14Gelfand–Naimark theorem, 26non-commutative, 28, 29

Gelfand-Raikov theorem, 92Gelfandrepresentation theorem, 22theory, 20transform, 25topology, 21

generalized kernel, 271generalized range, 220, 271Gleason, 79group algebras, 11, 86, 87group, Burnside, 127

Haar measure, 81homomorphismabsolutely continuous, 150singular, 150type C0, 155

hull, 27, 247hull-kernel topology, 27hyperinvariant, 141subspaces, 138, 154

idempotents, 34, 43index, 231theorem, 231, 300

inner derivations, 52invariant subspace, 138analytic invariant subspace,182

problem, 137inverse, 8invertible, 8involution, 26

Johnson, 41, 60, 63joint spectrum, 37jump, 233, 303

Kawada-Ito theorem, 82, 119kernel, 27, 247

lattice, rich, 172Laurent series, 77left Haar measure, 11left regular representation, 91left translate, 82Lie group, 77local resolvent, 212, 286locally compact groups, 75, 76abelian, 25symmetric, 106

Lomonosov, 139

main boundedness theorem, 42, 121matrix, symmetric, 201maximal ideals, 20mean, 63left-invariant, 63, 109topologically invariant, 109

measureabsolutely continuous, 150, 188algebra, 66Henkin, 151, 158singular, 150, 188spectral, 202

Michael’s problem, 28minimal, 187central idempotent, 99

modular function, 81module, 48dual, 51homomorphism, 50left, 48right, 48simple, 49

Montel sequence, 148Montgomery and Zippin, 79multiplier, 241

natural, 24norm, 3algebra, 3projective, 54

Olin and Thomson, 178operatoranalytical core, 270A-scalar, 216ascent, 298Cesaro, 311decomposable, 203, 212, 228defect, 300descent, 298essentially semi-regular, 303Fredholm, 215, 230, 267, 302generalized scalar, 204, 222hyponormal, 171, 240index, 300

Subject index 323

intertwines, 230Kato decomposition, 303lower semi-Fredholm, 302norm, 5normal, 202nullity, 300paranormal, 313quasi-affinity, 231quasi-nilpotent part, 280reflexive, 178Riesz, 215Riesz–Schauder, 306self-adjoint, 202semi-regular, 269semi-Fredholm, 298, 302similar, 230sub-decomposable, 171, 227subnormal, 158, 166, 189super-decomposable, 214totally paranormal, 313trace-class, 147upper semi-Fredholm, 302

orthogonal, 34

paradoxical decomposition, 112partitions of unity, 216point derivations, 52pointwise boundedly dense, 148polynomial convex hull, 30, 207polynomial generator, 27Pontryagin duality theorem, 25primitive ideal, 49PropertyPp , 114(β), 205, 285(δ), 206, 285

punctured disc theorem, 232

quasi-nilpotent, 13, 280

radical, 16, 19random walks, 119Ransford, 41regular Banach algebra, 27, 103, 247Reiter, 114representation, 49resolvent set, 12Kato, 274

richly endowed, 216Riemann–Lebesgue Lemma, 25Riesz functional calculus, 35right shift, 18, 231, 296, 309right translate, 82

Schauder, 139Scott Brown technique, 143, 163semisimple, 16

separating space, 39set of synthesis, 103, 108Silov’s idempotent theorem, 37Sinclair, 53Singer and Wermer, 52single-valued extension property, 106, 205,

267, 285, 286single-variable analytic functional calculus,

32singularity set, 239spectral mapping theorem, 34, 37spectral radius, 13, 280spectral radius formula, 14spectral radius, local, 297spectral subspace, algebraic, 220, 270spectral space, glocal, 205, 288local analytic, 172, 212, 270, 287

spectral synthesis, 103spectrum, 12approximate point, 213, 308Browder, 312essential, 161, 230, 310Harte, 189joint, 186Kato, 274, 282left essential, 161local, 212, 286lower semi-Fredholm, 310point, 290right essential, 161semi-Fredholm, 310surjectivity, 213, 308upper semi-Fredholm, 310Weyl, 312

spherical dilation, 187minimal, 187

spherical unitary, 187spherical contraction, 186state space, 21Stone–Cech compactification, 28strong Wedderburn decomposition, 67

tensor product, 53projective, 54

thick, 171topological group, 76topologically simple, 46topologyhull-kernel, 247ultraweak operator, 147weak operator, 148

translation-invariant, 81

uniform algebra, 11unilateral shift, 137unimodular group, 81unique complete norm, 39

324 Subject index

unitary dilation, 143, 144minimal, 145

unitary representation, 90

van Dantzig, 79Volterra algebra, 10, 18Volterra operator, 10, 137, 221,

296von Neumann’s inequality, 145, 186

weak containment, 115weight, 6Wendel, 88, 89Wiener algebra, 104Wiener–Levy theorem, 34

Yamabe, 79

zero-jump theorem, 233

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Introduction to Banach Algebras, Operators, and Harmonic Analysis

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