Table Of ContentFREDHOLM AND LOCAL SPECTRAL THEORY,
WITH APPLICATIONS TO MULTIPLIERS
Fredholm and Local Spectral Theory,
with Applications to Multipliers
by
Pietro Aiena
Dipartimento di Matematica ed Applicazioni,
Università di Palermo, Palermo, Italy
KLUWER ACADEMIC PUBLISHERS
NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW
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A mia madre e a mia sorella Mariella
Contents
ix
Preface
Chapter 1. The Kato decomposition property 1
1. Hyper-kernel and hyper-range of an operator 2
2. Semi-regular operators on Banach spaces 7
3. Analytical core of an operator 11
4. The semi-regular spectrum of an operator 14
5. The generalized Kato decomposition 24
6. Semi-Fredholm operators 33
7. Quasi-nilpotent part of an operator 43
8. Two spectral mapping theorems 48
Chapter 2. The single-valued extension property 55
1. Local spectrum and SVEP 56
2. The SVEP at a point 69
3. A local spectral mapping theorem 76
4. Algebraic spectral subspaces 90
5. Weighted shift operators and SVEP 99
Chapter 3. The SVEP and Fredholm theory 109
1. Ascent, descent, and the SVEP 110
2. The SVEP for operators of Kato type 119
3. The SVEP on the components of ρ (T) 127
k
4. The Fredholm, Weyl, and Browder spectra 132
5. Compressions 143
6. Some spectral mapping theorems 146
7. Isolated points of the spectrum 157
8. Weyl’s theorem 164
9. Riesz operators 179
10. The spectra of some operators 182
Chapter 4. Multipliers of commutative Banach algebras 191
1. Definitions and elementary properties 192
2. The Helgason–Wang function 200
3. First spectral properties of multipliers 210
4. Multipliers of group algebras 218
5. Multipliers of Banach algebras with orthogonal basis 225
vii
viii CONTENTS
6. Multipliers of commutative H∗ algebras 233
Chapter 5. Abstract Fredholm theory 239
1. Inessential ideals 240
2. The socle 244
3. The socle of semi-prime Banach algebras 249
4. Riesz algebras 255
5. Fredholm elements of Banach algebras 264
6. Compact multipliers 271
7. Weyl multipliers 279
8. Multipliers of Tauberian regular commutative algebras 286
9. Some concrete cases 292
10. Browder spectrum of a multiplier 298
Chapter 6. Decomposability 309
1. Spectral maximal subspaces 310
2. Decomposable operators on Banach spaces 314
3. Super-decomposable operators 323
4. Decomposable right shift operators 331
5. Decomposable multipliers 338
6. Riesz multipliers 357
7. Decomposable convolution operators 361
Chapter 7. Perturbation classes of operators 367
1. Inessential operators between Banach spaces 368
2. Ω and Ω operators 382
+ −
3. Strictly singular and strictly cosingular operators 388
4. Improjective operators 401
5. Incomparability between Banach spaces 415
Bibliography 423
Index 437
Preface
A significant sector of the development of spectral theory outside the
classical area of Hilbert space may be found amongst at multipliers defined
on a complex commutative Banach algebra A. Although the general theory
of multipliers for abstract Banach algebras has been widely investigated by
several authors, it is surprising how rarely various aspects of the spectral
theory, for instance Fredholm theory and Riesz theory, of these important
classes of operators have been studied. This scarce consideration is even
more surprising when one observes that the various aspects of spectral the-
ory mentioned above are quite similar to those of a normal operator defined
on a complex Hilbert space.
Inthelasttenyearstheknowledgeofthespectralpropertiesofmultipli-
ers of Banach algebras has increased considerably, thanks to the researches
undertaken by many people working in local spectral theory and Fredholm
theory. This research activity recently culminated with the publication of
the book of Laursen and Neumann [214], which collects almost every thing
that is known about the spectral theory of multipliers.
The beautiful book of Laursen and Neumann has as a main motivation
that of providing a modern introduction to local spectral theory and a par-
ticular emphasis is placed on the applications of the general local spectral
theory to convolution operators on group and measure algebras. This book
contains also several results on Fredholm theory. However, most of the as-
pects of Fredholm theory which are developed in this book are those that
may be particularly approached through the methods of local spectral the-
ory.
In this book I will try, in a certain sense, to reverse this process. In-
deed, our first major concern is with the Fredholm theory, in particular the
interested reader will find here a distinct flavor of this theory, which is em-
phasizedbythechaptersdevotedtotheKatodecomposition,totheabstract
Fredholmtheoryinsemi-primeBanachalgebras,andbythechapterdevoted
to inessential operators between Banach spaces.
Asecondconcernofthismonographisthatofshowinghowtheinterplay
between Fredholm theory and local spectral theory is significant and beau-
tiful. Itshouldbeclearthatrelativetothepartonlocalspectraltheory,the
content is rather limited, and the notions developed in this monograph are
thosewhichareofinterestfortheapplicationstomultipliersofcommutative
semi-simple Banach algebras.
ix
x
PREFACE
The deep interaction between the Fredholm theory and local spectral
theory becomes evident when one consider the so called single-valued ex-
tension property. This property, which dates back to the early days of local
spectral theory, was first introduced in 1952 by Dunford and has received
a systematic treatment in the classical Dunford and Schwartz book [97]. It
also plays an important role in the book of Colojoara˘ and C. Foia¸s [83],
in Vasilescu [309], and in the recent book of Laursen and Neumann [214].
This single-valued extension property is one of the major unifying themes
forawidevarietyoflinear bounded operatorsin thespectraldecomposition
property. This property is, for instance, satisfied by normal operators on
Hilbert spaces and more generally by decomposable operators. Examples
of non-decomposable operators having this property may be found amongst
the class of all multipliers of semi-simple Banach algebras. In fact, every
multiplier of a semi-prime Banach algebra has the single-valued extension
property and there exist examples of convolution operators which are not
decomposable.
A third aim of this book is to present some important progress, made
in recent years, in the study of perturbation theory for classes of operators
which occur in Fredholm theory. This subject is covered by several excel-
lent books, such as the monographies by Kato [183], by Heuser [160], by
Caradus,Pfaffenberger, andYood[76],byPietsch[263], butnaturallythey
do not present the more recent results. Some of these new results solve very
old open problems in operator theory and this book may be intended as a
complement of the books mentioned above.
For those unacquainted with the subject matter, examples and motiva-
tions for certain definitions are mentioned in order to give some feeling for
what is going on.
Nowwedescribeinmoredetailthearchitectureofthismonograph. This
book consists of seven chapters.
The first chapter is devoted to the Kato decomposition for bounded
operator on Banach spaces. This decomposition property arises from the
classical treatment of perturbation theory of Kato [182], and its flourishing
hasgreatlybenefitedfromtheworkofmanyauthorsinthelasttenyears,in
particular from the work of Mbekhta [226], [227], Mu¨ller [240], Rakoˇcevi´c
[274] and Schmoeger [290]. The operators which satisfy this property form
a class which includes the class of semi-Fredholm operators. This is an im-
portant result of Kato and here we present the proof given by West [324].
We shall develop, quite systematically, the properties of some important
subspaces which play an important role in this decomposition theory, such
astheanalyticalcore,thequasi-nilpotentpart,andtheKatoresolventofan
operator. We shall also introduce the concepts of semi-regular, essentially
semi-regular,andKatotypeofoperators. Alltheseconceptsgeneratedistin-
guished spectra, and particular emphasis is placed on the spectral mapping
theorems for these spectra.
The second chapter deals with a localized version of the single-valued
xi
PREFACE
extension property at a point and relates it, in the third chapter, to the
finiteness of some classical quantities associated with an operator. These
properties, such as the ascent and the descent of an operator, are the ba-
sic bricks of Fredholm theory, so the third chapter may be viewed as the
part of the book in which the interaction between local spectral theory and
Fredholm theory comes into focus. In fact, for semi-Fredholm operators the
single-valued extension property at a point may be characterized in several
ways,andsomeofthesecharacterizationsleadtothelocalizationofsomedis-
tinguished part of the spectra originating from Fredholm theory. Moreover,
some classical spectral mapping theorems related to semi-Browder spectra
and to the Browder spectrum, as well as the classification of the open con-
nected components of the Fredholm, or the semi-Fredholm, region being
able to be obtained via the localized single-valued extension property. The
third chapter also deals with another important class of operators, the class
of Riesz operators. Many of the results related to these operators are here
revisited through the single-valued extension property at a point.
The fourth chapter is devoted to the basic ingredients of the theory of
multipliers of commutative semi-simple Banach algebras. Our intention is
to make this book as much self-contained as possible, so we shall give in
our exposition of the elementary theory of multipliers the proofs of all the
results needed in the subsequent chapters. Of course, part of the material
concerning the basic theory of multipliers may be found in some books on
the subject, for instance in the monograph by Larsen [199], or the book by
Laursen and Neumann [214]. However, some parts of this chapter seems to
be new, and the basic material treated here is that strictly related to the
study of spectral properties of multipliers. The theory of multipliers may
also be developed for non-commutative Banach algebras. We shall not in-
vestigatethiscaseextensivelybutinsteadshallrestrictourstudyessentially
tothecommutativealgebras. InthiscasetheGelfandtheorypermitstorep-
resent every multiplier on a commutative semi-simple Banach algebra as a
bounded continuous complex-valued function, the Helgason–Wang function
defined on the regular maximal ideal space of the algebra. By means of this
representation it appears evident how the spectral properties of a multiplier
are related, in a certain measure, to the range of the Helgason–Wang func-
tion. In this chapter we shall also give an account of the multiplier theory
of various algebras: group algebras, Banach algebras with an orthogonal
basis, commutative H∗ algebras and commutative C∗ Banach algebras. In
particular, we describe the multiplier algebras and the ideal of all compact
multipliers of these algebras. The characterizations of compact multipliers
will permit to us to characterizethe Fredholm operators which are multipli-
ers for several concrete Banach algebras.
The initial part of the fifth chapter is devoted to the introduction of
some basic ingredients of the abstract Fredholm theory on a not necessarily
commutative Banach algebras. It should be mentioned that the interested
readermayfindamorecompletetreatmentofthistheoryinthemonograph
xii
PREFACE
by Barnes, Murphy, Smyth, and West [62], although our approach to the
abstract Fredholm theory is somewhat different. The central notion of this
chapteristhatofinessentialideals. Theseidealswillbeintroducedbyusing
some ideas of Aupetit [53].
An abstract Fredholm theory on a Banach algebra is essentially the
A
theoryofallinvertibleelementsof moduloan essentialideal, i.e., an ideal
A
of forwhichthespectrumofeveryelementisafiniteoradenumerableset.
A
The classical Fredholm theory for operators on Banach spaces corresponds
to the Fredholm theory of the semi-simple Banach algebra :=L(X) of all
A
boundedoperatorsonaBanachspaceX withrespecttotheinessentialideal
ofallboundedfiniterankoperators. However,theabstractFredholmtheory
in Banach algebras is not merely a generalization of the classical Fredholm
theory for operators. An illuminating example which strengthens this as-
sertion is given, in fact, by the theory of multipliers of semi-prime Banach
algebra. Indeed, in a semi-prime Banach algebra, the Fredholm theory of
multipliersisexactlytheFredholmtheoryofthesemi-primemultiplieralge-
brawithrespecttothesocleofthealgebra,or,whichisthesamething,with
respect to the socle of the multiplier algebra. This result allows us to give
a description of multipliers which are Fredholm operators by involving only
objects of the same class, not involving operators which are not multipliers.
As a consequence we can obtain a very clear description of convolution op-
erators of group algebras which are Fredholm.
We shall focus our attention on the inessential ideal of the socle, which
corresponds in the case of the Banach algebra of all bounded linear oper-
ators to the ideal of all bounded finite rank operators. Subsequently we
shall consider the theory of Riesz algebras of Banach algebras, developed
by Smyth [304]. We shall give further information on the Fredholm theory
of multipliers of commutative regular Tauberian Banach algebras. These
results deal to the complete description of Fredholm convolution operators
acting on group algebras.
The content of the sixth chapter concerns some other important tools
of local spectral theory. The first part of the chapter concerns the class of
all decomposable operators, but we shall avoid the duality theory of these
operators, since this theory requires two functional models for operators on
Banach spaces, and to develop this argument in this book would lead too
far afield. Another important part of Chapter 6 concerns the study of the
propertyofdecomposabilitywithinthetheoryofmultipliersofcommutative
semi-simple Banach algebras. In this framework the Fredholm theory, the
Gelfand theory, the local spectral theory, and harmonic analysis are closely
intertwined. This part is strongly based on the work of Laursen, Neumann
[210], [243], and Eschmeier, Laursen, and Neumann [112].
The concluding chapter of the book concerns some perturbation classes
of operators which occur in Fredholm theory. In this theory we find two
fundamentally different classes of operators: semi-groups, such as the class
