Tauberian Operators

252 Pages·2010·1.878 MB·English

by  Manuel González, Antonio Martínez-Abejón (auth.)

Operator Theory: Advances and Applications
Vol. 194
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Tauberian Operators
Manuel González
Antonio Martínez-Abejón
Birkhäuser
Basel · Boston · Berlin
Authors:
Manuel González Antonio Martínez-Abejón
Departamento de Matemáticas, Departamento de Matemáticas
Estadística y Computación Universidad de Oviedo
Universidad de Cantabria Calle Calvo Sotelo s/n
Avenida de los Castros s/n 33007 Oviedo
39071 Santander Spain
Spain e-mail: [email protected]
e-mail: [email protected]
2000 Mathematics Subject Classification: 46B03, 46B10, 47A53, 47L20
Library of Congress Control Number: 2009938295
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Contents
Preface vii
Notation xi
1 The origins of tauberian operators 1
1.1 Tauberian conditions in summability theory . . . . . . . . . . . . . 1
1.2 Tauberian matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Tauberian operators . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Notes and Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Tauberian operators. Basic properties 9
2.1 Basic facts about tauberian operators . . . . . . . . . . . . . . . . 10
2.2 Main characterizations of tauberian operators . . . . . . . . . . . . 16
2.3 Preservationof the closedness of closed convex sets . . . . . . . . . 21
2.4 Action of tauberian operators on basic sequences . . . . . . . . . . 25
2.5 Notes and Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Duality and examples of tauberian operators 35
3.1 Cotauberian operators . . . . . . . . . . . . . . . . . . . . . . . . . 36
The residuum operator . . . . . . . . . . . . . . . . . . . . . . . . . 37
A counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 The DFJP factorization of operators . . . . . . . . . . . . . . . . . 46
3.3 Variations of the DFJP factorization . . . . . . . . . . . . . . . . . 52
An isometric variation of the DFJP factorization . . . . . . . . . . 53
Connections with real interpolation methods . . . . . . . . . . . . 54
A conditional variation of the DFJP factorization . . . . . . . . . . 56
3.4 Inclusions of vector-valuedfunction spaces . . . . . . . . . . . . . . 59
3.5 Tauberian-like classes of operators defined in terms of sequences . 61
3.6 Notes and Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
vi Contents
4 Tauberian operators on spaces of integrable functions 73
4.1 Tauberian operators on L1(μ) spaces . . . . . . . . . . . . . . . . . 74
4.2 Ultrapowers of tauberian operators on L1(μ) spaces . . . . . . . . 79
4.3 The perturbation class of T(L1(μ),Y) . . . . . . . . . . . . . . . . 82
On the operators in T(L1(μ)) . . . . . . . . . . . . . . . . . . . . . 86
4.4 Biconjugates of tauberian operators on L1(μ) . . . . . . . . . . . . 88
4.5 Notes and Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Some applications 93
5.1 EquivalenceoftheRadon-Nikody´mpropertyandtheKrein-Milman
property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Preservationof isomorphic properties . . . . . . . . . . . . . . . . . 98
Properties of spaces preserved by tauberian operators . . . . . . . 98
Properties of bounded subsets preserved by tauberian operators . . 99
5.3 Operator ideals and factorization . . . . . . . . . . . . . . . . . . . 100
5.4 Uniform factorization for compact sets of operators . . . . . . . . . 104
5.5 Factorization of holomorphic mappings . . . . . . . . . . . . . . . . 109
The surjective case . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
The injective case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.6 Approximation properties and Calkin algebras . . . . . . . . . . . 113
5.7 Notes and Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Tauberian-like classes of operators 121
6.1 Operator ideals and semigroups . . . . . . . . . . . . . . . . . . . . 122
6.2 Strongly tauberian operators . . . . . . . . . . . . . . . . . . . . . 131
Strongly tauberian operators on L-embedded spaces . . . . . . . . 136
6.3 Finite representability of operators . . . . . . . . . . . . . . . . . . 139
6.4 Ultrapower-stable classes of operators . . . . . . . . . . . . . . . . 157
6.5 Supertauberian operators . . . . . . . . . . . . . . . . . . . . . . . 162
6.6 Cosupertauberian operators . . . . . . . . . . . . . . . . . . . . . . 173
Local supportability versus local representability . . . . . . . . . . 182
6.7 Notes and Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 185
A Basic concepts 191
A.1 Semi-Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . 191
A.2 Operator ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.3 Bases and basic sequences . . . . . . . . . . . . . . . . . . . . . . . 197
A.4 Ultraproducts in Banach space theory . . . . . . . . . . . . . . . . 200
A.5 Reflexivity and super-reflexivity . . . . . . . . . . . . . . . . . . . . 212
A.6 Ultraproducts of L1(μ) spaces . . . . . . . . . . . . . . . . . . . . . 216
Bibliography 231
Index 243
Preface
Tauberian operators were introduced to investigate a problem in summability
theory from an abstract point of view. Since that introduction, they have made a
deep impact on the isomorphic theory of Banach spaces. In fact, these operators
havebeenusefulinseveralcontextsofBanachspacetheorythathavenoapparent
or obvious connections. For instance, they appear in the famous factorization of
Davis, Figiel, Johnson and Pel(cid:2)czyn´ski [49] (henceforth the DFJP factorization),
in the study of exact sequences of Banach spaces [174], in the solution of certain
summabilityproblemsoftauberiantype[63,115],intheproblemoftheequivalence
between the Krein-Milman property and the Radon-Nikody´m property [151], in
certain sequels of James’ characterization of reflexive Banach spaces [135], in the
construction of hereditarily indecomposable Banach spaces [13], in the extension
ofthe principle oflocalreflexivityto operators[27],inthe study ofcertainCalkin
algebrasassociatedwith the weakly compact operators [16], etc. Since the results
concerningtauberianoperatorsappearscatteredthroughouttheliterature,inthis
bookwegiveaunifiedpresentationoftheirpropertiesandtheirmainapplications
infunctionalanalysis.We alsodescribesomequestionsabouttauberianoperators
that remain open.
This book has six chapters and an appendix. In Chapter 1 we show how the
concept of tauberian operator was introduced in the study of a classical problem
in summability theory–the characterization of conservative matrices that sum no
bounded divergent sequences–by means of functional analysis techniques. One of
thosesolutionsisduetoCrawford[45],whoconsideredthesecondconjugateofthe
operatorassociatedwithoneofthosematrices.Crawford’ssolutionledKaltonand
Wilanskytointroducetauberianoperatorsin[115]asthoseoperatorsT: X −→Y
actingbetweenBanachspacesforwhichT∗∗(X∗∗\X)⊂Y∗∗\Y,whereT∗∗denotes
the second conjugate of T.
Chapter 2 displays the basic structural properties of the class of tauberian
operators; in particular, the links between tauberian operators, weakly compact
operators and reflexivity. We present some basic examples and describe the most
important characterizations of tauberian operators: the sequential characteriza-
tion of Kalton and Wilansky [115], the geometrical characterizations obtained by
Neidinger and Rosenthal [135], the characterization in terms of reflexivity of the
viii Preface
kernel of the compact perturbations given in [92], the algebraic characterization
obtained in [92], and some characterizations in terms of the action of tauberian
operators upon basic sequences proved by Holub [103].
We begin Chapter 3 by introducing the cotauberian operators as those op-
erators T such that T∗ is tauberian. Next we give the main properties of these
operators.Several results show that cotauberianoperators form the right class to
betakenasthedualclassofthetauberianoperators.However,thisrelationshipof
dualityisnotperfect:wegiveanexample,obtainedin[8],ofatauberianoperator
T suchthatT∗ isnotcotauberian.Wealsoincludeaperturbativecharacterization
and an algebraic characterization for the cotauberian operators similar to those
obtained for the tauberian operators in the previous chapter.
We describeanimprovedversionoftheDFJPfactorization,obtainedin[68],
whichallowsustoshowplentyofexamplesoftauberianandcotauberianoperators:
every operator T: X −→ Y can be factorized as T = jUk, with j tauberian, k
cotauberian and U a bijective isomorphism. Moreover, this version behaves well
under duality.
The DFJP factorizationhasreceivedalotofattention.Inparticular,several
variations of it have been introduced. We describe an isometric variation and a
conditionalvariation.ThefirstonewasintroducedbyLima,NygaardandOja[119]
to study the approximation property of Banach spaces, and the second one was
introduced by Argyros and Felouzis [13] to construct examples of hereditarily
indecomposable Banach spaces. Moreover, following Beauzamy’s exposition [21],
we show that the intermediate space in the DFJP factorization can be identified
with a real interpolation space for certain values of the interpolation parameters.
We treatothersituationsinwhichtauberianoperatorsappear.Forexample,
following [32,35], we show that the natural embedding of certain Orlicz function
spaces LΦ(μ) into L1(μ) is a tauberian operator if and only if for every Banach
space X the natural embedding of the space of vector-valued functions LΦ(μ,X)
into L1(μ,X) is a tauberian operator.
The aforementioned characterizations show that the tauberian and the co-
tauberianoperatorsarecloselylinkedtothe operatoridealofthe weaklycompact
operators.Following [89,90,92],we consider four other operatorideals that admit
sequential characterizations.We show that each one has two classes of associated
operatorssimilartothetauberianandthecotauberianoperators.Thefirstofthese
classes is defined in terms of sequences and the second one is defined by duality.
Weshowthatbothclassesadmitaperturbativecharacterizationandanalgebraic
characterization.
Chapter 4 is devoted to the study of tauberian operators T: L1(μ) −→ Y,
where μ is a finite measure and Y is a Banach space [75]. The characterizations
of relatively weakly compact subsets of L1(μ) are applied to obtain some useful
characterizations of these tauberian operators and show that their properties are
better than those of the general tauberian operators. For example, the set of
tauberianoperatorsfromL1(μ) into Y is open inthe set ofalloperators,andone
Preface ix
of these operators T is tauberian if and only if so is its second conjugate.
In Chapter 5 we describe the main applications of tauberian operators in
Banachspacetheory.FollowingSchachermayerin[151]andthe expositionin[67],
we show that, for a Banach space X for which there exists a tauberian operator
T: X ×X −→X, the Radon-Nikody´m property and the Krein-Milman property
are equivalent. We also show that tauberian operators preserve some isomorphic
properties: following Neidinger’s thesis [133], we show that, given a tauberian
operator T: X −→Y and a bounded subset C of X, some isomorphic properties
of the set T(C) are inherited by C and some isomorphic properties of the space
Y are inherited by X.
UsingtheversionoftheDFJPfactorizationpresentedinChapter3,weshow
thatsomeoperatoridealsApossessthefactorizationproperty:eachoperatorinA
factorsthroughaBanachspacewhoseidentitybelongstoA.Hereweincludesome
results of Heinrich [100]and some extensions of these results obtained in [68].We
also show that these factorization results can be extended in two directions: one
ofthembyshowingthatwecanobtainauniformfactorizationofthis kindforthe
operators of a compact set of operators [73], and the other one (see [71,72]) by
showingthatthedefinitionofsomeoperatoridealscanbeextendedtoholomorphic
mappings f: X −→Y acting between Banach spaces X and Y, and that in some
cases these maps can be written as f = T ◦g or f = g ◦T, where g is another
holomorphic mapping and T is an operator that belongs to the same operator
ideal as f.
We also give some applications of the isometric variation of the DFJP fac-
torization to study the approximation property of Banach spaces, due to Lima,
Nygaard and Oja [119], and following Astala and Tylli [16], we characterize the
weakly compact approximation property of Banach spaces in terms of the weak
Calkin algebra.
InChapter6weconsidersomeclassesofoperatorsthathaveasimilarbehav-
ior to that of tauberian operators. Some of these classes were named semigroups
in[89,90,92],followingLebowandSchechter[118]whodiditforthesemi-Fredholm
operators. Finally, the notion of an operator semigroup was axiomatized in [1] as
a counterpart to Pietsch’s concept of an operator ideal [139].
EveryoperatoridealAhastwosemigroupsA+andA−associatedinasimilar
way as the weakly compact operators have the tauberian and the cotauberian
operators. We summarize the main properties of these two operator semigroups
and show other general constructions that provide semigroups.
We describe a strongly tauberian operator and its dual class, introduced by
Rosenthal [147]. Moreover, we show how tauberian operators have been useful in
distinguishingbetweenthe differentconcepts oflocalrepresentabilityofoperators
that have appeared in the literature.
We studyinsomedetailtheultrapower-stableoperatorsemigroups.Forthat
purpose,weconsidertwodifferenttypesoffinite representabilityforoperators:lo-
calrepresentabilityandlocalsupportability.As anapplication,we investigatethe
x Preface
classofsupertauberianoperators,whichisthelargestultrapower-stablesemigroup
contained in the class of tauberian operators, and their dual class: the cosuper-
tauberian operators.
Each chapter ends with a section of Notes and Remarks where we include
some comments, complementary results and bibliographical references.
This book is addressed to graduate students and researchers interested in
functionalanalysisandoperatortheory.Theprerequisitesforreadingthisbookare
a basic knowledge of functional analysis,including the consequencesof the Hahn-
Banach theorem and the open mapping theorem. Familiarity with the rudiments
of Fredholm theory for operators and some parts of Banach space theory, like
criteriaforthe existence ofbasicsubsequencesfroma givensequence,Rosenthal’s
(cid:2)1-theorem, ultraproducts and the principle of local reflexivity would be helpful.
Fortheconvenienceofthereader,abriefexpositionoftheseprerequisiteshasbeen
included in Appendix A.
Our intention has been to present a self-contained exposition of the funda-
mentalresultsofthesubject.Whendescribingtheapplications,sometimeswegive
a reference instead of a complete proof.
Many people gave us information or advice during the preparation of the
manuscript. Thanks are due in particular to Teresa A´lvarez, Joe Diestel, Hans
Jarchow, Antonio Martino´n, Olav Nygaard, Eve Oja, Javier Pello, Antonia Salas
and Hans-Olav Tylli for their useful comments. Thanks should also go to our
colleaguesoftheMathematicsDepartmentoftheUniversidaddeCantabriaandof
the Universidadde Oviedo for a friendly working environment. Financial support
from the Spanish government institution for the promotion of research (Grants
MTM2005-03831and MTM2007-67994)is duly acknowledged.
Manuel Gonza´lez and Antonio Mart´ınez-Abej´on
Santander and Oviedo, April 2009