Handbook of analytic operator theory

370 Pages·2019·1.891 MB·English

by  Zhu K (ed.)

Handbook of Analytic
Operator Theory
CRC Press/Chapman and Hall
Handbooks in Mathematics
Series
Series Editor: Steven G. Krantz
Handbook of Analytic Operator Theory
Kehe Zhu, Editor
https://www.crcpress.com/CRC-PressChapman-and-
Hall-Handbooks-in-Mathematics-Series/book-series/
CRCCHPHBKMTH
Handbook of Analytic
Operator Theory
Edited by
Kehe Zhu
CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Zhu, Kehe, 1961- author.
Title: Handbook of analytic operator theory / Kehe Zhu.
Description: Boca Raton : CRC Press, Taylor & Francis Group, 2019. | Includes
bibliographical references and index.
Identifiers: LCCN 2019004988 | ISBN 9781138486416
Subjects: LCSH: Operator theory. | Holomorphic functions. | Function spaces.
Classification: LCC QA329 .Z475 2019 | DDC 515/.724--dc23
LC record available at https://lccn.loc.gov/2019004988
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Contents
Preface ix
1 FockSpace,theHeisenbergGroup,HeatFlow,andToeplitzOperators 1
LewisA.Coburn
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ToeplitzoperatorsandtheHeisenberggroup . . . . . . . . . . . . 2
1.3 ToeplitzoperatorsandtheBargmanntransform . . . . . . . . . . 4
1.4 Limitbehaviorof T(t) and H(t) ast 0 . . . . . . . . . . . 7
k f kt k f kt !
1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Two-VariableWeightedShiftsinMultivariableOperatorTheory 17
Rau´lE.Curto
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 2-variableweightedshifts . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Theliftingproblemforcommutingsubnormals . . . . . . . . . . 21
2.4 Hyponormality, 2-hyponormality and subnormality for 2-variable
weightedshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Existenceofnonsubnormalhyponormal2-variableweightedshifts 24
2.6 Propagationinthe2-variablehyponormalcase . . . . . . . . . . . 32
2.7 Ameasure-theoreticnecessary(butnotsufficient!)conditionforthe
existenceofalifting . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.8 ReconstructionoftheBergermeasurefor2-variableweightedshifts
whosecoreisoftensorform . . . . . . . . . . . . . . . . . . . . 37
2.9 Thesubnormalcompletionproblemfor2-variableweightedshifts 39
2.10 Spectralpictureofhyponormal2-variableweightedshifts . . . . . 43
2.11 Abridgebetween2-variableweightedshiftsandshiftsondirected
trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.12 ThesphericalAluthgetransform . . . . . . . . . . . . . . . . . . 49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3 Commutants,ReducingSubspaces,andvonNeumannAlgebras 65
KunyuGuoandHansongHuang
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Commutants and reducing subspaces for multiplication operators
ontheHardyspaceH2(D) . . . . . . . . . . . . . . . . . . . . . 68
v
vi
3.3 ThecaseoftheBergmanspaceLa2(D) . . . . . . . . . . . . . . . 71
3.4 ThecaseofBergmanspaceoverapolygon . . . . . . . . . . . . . 74
3.5 ThecaseoftheBergmanspaceoverhighdimensionaldomains . . 76
3.6 Furtherquestions . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 OperatorsintheCowen-DouglasClassandRelatedTopics 87
GadadharMisra
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 Somefuturedirectionsandfurtherthoughts . . . . . . . . . . . . 104
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 ToeplitzOperatorsandToeplitzC -Algebras 139
(cid:3)
HaraldUpmeier
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2 ToeplitzoperatorsonHilbertspacesofmulti-variableholomorphic
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3 Stronglypseudoconvexdomains . . . . . . . . . . . . . . . . . . 143
5.4 SymmetricdomainsandJordantriples . . . . . . . . . . . . . . . 146
5.5 Holomorphicfunctionspacesonsymmetricdomains . . . . . . . 152
5.6 ToeplitzC -algebrasonsymmetricdomains . . . . . . . . . . . . 156
(cid:3)
5.7 HilbertquotientmodulesandKeplervarieties . . . . . . . . . . . 160
5.8 ToeplitzoperatorsonReinhardtdomains . . . . . . . . . . . . . . 164
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6 Mo¨biusInvariantQ andQ Spaces 171
p K
HasiWulan
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.3 BasicpropertiesofQ spaces . . . . . . . . . . . . . . . . . . . 174
p
6.4 Carlesonmeasures . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.5 Theboundaryvaluecharacterizations . . . . . . . . . . . . . . . 179
6.6 Q spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
K
6.7 K-Carlesonmeasures . . . . . . . . . . . . . . . . . . . . . . . . 184
6.8 BoundaryQ spaces . . . . . . . . . . . . . . . . . . . . . . . . 191
K
6.9 CompositionoperatorsonQ andQ spaces . . . . . . . . . . . 193
p K
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7 AnalyticalAspectsoftheDrury-ArvesonSpace 203
QuanleiFangandJingboXia
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.2 vonNeumanninequalityforrowcontractions . . . . . . . . . . . 204
7.3 Themultipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.4 Afamilyofreproducing-kernelHilbertspaces . . . . . . . . . . . 210
7.5 Essentialnormality . . . . . . . . . . . . . . . . . . . . . . . . . 212
vii
7.6 ExpandingonDrury’sidea . . . . . . . . . . . . . . . . . . . . . 214
7.7 Closureofthepolynomials . . . . . . . . . . . . . . . . . . . . . 217
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8 ABriefSurveyofOperatorTheoryinH2(D2) 223
RongweiYang
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.3 Nagy-FoiastheoryinH2(D2) . . . . . . . . . . . . . . . . . . . . 228
8.4 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.5 Two-variableJordanblock . . . . . . . . . . . . . . . . . . . . . 236
8.6 Fredholmnessofthepairs(R ;R )and(S ;S ) . . . . . . . . . . 239
1 2 1 2
8.7 Essentialnormalityofquotientmodule . . . . . . . . . . . . . . . 241
8.8 Twosinglecompanionoperators . . . . . . . . . . . . . . . . . . 243
8.9 Congruentsubmodulesandtheirinvariants . . . . . . . . . . . . . 248
8.10 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . 251
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9 WeightedCompositionOperatorsonSomeAnalyticFunctionSpaces 259
RuhanZhao
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
9.3 Carlesonmeasures . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.4 Weighted composition operators between weighted Bergman
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.5 WeightedcompositionoperatorsbetweenHardyspaces . . . . . . 271
9.6 Weighted composition operators between weighted spaces of ana-
lyticfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
9.7 WeightedcompositionoperatorsbetweenBlochtypespaces . . . 277
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10 ToeplitzOperatorsandtheBerezinTransform 287
XianfengZhaoandDechaoZheng
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
10.2 BasicpropertiesofToeplitzoperatorsandtheBerezintransform . 289
10.3 PositivityofToeplitzoperatorsviatheBerezintransform . . . . . 293
10.4 InvertibilityofToeplitzoperatorsviatheBerezintransform . . . . 303
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
11 TowardsaDictionaryfortheBargmannTransform 319
KeheZhu
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
11.2 Hermitepolynomials . . . . . . . . . . . . . . . . . . . . . . . . 321
11.3 TheFouriertransform . . . . . . . . . . . . . . . . . . . . . . . . 322
11.4 Dilation,translation,andmodulationoperators . . . . . . . . . . 325
viii
11.5 Gaborframes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
11.6 Thecanonicalcommutationrelation . . . . . . . . . . . . . . . . 334
11.7 Uncertaintyprinciples . . . . . . . . . . . . . . . . . . . . . . . . 336
11.8 TheHilberttransform . . . . . . . . . . . . . . . . . . . . . . . . 338
11.9 Pseudo-differentialoperators . . . . . . . . . . . . . . . . . . . . 343
11.10 Furtherresultsandremarks . . . . . . . . . . . . . . . . . . . . . 345
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Index 351
Preface
Therehasbeenbroadinterestinanalyticfunctionspacesandrelatedoperatortheory
overthepastfewdecades.Inparticular,severalmonographsappearedinthisperiod,
including“CompositionOperatorsonSpacesofAnalyticFunctions”byCowenand
MacCluer,“BergmanSpaces”byDurenandSchuster,“InterpolationandSampling
inSpacesofAnalyticFunctions”bySeip,and“AnalysisonFockSpaces”,“Spaces
ofHolomorphicFunctionsintheUnitBall”,“OperatorTheoryinFunctionSpaces”
allbyZhu.
The present volume consists of eleven articles in the general area of analytic
function spaces and operators on them, which nicely complement the monographs
mentionedabove.Theoverlapbetweenthisvolumeandexistingbooksintheareais
minimal.Thematerialontwo-variableweightedshiftsbyCurto,theDrury-Arveson
spacebyFangandXia,theCowen-DouglasclassbyMisra,andoperatortheoryon
thebi-diskbyYanghasneverappearedinbookformbefore.
All the papers here are surveys in nature, although some of them do contain
results that have not been published before. Toeplitz operators on various function
spaces have been studied extensively in the literature and discussions about them
canbefoundinmanybooks.However,theexpositioninthepapersbyCoburn,Up-
meier,andZhaoandZhengpurposelyavoidedmaterialthathasalreadybeenwidely
disseminated.
Thetopicscoveredinthevolumearediverse,soIwillbehappyifeveryreader
finds one article (or more) interesting and useful. There are also several important
topics that should have been included here but are missing, for example, function
theoryandoperatortheoryonHilbertspacesofDirichletseriesonthecomplexplane.
Iinvitedexpertsintheseareastomakecontributionstothevolume,butforonereason
oranother,Ididnoteventuallyreceivethesearticles.
IwanttothankStevenKrantzforinvitingmetoeditthisvolume.Ialsowantto
thank all the contributors for your hard work and great papers. I enjoyed working
withyouall!
KeheZhu
Albany,NY
March,2019
ix