The theory of H(b) spaces. Vol.1

704 Pages·2016·3.307 MB·English

by  Emmanuel Fricain, Javad Mashreghi

H
The Theory of (b) Spaces
Volume1
An H(b) space is defined as a collection of analytic functions that are in the
image of an operator. The theory of H(b) spaces bridges two classical sub-
jects,complexanalysisandoperatortheory,whichmakesitbothappealingand
demanding.
Volume1ofthiscomprehensivetreatmentisdevotedtothepreliminarysub-
jects required to understand the foundation of H(b) spaces, such as Hardy
spaces,Fourieranalysis,integralrepresentationtheorems,Carlesonmeasures,
ToeplitzandHankeloperators,varioustypesofshiftoperatorsandClarkmea-
sures. Volume 2 focuses on the central theory. Both books are accessible to
graduate students as well as researchers: each volume contains numerous
exercisesandhints,andfiguresareincludedthroughouttoillustratethetheory.
Together,thesetwovolumesprovideeverythingthereaderneedstounderstand
andappreciatethisbeautifulbranchofmathematics.
EmmanuelFricainisProfessorofMathematicsatLaboratoirePaulPainleve´,
Universite´ Lille 1, France. Part of his research focuses on the interaction
between complex analysis and operator theory, which is the main content of
thisbook.Hehasawealthofexperienceteachingnumerousgraduatecourses
on different aspects of analytic Hilbert spaces, and he has published several
papersonH(b)spacesinhigh-qualityjournals,makinghimaworldspecialist
inthissubject.
JavadMashreghiisaProfessorofMathematicsatUniversite´Laval,Que´bec,
Canada,wherehehasbeenselectedStarProfessoroftheYearseventimesfor
excellenceinteaching.Hismainfieldsofinterestarecomplexanalysis,oper-
ator theory and harmonic analysis. He is the author of several mathematical
textbooks, monographs and research articles. He won the G. de B. Robinson
Award,thepublicationprizeoftheCanadianMathematicalSociety,in2004.
NEW MATHEMATICAL MONOGRAPHS
EditorialBoard
Be´laBolloba´s,WilliamFulton,AnatoleKatok,FrancesKirwan,PeterSarnak,BarrySimon,BurtTotaro
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.Fora
completeserieslistingvisitwww.cambridge.org/mathematics.
1. M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductiveGroups
2. J.B.GarnettandD.E.MarshallHarmonicMeasure
3. P.CohnFreeIdealRingsandLocalizationinGeneralRings
4. E.BombieriandW.GublerHeightsinDiophantineGeometry
5. Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns
6. S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures
7. A.ShlapentokhHilbert’sTenthProblem
8. G.MichlerTheoryofFiniteSimpleGroupsI
9. A.BakerandG.Wu¨stholzLogarithmicFormsandDiophantineGeometry
10. P.KronheimerandT.MrowkaMonopolesandThree-Manifolds
11. B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T)
12. J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory
13. M.GrandisDirectedAlgebraicTopology
14. G.MichlerTheoryofFiniteSimpleGroupsII
15. R.SchertzComplexMultiplication
16. S.BlochLecturesonAlgebraicCycles(2ndEdition)
17. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups
18. T.DownarowiczEntropyinDynamicalSystems
19. C.SimpsonHomotopyTheoryofHigherCategories
20. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesI
21. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesII
22. J.Goubault-LarrecqNon-HausdorffTopologyandDomainTheory
23. J.S´niatyckiDifferentialGeometryofSingularSpacesandReductionofSymmetry
24. E.RiehlCategoricalHomotopyTheory
25. B.A.MunsonandI.Volic´CubicalHomotopyTheory
26. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups(2ndEdition)
27. J.Heinonen,P.Koskela,N.ShanmugalingamandJ.T.TysonSobolevSpacesonMetricMeasureSpaces
28. Y.-G.OhSymplecticTopologyandFloerHomologyI
29. Y.-G.OhSymplecticTopologyandFloerHomologyII
The Theory of H(b) Spaces
Volume 1
EMMANUEL FRICAIN
Universite´Lille1
JAVAD MASHREGHI
Universite´Laval,Que´bec
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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(cid:2)c EmmanuelFricainandJavadMashreghi2016
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Firstpublished2016
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LibraryofCongressCataloguinginPublicationdata
Fricain,Emmanuel,1971–author.
ThetheoryofH(b)spaces/EmmanuelFricain,JavadMashreghi.
2volumes;cm.–(Newmathematicalmonographs;v.20–21)
ISBN978-1-107-02777-0(Hardback)
1. Hilbertspace. 2. Hardyspaces. 3. Analyticfunctions. 4. Linear
operators. I. Mashreghi,Javad,author. II. Title.
QA322.4.F732014
515(cid:2).733–dc23 2014005539
ISBN–2VolumeSet978-1-107-11941-3Hardback
ISBN–Volume1 978-1-107-02777-0Hardback
ISBN–Volume2 978-1-107-02778-7Hardback
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Contents for Volume 1
Preface pagexvii
1 Normedlinearspacesandtheiroperators 1
1.1 Banachspaces 1
1.2 Boundedoperators 9
1.3 Fourierseries 14
1.4 TheHahn–Banachtheorem 15
1.5 TheBairecategorytheoremanditsconsequences 21
1.6 Thespectrum 26
1.7 Hilbertspaceandprojections 30
1.8 Theadjointoperator 40
1.9 Tensorproductandalgebraicdirectsum 45
1.10 Invariantsubspacesandcyclicvectors 49
1.11 Compressionsanddilations 52
1.12 Anglebetweentwosubspaces 54
NotesonChapter1 57
2 Somefamiliesofoperators 60
2.1 Finite-rankoperators 60
2.2 Compactoperators 62
2.3 Subdivisionsofspectrum 65
2.4 Self-adjointoperators 70
2.5 Contractions 77
2.6 Normalandunitaryoperators 78
2.7 Forwardandbackwardshiftoperatorson(cid:2)2 80
2.8 ThemultiplicationoperatoronL2(μ) 83
2.9 DoublyinfiniteToeplitzandHankelmatrices 86
NotesonChapter2 92
vii
viii Contents
3 Harmonicfunctionsontheopenunitdisk 96
3.1 Nontangentialboundaryvalues 96
3.2 Angularderivatives 98
3.3 Somewell-knownfactsinmeasuretheory 101
3.4 BoundarybehaviorofPμ 106
3.5 IntegralmeansofPμ 110
3.6 BoundarybehaviorofQμ 112
3.7 IntegralmeansofQμ 113
3.8 Subharmonicfunctions 116
3.9 SomeapplicationsofGreen’sformula 117
NotesonChapter3 120
4 Hardyspaces 122
4.1 Hyperbolicgeometry 122
4.2 ClassicHardyspacesHp 124
4.3 TheRieszprojectionP 130
+
4.4 KernelsofP+andP− 135
4.5 DualandpredualofHpspaces 137
4.6 Thecanonicalfactorization 141
4.7 TheSchwarzreflectionprincipleforH1functions 148
4.8 Propertiesofouterfunctions 149
4.9 Auniquenesstheorem 154
4.10 MoreonthenorminHp 157
NotesonChapter4 163
5 Morefunctionspaces 166
5.1 TheNevanlinnaclassN 166
5.2 Thespectrumofb 171
5.3 ThediskalgebraA 173
5.4 ThealgebraC(T)+H ∞ 181
5.5 GeneralizedHardyspacesHp(ν) 183
5.6 Carlesonmeasures 187
5.7 EquivalentnormsonH2 198
5.8 Thecoronaproblem 202
NotesonChapter5 211
6 Extremeandexposedpoints 214
6.1 Extremepoints 214
6.2 ExtremepointsofLp(T) 217
6.3 ExtremepointsofHp 219
6.4 Strictconvexity 224
6.5 ExposedpointsofB(X) 227
6.6 StronglyexposedpointsofB(X) 230
6.7 EquivalenceofrigidityandexposedpointsinH1 232