Operator Analysis: Hilbert Space Methods in Complex Analysis

393 Pages·2020·2.607 MB·English

by  Jim Agler, John Edward McCarthy, Nicholas John Young

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Operator Analysis
Hilbert Space Methods in Complex Analysis
JIM AGLER
UniversityofCalifornia,SanDiego
JOHN EDWARD McCARTHY
WashingtonUniversityinSt.Louis
NICHOLAS YOUNG
LeedsUniversityandNewcastleUniversity
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DOI:10.1017/9781108751292
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Contents
Preface pagexiii
Acknowledgments xv
PARTONE COMMUTATIVETHEORY 1
1 TheOriginsofOperator-TheoreticApproaches
toFunctionTheory 3
1.1 Operators 3
1.2 FunctionalCalculi 4
1.3 OperatorsonHilbertSpace 9
1.4 TheSpectralTheorem 12
1.5 HardySpaceandtheUnilateralShift 13
1.6 InvariantSubspacesoftheUnilateralShift 15
1.7 VonNeumann’sTheoryofSpectralSets 16
1.8 TheSchurClassandSpectralDomains 19
1.9 TheSz.-NagyDilationTheorem 21
1.10 Andoˆ’sDilationTheorem 22
1.11 TheSz.-Nagy–FoiasModelTheory 23
1.12 TheSarasonInterpolationTheorem 25
1.13 HistoricalNotes 30
2 OperatorAnalysisonD:ModelFormulas,Lurking
Isometries,andPositivityArguments 33
2.1 Overview 33
2.2 AModelFormulaforS(D) 33
2.3 ReproducingKernelHilbertSpaces 37
2.4 LurkingIsometries 39
2.5 TheNetworkRealizationFormula(ScalarCase)
viaModelTheory 44
2.6 InterpolationviaModelTheory 52
vii
viii Contents
2.7 TheMu¨ntz–Sza´szInterpolationTheorem 56
2.8 PositivityArguments 61
2.9 HistoricalNotes 70
3 FurtherDevelopmentofModelsontheDisc 71
3.1 AModelFormulaforSB(H,K)(D) 71
3.2 LurkingIsometriesRevisited 74
3.3 TheNetworkRealizationFormula 75
3.4 TensorProductsofHilbertSpaces 78
3.5 TensorProductsofOperators 79
3.6 RealizationofRationalMatrixFunctionsand
theMcMillanDegree 81
3.7 PickInterpolationRevisited 83
3.8 TheCoronaProblem 85
3.9 HistoricalNotes 91
4 OperatorAnalysisonD2 93
4.1 TheSpaceofHereditaryFunctionsonD2 93
4.2 TheHereditaryFunctionalCalculusonD2 95
4.3 ModelsonD2 99
4.4 ModelsonD2viatheDualityConstruction 104
4.5 TheNetworkRealizationFormulaforD2 106
4.6 Nevanlinna–PickInterpolationonD2 109
4.7 ToeplitzCoronafortheBidisc 112
4.8 Operator-ValuedFunctionsonD2 113
4.9 ModelsofOperator-ValuedFunctionsonD2 116
4.10 HistoricalNotes 129
5 Carathe´odory–JuliaTheoryontheDiscandtheBidisc 131
5.1 TheOne-VariableResults 131
5.2 TheModelApproachtoRegularityonD:B-pointsandC-points 133
5.3 AProofoftheCarathe´odory–JuliaTheoremonDviaModels 137
5.4 PickInterpolationontheBoundary 141
5.5 Regularity,B-pointsandC-pointsontheBidisc 143
5.6 TheMissingLink 146
5.7 HistoricalNotes 147
6 HerglotzandNevanlinnaRepresentationsinSeveral
Variables 148
6.1 Overview 148
6.2 TheHerglotzRepresentationonD2 149
6.3 NevanlinnaRepresentationsonHviaOperatorTheory 153
Contents ix
6.4 TheNevanlinnaRepresentationsonH2 156
6.5 AClassificationSchemeforNevanlinnaRepresentations
inTwoVariables 160
6.6 TheTypeofaFunction 165
6.7 HistoricalNotes 168
7 ModelTheoryontheSymmetrizedBidisc 169
7.1 AddingSymmetrytotheFundamentalTheoremforD2 170
7.2 HowtoDefineModelsontheSymmetrizedBidisc 173
7.3 TheNetworkRealizationFormulaforG 176
7.4 TheHereditaryFunctionalCalculusonG 177
7.5 WhenIsGaSpectralDomain? 182
7.6 GSpectralImpliesGCompleteSpectral 185
7.7 TheSpectralNevanlinna–PickProblem 185
7.8 HistoricalNotes 187
8 SpectralSets:ThreeCaseStudies 189
8.1 VonNeumann’sInequalityandthePseudo-Hyperbolic
MetriconD 189
8.2 SpectralDomainsandtheCarathe´odoryMetric 192
8.3 BackgroundMaterial 194
8.4 Lempert’sTheorem 195
8.5 TheCarathe´odoryDistanceonG 203
8.6 VonNeumann’sInequalityonSubvarietiesoftheBidisc 205
8.7 HistoricalNotes 211
9 CalcularNorms 213
9.1 TheTaylorSpectrumandFunctionalCalculus 213
9.2 CalcularNormsandAlgebras 215
9.3 Halmos’sConjectureandPaulsen’sTheorem 223
9.4 TheDouglas–PaulsenNorm 226
9.5 TheB.andF.DelyonNormandCrouzeix’sTheorem 233
9.6 TheBadea–Beckermann–CrouzeixNorm 235
9.7 ThePolydiscNorm 243
9.8 TheOkaExtensionTheoremandCalcularNorms 245
9.9 HistoricalNotes 252
10 OperatorMonotoneFunctions 254
10.1 Lo¨wner’sTheorems 254
10.2 AnInterludeonLinearProgramming 262
10.3 LocallyMatrixMonotoneFunctionsind Variables 269
10.4 TheLo¨wnerClassind Variables 278