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TheRandomMatrixTheoryoftheClassicalCompactGroups.ByE.S.Meckes Operator Analysis Hilbert Space Methods in Complex Analysis JIM AGLER UniversityofCalifornia,SanDiego JOHN EDWARD McCARTHY WashingtonUniversityinSt.Louis NICHOLAS YOUNG LeedsUniversityandNewcastleUniversity UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi110025,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversitysmissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108485449 DOI:10.1017/9781108751292 ©JimAgler,JohnEdwardMcCarthy,andNicholasYoung2020 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2020 PrintedintheUnitedKingdombyTJInternationalLtd,PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-108-48544-9Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. To Sarah,Suzanne,and 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