LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor:ProfessorEndreSu¨li,MathematicalInstitute,UniversityofOxford, WoodstockRoad,OxfordOX26GG,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat www.cambridge.org/mathematics 372 Moonshine:Thefirstquartercenturyandbeyond, J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds) 373 Smoothness,regularityandcompleteintersection, J.MAJADAS&A.G.RODICIO 374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction, S.ALINHAC 375 Triangulatedcategories, T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds) 376 Permutationpatterns, S.LINTON,N.RUSˇKUC&V.VATTER(eds) 377 AnintroductiontoGaloiscohomologyanditsapplications, G.BERHUY 378 Probabilityandmathematicalgenetics, N.H.BINGHAM&C.M.GOLDIE(eds) 379 Finiteandalgorithmicmodeltheory, J.ESPARZA,C.MICHAUX&C.STEINHORN(eds) 380 Realandcomplexsingularities, M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds) 381 Symmetriesandintegrabilityofdifferenceequations, D.LEVI,P.OLVER,Z.THOMOVA& P.WINTERNITZ(eds) 382 Forcingwithrandomvariablesandproofcomplexity, J.KRAJ´ICˇEK 383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI, R.CLUCKERS,J.NICAISE&J.SEBAG(eds) 384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII, R.CLUCKERS,J.NICAISE&J.SEBAG(eds) 385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems, B.MARCUS,K.PETERSEN &T.WEISSMAN(eds) 386 Independence-friendlylogic, A.L.MANN,G.SANDU&M.SEVENSTER 387 GroupsStAndrews2009inBathI, C.M.CAMPBELLetal(eds) 388 GroupsStAndrews2009inBathII, C.M.CAMPBELLetal(eds) 389 Randomfieldsonthesphere, D.MARINUCCI&G.PECCATI 390 Localizationinperiodicpotentials, D.E.PELINOVSKY 391 Fusionsystemsinalgebraandtopology, M.ASCHBACHER,R.KESSAR&B.OLIVER 392 Surveysincombinatorics2011, R.CHAPMAN(ed) 393 Non-abelianfundamentalgroupsandIwasawatheory, J.COATESetal(eds) 394 Variationalproblemsindifferentialgeometry, R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds) 395 Howgroupsgrow, A.MANN 396 Arithmeticdifferentialoperatorsoverthep-adicintegers, C.C.RALPH&S.R.SIMANCA 397 Hyperbolicgeometryandapplicationsinquantumchaosandcosmology, J.BOLTE&F.STEINER(eds) 398 Mathematicalmodelsincontactmechanics, M.SOFONEA&A.MATEI 399 Circuitdoublecoverofgraphs, C.-Q.ZHANG 400 Densespherepackings:ablueprintforformalproofs, T.HALES 401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory, B.DENG,J.DU&Q.FU 402 Mathematicalaspectsoffluidmechanics, J.C.ROBINSON,J.L.RODRIGO&W.SADOWSKI(eds) 403 Foundationsofcomputationalmathematics,Budapest2011, F.CUCKER,T.KRICK,A.PINKUS& A.SZANTO(eds) 404 Operatormethodsforboundaryvalueproblems, S.HASSI,H.S.V.DESNOO&F.H.SZAFRANIEC(eds) 405 Torsors,e´talehomotopyandapplicationstorationalpoints, A.N.SKOROBOGATOV(ed) 406 Appalachiansettheory, J.CUMMINGS&E.SCHIMMERLING(eds) 407 Themaximalsubgroupsofthelow-dimensionalfiniteclassicalgroups, J.N.BRAY,D.F.HOLT& C.M.RONEY-DOUGAL 408 Complexityscience:theWarwickmaster’scourse, R.BALL,V.KOLOKOLTSOV&R.S.MACKAY(eds) 409 Surveysincombinatorics2013, S.R.BLACKBURN,S.GERKE&M.WILDON(eds) 410 Representationtheoryandharmonicanalysisofwreathproductsoffinitegroups, T.CECCHERINI-SILBERSTEIN,F.SCARABOTTI&F.TOLLI 411 Modulispaces, L.BRAMBILA-PAZ,O.GARC´IA-PRADA,P.NEWSTEAD&R.P.THOMAS(eds) 412 Automorphismsandequivalencerelationsintopologicaldynamics, D.B.ELLIS&R.ELLIS 413 Optimaltransportation, Y.OLLIVIER,H.PAJOT&C.VILLANI(eds) 414 AutomorphicformsandGaloisrepresentationsI, F.DIAMOND,P.L.KASSAEI&M.KIM(eds) 415 AutomorphicformsandGaloisrepresentationsII, F.DIAMOND,P.L.KASSAEI&M.KIM(eds) 416 Reversibilityindynamicsandgrouptheory, A.G.O’FARRELL&I.SHORT 417 Recentadvancesinalgebraicgeometry, C.D.HACON,M.MUSTAT¸A˘&M.POPA(eds) 418 TheBloch–KatoconjecturefortheRiemannzetafunction, J.COATES,A.RAGHURAM,A.SAIKIA& R.SUJATHA(eds) 419 TheCauchyproblemfornon-Lipschitzsemi-linearparabolicpartialdifferentialequations, J.C.MEYER& D.J.NEEDHAM 420 Arithmeticandgeometry, L.DIEULEFAITetal(eds) 421 O-minimalityandDiophantinegeometry, G.O.JONES&A.J.WILKIE(eds) 422 GroupsStAndrews2013, C.M.CAMPBELLetal(eds) 423 Inequalitiesforgrapheigenvalues, Z.STANIC´ 424 Surveysincombinatorics2015, A.CZUMAJetal(eds) 425 Geometry,topologyanddynamicsinnegativecurvature, C.S.ARAVINDA,F.T.FARRELL& J.-F.LAFONT(eds) 426 Lecturesonthetheoryofwaterwaves, T.BRIDGES,M.GROVES&D.NICHOLLS(eds) 427 RecentadvancesinHodgetheory, M.KERR&G.PEARLSTEIN(eds) 428 GeometryinaFre´chetcontext, C.T.J.DODSON,G.GALANIS&E.VASSILIOU 429 Sheavesandfunctionsmodulop, L.TAELMAN 430 RecentprogressinthetheoryoftheEulerandNavier–Stokesequations, J.C.ROBINSON,J.L.RODRIGO, W.SADOWSKI&A.VIDAL-LO´PEZ(eds) 431 Harmonicandsubharmonicfunctiontheoryontherealhyperbolicball, M.STOLL 432 Topicsingraphautomorphismsandreconstruction(2ndEdition), J.LAURI&R.SCAPELLATO 433 RegularandirregularholonomicD-modules, M.KASHIWARA&P.SCHAPIRA 434 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems(2ndEdition), K.TAIRA 435 GradedringsandgradedGrothendieckgroups, R.HAZRAT 436 Groups,graphsandrandomwalks, T.CECCHERINI-SILBERSTEIN,M.SALVATORI& E.SAVA-HUSS(eds) 437 Dynamicsandanalyticnumbertheory, D.BADZIAHIN,A.GORODNIK&N.PEYERIMHOFF(eds) 438 Randomwalksandheatkernelsongraphs, M.T.BARLOW 439 Evolutionequations, K.AMMARI&S.GERBI(eds) 440 Surveysincombinatorics2017, A.CLAESSONetal(eds) 441 Polynomialsandthemod2SteenrodalgebraI, G.WALKER&R.M.W.WOOD 442 Polynomialsandthemod2SteenrodalgebraII, G.WALKER&R.M.W.WOOD 443 Asymptoticanalysisingeneralrelativity, T.DAUDE´,D.HA¨FNER&J.-P.NICOLAS(eds) 444 Geometricandcohomologicalgrouptheory, P.H.KROPHOLLER,I.J.LEARY,C.MART´INEZ-PE´REZ& B.E.A.NUCINKIS(eds) 445 Introductiontohiddensemi-Markovmodels, J.VANDERHOEK&R.J.ELLIOTT 446 Advancesintwo-dimensionalhomotopyandcombinatorialgrouptheory, W.METZLER& S.ROSEBROCK(eds) 447 Newdirectionsinlocallycompactgroups, P.-E.CAPRACE&N.MONOD(eds) 448 Syntheticdifferentialtopology, M.C.BUNGE,F.GAGO&A.M.SANLUIS 449 Permutationgroupsandcartesiandecompositions, C.E.PRAEGER&C.SCHNEIDER 450 Partialdifferentialequationsarisingfromphysicsandgeometry, M.BENAYEDetal(eds) 451 Topologicalmethodsingrouptheory, N.BROADDUS,M.DAVIS,J.-F.LAFONT&I.ORTIZ(eds) 452 Partialdifferentialequationsinfluidmechanics, C.L.FEFFERMAN,J.C.ROBINSON& J.L.RODRIGO(eds) 453 Stochasticstabilityofdifferentialequationsinabstractspaces, K.LIU 454 Beyondhyperbolicity, M.HAGEN,R.WEBB&H.WILTON(eds) 455 GroupsStAndrews2017inBirmingham, C.M.CAMPBELLetal(eds) 456 Surveysincombinatorics2019, A.LO,R.MYCROFT,G.PERARNAU&A.TREGLOWN(eds) 457 Shimuravarieties, T.HAINES&M.HARRIS(eds) 458 IntegrablesystemsandalgebraicgeometryI, R.DONAGI&T.SHASKA(eds) 459 IntegrablesystemsandalgebraicgeometryII, R.DONAGI&T.SHASKA(eds) 460 Wigner-typetheoremsforHilbertGrassmannians, M.PANKOV 461 Analysisandgeometryongraphsandmanifolds, M.KELLER,D.LENZ&R.K.WOJCIECHOWSKI 462 ZetaandL-functionsofvarietiesandmotives, B.KAHN 463 Differentialgeometryinthelarge, O.DEARRICOTTetal(eds) 464 Lecturesonorthogonalpolynomialsandspecialfunctions, H.S.COHL&M.E.H.ISMAIL(eds) 465 ConstrainedWillmoresurfaces, A´.C.QUINTINO 466 Invarianceofmodulesunderautomorphismsoftheirenvelopesandcovers, A.K.SRIVASTAVA, A.TUGANBAEV&P.A.GUILASENSIO 467 ThegenesisoftheLanglandsprogram, J.MUELLER&F.SHAHIDI 468 (Co)endcalculus, F.LOREGIAN 469 Computationalcryptography, J.W.BOS&M.STAM(eds) 470 Surveysincombinatorics2021, K.K.DABROWSKIetal(eds) 471 Matrixanalysisandentrywisepositivitypreservers, A.KHARE 472 FacetsofalgebraicgeometryI, P.ALUFFIetal(eds) 473 FacetsofalgebraicgeometryII, P.ALUFFIetal(eds) 474 Equivarianttopologyandderivedalgebra, S.BALCHIN,D.BARNES,M.KE˛DZIOREK& M.SZYMIK(eds) 475 EffectiveresultsandmethodsforDiophantineequationsoverfinitelygenerateddomains, J.-H.EVERTSE&K.GYO˝RY 476 Anindefiniteexcursioninoperatortheory, A.GHEONDEA 477 Ellipticregularitytheorybyapproximationmethods, E.A.PIMENTEL 478 Recentdevelopmentsinalgebraicgeometry, H.ABBAN,G.BROWN,A.KASPRZYK&S.MORI(eds) 479 Boundedcohomologyandsimplicialvolume, C.CAMPAGNOLO,F.FOURNIER-FACIO,N.HEUER& M.MORASCHINI(eds) 480 StacksProjectExpositoryCollection(SPEC), P.BELMANS,W.HO&A.J.DEJONG(eds) 481 Surveysincombinatorics2022, A.NIXON&S.PRENDIVILLE(eds) LondonMathematicalSocietyLectureNoteSeries:476 An Indefinite Excursion in Operator Theory Geometric and Spectral Treks in Kre˘ın Spaces AURELIAN GHEONDEA BilkentUniversity,Ankara andIMAR,Bucharest UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108969031 DOI:10.1017/9781108979061 ©AurelianGheondea2022 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 PrintedintheUnitedKingdombyTJBooksLimited,PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-108-96903-1Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. ToTiberiuConstantinescu(1955–2005)andPeterJonas(1941–2007) whosharedwithmethejoyoftrekkingthrough theindefiniterealmofoperatortheory aswellas Tomybeautifulandgenerouscountry,Roumania Contents Preface xi 1 InnerProductSpaces 1 1.1 BasicDefinitionsandProperties . . . . . . . . . . . . . . . . . . . . . . 1 1.2 TheWeakTopology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 NormedTopologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Kre˘ınSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Pre-Kre˘ınSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 AngularOperators 28 2.1 SemidefiniteSubspacesandAngularOperators . . . . . . . . . . . . . . 28 2.2 ExtensionsofSemidefiniteSubspaces . . . . . . . . . . . . . . . . . . . 35 2.3 Intermezzo:MaximalAccretiveandSelfadjointExtensions . . . . . . . . 45 2.4 AUniversalityPropertyofKre˘ınSpaces . . . . . . . . . . . . . . . . . . 55 2.5 GeneralisedAngularOperators . . . . . . . . . . . . . . . . . . . . . . . 57 2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3 SubspacesofKre˘ınSpaces 62 3.1 RegularSubspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2 Pseudo-RegularSubspaces . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 StrongDualityofSubspaces . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4 FredholmSubspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5 IndexFormulaeforLinearRelations . . . . . . . . . . . . . . . . . . . . 81 3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4 LinearOperatorsonKre˘ınSpaces 85 4.1 TheAdjointOperator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 SomeClassesofBoundedOperators . . . . . . . . . . . . . . . . . . . . 90 4.3 ContinuityofIsometricOperators . . . . . . . . . . . . . . . . . . . . . 96 vii viii CONTENTS 4.4 DissipativeOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5 CayleyTransformations. . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5 SelfadjointProjectionsandUnitaryOperators 113 5.1 SelfadjointProjections . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 MonotoneNetsofSelfadjointProjections . . . . . . . . . . . . . . . . . 118 5.3 UnitaryOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.4 DenseOperatorRanges . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6 TechniquesofInducedKre˘ınSpaces 134 6.1 Kre˘ınSpacesInducedbySelfadjointOperators . . . . . . . . . . . . . . 134 6.2 NevanlinnaTypeRepresentations. . . . . . . . . . . . . . . . . . . . . . 143 6.3 LinearisationofSelfadjointOperatorPencils. . . . . . . . . . . . . . . . 153 6.4 Carathe´odoryTypeRepresentations . . . . . . . . . . . . . . . . . . . . 156 6.5 ElementaryRotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.6 IsometricandUnitaryDilations. . . . . . . . . . . . . . . . . . . . . . . 178 6.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7 Plus/Minus-Operators 182 7.1 SpaceswithTwoInnerProducts . . . . . . . . . . . . . . . . . . . . . . 182 7.2 Minus-Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.3 UniformMinus-Operators . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.4 ExtensionsofUniformMinus-Operators . . . . . . . . . . . . . . . . . . 194 7.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 8 GeometryofContractiveOperators 199 8.1 ContractionsinKre˘ınSpaces . . . . . . . . . . . . . . . . . . . . . . . . 199 8.2 BoundednessofContractionsinKre˘ınSpaces . . . . . . . . . . . . . . . 204 8.3 TheAdjointofaContraction . . . . . . . . . . . . . . . . . . . . . . . . 205 8.4 TheScatteringTransform . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.5 LinearFractionalTransformations . . . . . . . . . . . . . . . . . . . . . 213 8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 CONTENTS ix 9 InvariantMaximalSemidefiniteSubspaces 219 9.1 QuestionsandDiscussions . . . . . . . . . . . . . . . . . . . . . . . . . 219 9.2 SpectralMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 9.3 FixedPointApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 9.4 FundamentalReducibility. . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.5 StrongStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 10 HankelOperatorsandInterpolationProblems 250 10.1 AGeneralisedNehariProblem . . . . . . . . . . . . . . . . . . . . . . . 250 10.2 MoreorLessClassicalHankelOperators . . . . . . . . . . . . . . . . . 256 10.3 IntertwiningDilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 10.4 GeneralisedInterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 263 10.5 TheBitangentialNevanlinna–PickProblem . . . . . . . . . . . . . . . . 268 10.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 11 SpectralTheoryforSelfadjointOperators 273 11.1 EigenvaluesandRootManifolds . . . . . . . . . . . . . . . . . . . . . . 273 11.2 JordanCanonicalForms . . . . . . . . . . . . . . . . . . . . . . . . . . 279 11.3 DefinitisableSelfadjointOperators . . . . . . . . . . . . . . . . . . . . . 285 11.4 Herglotz’sTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 11.5 TheResolventFunctionRepresentation . . . . . . . . . . . . . . . . . . 298 11.6 StieltjesInversionFormulae . . . . . . . . . . . . . . . . . . . . . . . . 303 11.7 TheSpectralFunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 11.8 DefinitisablePositiveOperators . . . . . . . . . . . . . . . . . . . . . . 318 11.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 12 Quasi-Contractions 323 12.1 GeometricPropertiesofQuasi-Contractions . . . . . . . . . . . . . . . . 323 12.2 DoubleQuasi-Contractions . . . . . . . . . . . . . . . . . . . . . . . . . 335 12.3 PolarDecompositionsofContractions . . . . . . . . . . . . . . . . . . . 339 12.4 ASpectralCharacterisationofDoubleQuasi-Contractions . . . . . . . . 342 12.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346