Eigenvalues of Self-Adjoint Toeplitz Operators with Respect to a Constrained Algebra

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Eigenvalues of Self-Adjoint Toeplitz Operators with Respect to a Constrained Algebra
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Broschinski, Adam E
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University of Florida
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
MCCULLOUGH,SCOTT A
Committee Co-Chair:
BLOCK,LOUIS S
Committee Members:
SHEN,LI C
JURY,MICHAEL THOMAS
TRICKEY,SAMUEL B

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annulus -- eigenvalues -- operators -- representations -- toeplitz
Mathematics -- Dissertations, Academic -- UF
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Abstract:
Let R be a finitely connected domain and A be a subalgebra of the Hardy space of bounded analytic function on R. By working with the collection of A invariant subspaces of the modulus automorphic Hardy spaces on R, conditions for eigenvalues in the bounded components complement of the essential range of the symbol -- the gaps -- of self-adjoint Toeplitz operators are studied. In the case that R is an annulus and A is the space of bounded analytic functions on R, the results improve upon and refine those of Abrahamse, Aryana, and Clancy on the existence of sequences of eigenvalues in the gaps. When R is a doubly connected domain and A is the space of bounded analytic functions on R, results complementary to those of Abrahamse, Aryana, and Clancy are obtained without the use of theta functions or symmetry. The techniques extend naturally to domains of higher connectivity as well. In the case where R is the unit disc, and A is the subalgebra of bounded analytic function on the unit disc consisting of those functions whose first derivative is 0 at 0 -- the Neil algebra -- necessary and sufficient conditions for the existence of eigenvalues in the gaps are found. Further, in contrast to classical Toeplitz operators on the unit disc, Toeplitz operators with respect to the Neil algebra have eigenvalues. Lastly, the unital pure completely contractive extremal representations of the Neil algebra are found and their relationship to Toeplitz operators exposed.
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by Adam E Broschinski.
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Thesis (Ph.D.)--University of Florida, 2014.
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Adviser: MCCULLOUGH,SCOTT A.
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Co-adviser: BLOCK,LOUIS S.

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EIGENVALUESOFSELF-ADJOINTTOEPLITZOPERATORSWITHRESPECTTOACONSTRAINEDALGEBRAByADAME.BROSCHINSKIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014AdamE.Broschinski 2

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Idedicatethistoeveryonewhohelpedme. 3

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ACKNOWLEDGMENTS IthankmyadvisorDr.McCulloughforhispatienceandadvice.Ithankmycommittee,Dr.Jury,Dr.Shen,Dr.Block,andDr.Trickey.Lastly,IthankmyteacherswhoinspiredmetotakeupMathematicsandScience. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................... 4 ABSTRACT ......................................... 7 CHAPTER 1CLASSICTOEPLITZOPERATORS ......................... 8 1.1Introduction ................................... 8 1.2TheHardy-HilbertSpaceontheUnitDisk ................... 8 1.3ToeplitzOperatorsontheUnitDisk ....................... 10 2GUIDINGEXAMPLES ................................. 13 2.1PreviousWork .................................. 13 2.2ToeplitzOperatorsontheAnnulus ....................... 13 2.2.1ProofofProposition 2.2.3 ........................ 18 2.2.2ProofofTheorem 2.2.4 ......................... 19 2.2.3ProofsoftheCorollaries ......................... 22 2.2.3.1ProofofCorollary 2.2.5 .................... 22 2.2.3.2ProofofCorollary 2.2.6 .................... 23 2.2.3.3ProofofCorollary 2.2.7 .................... 23 2.3ToeplitzOperatorsontheNeileParabola .................... 24 2.3.1ProofofLemma 2.3.1 .......................... 27 2.3.2ProofofProposition 2.3.2 ........................ 28 2.3.3ProofofTheorem 2.3.3 ......................... 28 2.3.4ProofsoftheCorollaries ......................... 30 2.3.4.1ProofofCorollary 2.3.4 .................... 30 2.3.4.2ProofofCorollary 2.3.5 .................... 31 2.3.4.3ProofofCorollary 2.3.6 .................... 32 3GENERALTHEOREM ................................ 34 3.1Introduction ................................... 34 3.2Sarason-Hardy-HilbertSpacesonMultiplyConnectedDomains ......... 34 3.3ExistenceofEigenvaluesTheorem ........................ 38 3.3.1ProofofTheorem 3.3.1 ......................... 40 3.3.2ProofofCorollaries ............................ 43 3.3.2.1ProofofCorollary 3.3.2 .................... 43 3.3.2.2ProofofCorollary 3.3.3 .................... 43 3.4ModelswithEigenvalues ............................. 43 5

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4NOVELEXAMPLES .................................. 46 4.1PreviousWork .................................. 46 4.2ToeplitzOperatorsonaMultiplyConnectedDomain .............. 46 4.2.1ProofsofLemmas ............................ 50 4.2.1.1ProofofLemma 4.2.2 .................... 50 4.2.1.2ProofofLemma 4.2.4 .................... 51 4.2.1.3ProofofLemma 4.2.6 .................... 51 4.2.1.4ProofofLemma 4.2.7 .................... 53 4.2.2ProofofProposition 4.2.5 ........................ 53 4.2.3ProofofTheorem 4.2.8 ......................... 54 4.2.4ProofofCorollary 4.2.9 ......................... 55 4.3SomeToeplitzOperatorsonaConstrainedDiskAlgebra ............ 56 4.3.1ProofsofLemmas ............................ 58 4.3.1.1ProofofLemma 4.3.1 ..................... 58 4.3.1.2ProofofLemma 4.3.2 ..................... 58 4.3.1.3ProofofLemma 4.3.3 ..................... 58 4.3.1.4ProofofLemma 4.3.4 ..................... 59 4.3.2ProofofTheorem 4.3.5 ......................... 60 5BUNDLESHIFTS ................................... 61 5.1PreviousWork .................................. 61 5.2Representations ................................. 61 5.3RepresentationsontheNeileAlgebra ...................... 63 5.3.1ProofofLemmas ............................. 64 5.3.1.1ProofofLemma 5.3.1 ..................... 64 5.3.1.2ProofofLemma 5.3.2 ..................... 65 5.3.2ProofofProposition 5.3.3 ........................ 66 5.3.3ProofofTheorem 5.3.5 ......................... 66 5.3.4ProofofCorollaries ............................ 68 5.3.4.1ProofofCorollary 5.3.4 .................... 68 5.3.4.2ProofofCorollary 5.3.6 .................... 68 6CONCLUSION ..................................... 69 REFERENCES ........................................ 70 BIOGRAPHICALSKETCH ................................. 73 6

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEIGENVALUESOFSELF-ADJOINTTOEPLITZOPERATORSWITHRESPECTTOACONSTRAINEDALGEBRAByAdamE.BroschinskiMay2014Chair:ScottMcCulloughMajor:MathematicsLetRbeanitelyconnecteddomainandAbeasubalgebraoftheHardyspaceofboundedanalyticfunctiononR.ByworkingwiththecollectionofAinvariantsubspacesofthemodulusautomorphicHardyspacesonR,conditionsforeigenvaluesintheboundedcomponentscomplementoftheessentialrangeofthesymbol{thegaps{ofself-adjointToeplitzoperatorsarestudied.InthecasethatRisanannulusandAisthespaceofboundedanalyticfunctionsonR,theresultsimproveuponandrenethoseofAbrahamse,Aryana,andClancyontheexistenceofsequencesofeigenvaluesinthegaps.WhenRisadoublyconnecteddomainandAisthespaceofboundedanalyticfunctionsonR,resultscomplementarytothoseofAbrahamse,Aryana,andClancyareobtainedwithouttheuseofthetafunctionsorsymmetry.Thetechniquesextendnaturallytodomainsofhigherconnectivityaswell.InthecasewhereRistheunitdisc,andAisthesubalgebraofboundedanalyticfunctionontheunitdiscconsistingofthosefunctionswhoserstderivativeis0at0{theNeilealgebra{necessaryandsucientconditionsfortheexistenceofeigenvaluesinthegapsarefound.Further,incontrasttoclassicalToeplitzoperatorsontheunitdisc,ToeplitzoperatorswithrespecttotheNeilealgebrahaveeigenvalues.Lastly,theunitalpurecompletelycontractiveextremalrepresentationsoftheNeilealgebraarefoundandtheirrelationshiptoToeplitzoperatorsexposed. 7

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CHAPTER1CLASSICTOEPLITZOPERATORS 1.1IntroductionLetRbeanicemultiplyconnecteddomainandletA(R)bethealgebraoffunctionsanalyticinRandcontinuouson R.Inthisdissertationeigenvaluesforself-adjointToeplitzoperatorswithrealsymbolsassociatedtoA(R).Additionally,eigenvaluesforself-adjointToeplitzoperatorsassociatedtoproperunitalsubalgebrasofA(R)areinvestigated.Specically,ToeplitzoperatorsassociatedtothesubalgebraofH1(D)consistingofthosefwhosederivativeat0is0,whichiscalledtheNeilealgebra,arestudied.Thedissertationproperisorganizedasfollows.TheremainderofthischapterdiscussestheclassicalHardy-HilbertspaceontheunitdiskandsomewellknownpropertiesofToeplitzoperatorsonthedisk.Chapter 2 introducesSarason'sgeneralizationoftheHardy-HilbertspacefortheannulusanddiscussesToeplitzoperatorsonthesespaces.Additionally,weintroduceasimilarfamilyofspacesfortheNeilealgebraandtheirassociatedToeplitzoperators.Chapter 3 introducesHilbertspacevaluedSarason-Hardy-HilbertspacesofamultiplyconnecteddomainandtheirassociatedToeplitzoperators,unifyingthetheoremsinChapter 2 .InChapter 4 weusetheresultsChapter 3 tostudyeigenvaluesofself-adjointToeplitzoperatorsonmultiplyconnecteddomains.Additionally,weusetheresultsofChapter 3 tostudyeigenvaluesofself-adjointToeplitzoperatorsassocatedwiththesubalgebraofH1(D)consistingofthosefwhoserstandsecondderivativesat0are0.Chapter 5 providesanadditionalrationaleforconsideringthechoiceoffamiliesofrepresentationsappearinginChapter 2 whenstudyingToeplitzoperatorsassociatedtotheNeilealgebra. 1.2TheHardy-HilbertSpaceontheUnitDiskLetD=fz2C:jzj<1g.Thisiscalledtheunitdisk.Theboundaryoftheunitdiscistheunitcircle,T=fz2C:jzj=1g. 8

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IfweletsbearclengthmeasureonT,thenwecandenethefollowingspacesL1(T)=f:T!C:fismeasureableandkfk1=ZTjfjds<1,L2(T)=f:T!C:fismeasureableandkfk22=ZTjfj2ds<1,andL1(T)=ff:T!C:fismeasureableandkfk1=esssupjfj<1g.ThespaceL2(T)isaHilbertspacewithinnerproducthf,gi=ZTf gds.WhenitisclearthatwearetalkingaboutthenormonL2(T)wewilljustwritekk.Wewillbeusingthefollowingtwofactsaboutthesespacesfreely,theproofscanbefoundin[ 17 ].Therstfactisthatifwelet(z)=z,thenusingtheFourierTransform,thespaceL2(T)canbeidentiedwithL2(T)=(f=1Xj=ajj:kfk22=1Xj=jajj2<1).Additionally,ifwedenotetheFouriertransformoff2L2(T)bybf,thenforeachj2Zbf(j)=aj.Thesecondfactisthatgivenf2L1(T),ifRTf ds=0forall2L1(T),thenf=0almosteverywhere.Wecannowdeneafewmoreimportantcollectionsoffunctions.TherstisA(D)=f: D!C:fiscontinuouson DandanalyticonD,whichiscalledthediskalgebra.ThespaceH2(T)= A(D)L2(T)iscalledtheHardy-Hilbertspaceontheunitdisk.ThefunctionsinH2(T)areonlydenedalmosteverywhereonT.Additionally,letH2(D)=(f(z)=1Xj=0ajzj:kfk22=1Xj=0jajj2). 9

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ThefunctionsinH2(D)areanalyticonDandhaveboundaryvaluesalmosteverywhereonT[ 29 ].Byidentifyingf2H2(D)withitsboundaryvaluesonT,f,whichisinH2(T)wegetanisometricisomorphismbetweenH2(D)andH2(T).Inparticular,forf=P1j=0ajzj2H2(D)theFouriertransformoffsatisesbf(j)=ajforallj2Z(whereajistakentobe0whenj<0).Conversely,forag2H2(T)whoseFouriercoecientsaregivenbybg(j)=ajforj2Z(withaj=0ifj<0)thefunctionf=P1j=0ajzjisinH2(D)anditsboundaryvaluesareequaltogalmosteverywhere.Therefore,wecancharacterizeH2(T)asasubspaceofL2(T)byf2H2(T)ifandonlyiff2L2(T)andforallj<0wehavethatbf(j)=0.ThischaracterizationallowsustoconsiderH2(D)asaclosedsubspaceofL2(T). 1.3ToeplitzOperatorsontheUnitDiskLet`2(N)=((aj)1j=0:fajgCand1Xj=0jajj2<1)andfbjg1j=C.ClassicallyaToeplitzoperatorisaninnitematrixofthefollowingformA=266666664b0b1b2...b)]TJ /F9 7.97 Tf 6.59 0 Td[(1b0b1...b)]TJ /F9 7.97 Tf 6.59 0 Td[(2b)]TJ /F9 7.97 Tf 6.59 0 Td[(1b0...............377777775whichactsboundedlyon`2(N)byleftmultiplication.ThematrixAisalsocalledaToeplitzmatrix.Thispointofviewhasapplicationsinprobabilityandstatistics[ 18 ].OurinterestinToepltizoperatorsarisesfromviewingthemasoperatorsonH2(D).First,notethatL2(T)isinvariantundermultiplicationbyfunctionsfromL1(T).If2L1(T),thentheoperator,MdenedbyMf=fforf2L2(T)isaboundedlinearoperatoronL2(T).Moreover,kMkop=kk1.LetPH2(D)betheprojectionfromL2(T)toH2(D).TheToeplitzoperatorwithsymbol,denotedbyT,onH2(D)isdenedbyTf=PH2(D)f. 10

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SincebothH2(D)and`2(N)areidentiedwithsquaresummablesequencesthereisanaturalisomorphicisomorphism,,betweenthosetwospacesdenedby)]TJ /F6 11.955 Tf 5.48 -9.69 Td[((aj)1j=0=1Xj=0ajzj.BecauseL1(T)L2(T)each2L1(T)hasaFouriertransformbandcanbewritten=1Xj=b(j)j.ToeachwecanassociateaToeplitzmatrixA=266666664b(0)b(1)b(2)b()]TJ /F6 11.955 Tf 9.3 0 Td[(1)b(0)b(1)...b()]TJ /F6 11.955 Tf 9.3 0 Td[(2)b()]TJ /F6 11.955 Tf 9.3 0 Td[(1)b(0)...............377777775.Infact,foranya=(aj)1j=02`2(N)wehavethatT(a)=(Aa).Conversely,ifAisaToeplitzmatrixwhichactsboundedlyon`2(N),thenthereexists2L1(T)suchthatforalla2`2(N)wehavethatT(a)=(Aa)[ 28 ].SothetwodenitionsofToeplitzoperatorsonH2(D)areequivalent.Asanexample,if= z,thenTisthebackwardshiftonH2(D);i.e.,T1Xj=0ajzj=1Xj=0aj+1zj.ThepropertiesofToeplitzoperatorsontheunitdiskhavebeenwellstudied.Thereferences[ 19 ],[ 20 ],[ 26 ],and[ 27 ]provideresultsforself-adjointToeplitzoperatorsand[ 13 ]providesanoverviewoftheresultsforgeneralToeplitzoperatorsonthedisk.OneofthemostimportantfactspresentedinDouglas[ 13 ],isthattherangeofthesymbol2L1(T)iscontainedinthespectrumoftheToeplitzoperatorwithsymbol.Infact,Abrahamseshowed 11

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thatthisistrueforToeplitzoperatorsonmultiplyconnecteddomains[ 1 ].ToeplitzoperatorsonmultiplyconnecteddomainswillbediscussedindetailinChapter 2 andChapter 4 .OntheunitdiskitisalsoknownthatthespectrumofaToeplitzoperatorisconnected.AbrahamseshowedthisisnottrueforToeplitzoperatorsonmultiplyconnecteddomains[ 1 ].Abrahamse'scounterexampleontheannulusmotivatedtheworkbyAryanaandClancyandtheworkpresentedinthisdissertation. 12

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CHAPTER2GUIDINGEXAMPLES 2.1PreviousWorkForthealgebraAoffunctionsanalyticontheannulusAandcontinuousontheclosureofA,theresultsobtainedinSection 2.2 givenerdetailthanthoseofAryanaandClancey[ 5 ],[ 6 ](seealso[ 4 ]and[ 11 ])intheirgeneralizationofaresultofAbrahamse[ 1 ].VersionsofAbrahamse'sandArayanaandClancy'sresultsarestatedinSection 2.2 .Theproofshereareaccessibletoreadersfamiliarwithbasicfunctionalanalysisandfunctiontheoryontheannulusasfoundineither[ 16 ]or[ 30 ];inparticular,theymakenouseofthetafunctions.PickinterpolationintheNeileAlgebra,i.e.,thefunctionsinH1(D)withf0(0)=0,andotherrelatedmoreelaboratesubalgebrasofH1,isacurrentactiveareaofresearchwith[ 7 ],[ 8 ],[ 9 ],[ 14 ],[ 12 ],[ 21 ],and[ 22 ]amongthereferences.Thischapterisanamplicationofworkoriginallypresentedin[ 10 ]. 2.2ToeplitzOperatorsontheAnnulusFix0
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ItiswellknownthatforAtheanalogoftheclassicalHilbertHardyspaceH2(D)onthediscisaoneparameterfamilyofHilbertspacesthatcanbedescribedinseveraldierentways[ 1 ],[ 2 ],or[ 30 ].Forourpurposesthefollowingisconvenient.Following[ 30 ]wewillusetheuniversalcoveringspaceoftheannulus,bA=f(r,t)2R2:q
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usedbySarasonanditsimpliesthenotation.LastlySarasonshowedthat,ifweletL2(\=f:)]TJ /F2 11.955 Tf 17.56 0 Td[(!C:fismeasurableandkfk22=Z)]TJ /F2 11.955 Tf 7.31 10.79 Td[(jfj2d,thenwecanidentifyeachF2H2(A;)withanfinL2(\byidentifyingF(r,t)withf(reit)=F(r,t)wherer2q,1andt2[0,2).Sarasonalsoestablishedthatifwelet(r,t)=reit,thenH2(A;)=fF:F2H2(A;0)g.NotethatisidentiedwithzonA.Moreover,SarasonshowedthattheLaurentpolynomialsaredenseinH2(A;0)andthusH2(A;0)admitsananalogtoFourierAnalysisonthedisk.LetC( A)denotetheBanachalgebra(intheuniformnorm)ofcontinuousfunctionsontheclosureofA.Theannulusalgebra,A(A),isthe(Banach)subalgebraofC( A)consistingofthosefwhichareanalyticinA.Additionally,letH1(A)betheclosureofA(A)underboundedpointwiselimits.IfweequipH1(A)withthenormgivenbykfk1=esssupf,thenH1(A)isaBanachspace.Themap:bA!AinducesamapfromfunctionsinH1(A)toboundedfunctionsonbAviaa2H1(A)liftstoa.Also,noticethatforany2[0,1),F2H2(A;),anda2H1(A)wehavethat(a)F2H2(A;).NowweturntomultiplicationandToeplitzoperatorsontheH2(A;)spaces.ItiseasytoseethateachH2(A;)spaceininvariantforA(A)inthesensethateacha2A(A)determinesaboundedlinearoperatorMaonH2(A;)denedbyMaF=(a)F.Moreover,themapping:A(A)!B(H2)denedby(a)=MaisanisometricunitalrepresentationofthealgebraA(A)intothespaceB(H2)ofboundedlinearoperatorsontheHilbertspaceH2.Lastly,becausewecanidentifyfunctionsinH2(A;)withfunctionsinL2(\ 15

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inaninjectivewayifafunctionf2L2(\isidentiedwithF2H2(A;)forsome2[0,1),thenwewillsayfisinH2(A;).Noticethatfor2[0,1),anyF2H2(A;)identiedwithf2L2(\,anda2H1(A)thefunction(a)Fisidentiedwithaf2H2(A;).FromnowonwewillimplicitlyusethisidenticationtoconsiderfunctionsinH2(A;)asfunctionsonA.Next,let2L1(\denoteareal-valuedfunctiononB.Thesymboldeterminesafamily,oneforeach,ofToeplitzoperators.Specically,letTdenotetheToeplitzoperatoronH2denedbyH23f7!Pf,wherePistheprojectionofL2(\ontoH2(A;).Givenan2[0,1)afunctiong2H2(A;)isouteriffag2H2(A;):a2A(A)gisdenseintheHilbertspaceH2(A;)(see[ 30 ,Theorem14]).AbrahamseshowedthatontheannulusthespectrumofaToeplitzoperatorneednotbeconnected. Theorem2.2.1(Abrahamse[ 1 ]). If=8>><>>:1on)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(0)]TJ /F6 11.955 Tf 9.3 0 Td[(1on)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(1,thenT0hasdiscretespectrumandhasasequenceofeigenvaluesapproaching1.ArayanaandClancygaveasucientconditionforaToeplitzoperatortohaveasequenceofeigenvalues. Theorem2.2.2(AryanaandClancy[ 6 ]). Letareal-valued2L1(\begiven.Ifesssupz2)]TJ /F13 5.978 Tf 4.82 -1.17 Td[(0(z)=m
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LetA(A)=fa:a2A(A)g.ToimproveupontheresultofArayanaandClancyweneedthefollowingpropositionwhichcharacterizesthemeasurethatannihilatesA(A)A(A). Proposition2.2.3. AmeasureabsolutelycontinuoustowhoseRadon-NikodymderivativewithrespecttoisinL1(\annihilatesA(A)A(A)ifandonlyifthereexistsac2Csuchthatd d=clogq)]TJ /F13 5.978 Tf 7.78 3.26 Td[(1 2.NowwecanstatethefollowingtheoremwhichgivesanecessaryandsucientconditionforTtohaveeigenvalueswhere2[0,1)and2L1(\isreal-valued. Theorem2.2.4. Fixareal-valued2L1(A).Let2[0,1)andanonzerog2H2(A;)begiven.IfTg=0,thengisouterandmoreoverthereexistsanonzeroc2Rsuchthat jgj2=clogq)]TJ /F9 7.97 Tf 6.58 0 Td[(1=2(2{1)almosteverywhere.IfgisouterinH2(A;)andEquation( 2{1 )holds,thenTg=0.Moreover,ifthereexistsac2Csuchthatclogjq)]TJ /F13 5.978 Tf 5.76 0 Td[(1=2j ispositive,logintegrable,andinL1(\,thenthereexistsan2[0,1)andanouterg2H2(A,)suchthatTg=0.Additionally,sinceistheindexofg,itiscongruentmodulo1to, 1 4logqZB0logjjd0)]TJ /F10 11.955 Tf 11.96 16.27 Td[(ZB1logjjd1.(2{2)Inparticular,thereexistsatmostonesuchthatThaseigenvalue0andthedimensionofthiseigenspaceisatmostone.WesaythatisaneigenvalueassociatedtorelativetoA(A)ifthereexistsan2[0,1)andnontrivialsolutiong2H2(A;)toTg=g. 17

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Corollary2.2.5. Letareal-valued2L1(\begiven.Ifesssupf(z):z2B0g=m<0
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Byreplacingnwith)]TJ /F3 11.955 Tf 9.3 0 Td[(ninthelastequationwecanseethatqnZ20d d(qeit)eintdt=q)]TJ /F12 7.97 Tf 6.58 0 Td[(nZ20d d(qeit)eintdt.Henceforalln2Z)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(qn)]TJ /F3 11.955 Tf 11.96 0 Td[(q)]TJ /F12 7.97 Tf 6.58 0 Td[(nZ20d d(qjeit)eintdt=0.Thatmeansthatforall06=m2Zandj=0,1Z20d d(qjeit)eimtdt=0.Sod dmustbeequaltoaconstant,a,almosteverywhereoneachboundarywitha=d d)]TJ /F13 5.978 Tf 4.82 -1.18 Td[(0=)]TJ /F9 7.97 Tf 16.24 4.71 Td[(d d)]TJ /F13 5.978 Tf 4.82 -1.17 Td[(1.Ifwechoosec2Rtobea logq)]TJ /F13 5.978 Tf 6.95 2.34 Td[(1 2,thend d=clogzq)]TJ /F13 5.978 Tf 7.79 3.25 Td[(1 2almosteverywhereon)]TJ /F1 11.955 Tf 6.77 0 Td[(.Nowletbeameasuresuchthatd d=clogzq)]TJ /F13 5.978 Tf 7.78 3.26 Td[(1 2almosteverywhereon)]TJ /F1 11.955 Tf 6.77 0 Td[(.Sincespanfzn, zn:n2ZgisdenseinA(A)A(A)itwillsucetoshowthatforn2ZZ)]TJ /F8 11.955 Tf 7.31 10.79 Td[(nd=Z)]TJ /F8 11.955 Tf 7.31 10.79 Td[(nclogq)]TJ /F13 5.978 Tf 7.78 3.26 Td[(1 2d=0andZ)]TJ ET q .478 w 144.16 -343.61 m 151.5 -343.61 l S Q BT /F8 11.955 Tf 144.16 -350.43 Td[(nd=Z)]TJ ET q .478 w 202.49 -343.61 m 209.82 -343.61 l S Q BT /F8 11.955 Tf 202.49 -350.43 Td[(nclogq)]TJ /F13 5.978 Tf 7.78 3.26 Td[(1 2d=0toprovethatannihilatesA(A)A(A).Ifn=0,thenZ)]TJ /F3 11.955 Tf 7.31 10.8 Td[(clogq)]TJ /F13 5.978 Tf 7.78 3.26 Td[(1 2d=Z)]TJ /F13 5.978 Tf 4.82 -1.17 Td[(0clogq)]TJ /F13 5.978 Tf 7.79 3.26 Td[(1 2d0+Z)]TJ /F13 5.978 Tf 4.82 -1.17 Td[(1clogq1 2d1=0.Ifn6=0,thenforj=0,1,Z)]TJ /F18 5.978 Tf 4.82 -1.41 Td[(jnclogq)]TJ /F13 5.978 Tf 7.78 3.25 Td[(1 2dj=clogqj)]TJ /F13 5.978 Tf 7.78 3.25 Td[(1 2Z)]TJ /F18 5.978 Tf 4.82 -1.41 Td[(jndj=0.AsimilarcomputationshowsthatR)]TJ ET q .478 w 180.98 -526.95 m 188.32 -526.95 l S Q BT /F8 11.955 Tf 180.98 -533.78 Td[(nclogq)]TJ /F13 5.978 Tf 7.78 3.26 Td[(1 2d=0. 2.2.2ProofofTheorem 2.2.4 Fixareal-valued2L1(A).Let2[0,1)andanonzerog2H2(A;)begiven.AdditionallysupposeTg=0.Usingthefactthat,ifa2A(A),thenag2H2(A;)itfollows 19

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that0=Tg,ag=Z)]TJ /F8 11.955 Tf 7.31 10.8 Td[(jgj2ad.Sincejgj2isreal-valuedZ)]TJ /F8 11.955 Tf 7.31 10.79 Td[(jgj2ad=0too.Thusjgj2annihilatesA(A)A(A).SobyProposition 2.2.3 thereexistssomec2Rsuchthatjgj2=clogq)]TJ /F13 5.978 Tf 7.78 3.25 Td[(1 2.Thenextobjectiveistoshowthatgisouter.Firstweneedthefollowingdenition.AfunctioninH2(A;)isinnerifjj=1on)]TJ /F1 11.955 Tf 6.77 0 Td[(.Sarasonin[ 30 ,Theorem7]provedaversionofinner-outerfactorizationfortheannulus:givenf2H2(A;),thereisa2[0,1)andaninnerfunction 2H2(A;)andouterfunctionF2H2(A;)]TJ /F8 11.955 Tf 11.95 0 Td[()suchthat f= F.(2{3)Letg= Fdenotetheinner-outerfactorizationofgasanH2(A;)functionasinEquation( 2{3 )andlet2[0,1)betheindexof .Sinceindex( )+index(F)=index(g),wehavethatF2H2.SimilarlywehavethatC=)]TJ /F14 7.97 Tf 6.59 0 Td[( 2H2(A;0)whichmeansthatitsrestrictionstoeachoftheboundarycomponents)]TJ /F9 7.97 Tf 6.78 -1.79 Td[(0and)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(1isrepresentableasaFourierserieswhosecoecientswewilldenotebybC0(n)andbC1(n)respectively.Moreoverby[ 30 ,Lemma1.1]weknowthatbC0(n)=q)]TJ /F12 7.97 Tf 6.58 0 Td[(nbC1(n).Sinceweshowedabovethatjgj2=clogq)]TJ /F13 5.978 Tf 7.78 3.26 Td[(1 2for 20

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somec2R,foranyn2Z 0=Tg,nF=Z)]TJ /F8 11.955 Tf 7.31 10.8 Td[(jgj2jj2)]TJ /F14 7.97 Tf 6.59 0 Td[( nd=Z)]TJ /F3 11.955 Tf 7.31 10.79 Td[(clog)]TJ 5.48 .48 Td[(q)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2jj2C nd=clog(q1=2)qn+2bC1(n))]TJ /F10 11.955 Tf 13.35 3.15 Td[(bC0(n)=clog(q)bC1(n))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(qn+2)]TJ /F3 11.955 Tf 11.95 0 Td[(q)]TJ /F12 7.97 Tf 6.58 0 Td[(n.(2{4)FromEquation( 2{4 )itfollowsthat,foreachn,eitherbC1(n)=0orn+=0.Since2[0,1)andbC1(m)6=0forsomem,itfollowsthatbC1(n)=0forn6=0and=0.Thus isaunitaryconstantandgisouter.Nextassumethatg2H2(A;)isouterandEquation( 2{1 )holds.ByProposition 2.2.3 ,Tg,ag=Z)]TJ /F8 11.955 Tf 7.31 10.79 Td[(jgj2 ad=Z)]TJ /F3 11.955 Tf 7.31 10.79 Td[(clogq)]TJ /F13 5.978 Tf 7.78 3.25 Td[(1 2 a=0,foreverya2A(D).Furthersincegisouterfagja2A(A)gisdenseinH2(A;).ThusTg=0whichprovesthesecondpart.Toprovethethirdpartofthetheoremnotethattheconditionthatclogq)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2 ispositive,inL1(\,andislogintegrableimpliesthatthereexistsanoutergsatasifyingEquation( 2{1 ).Therefore,wecanapplythepreviouspartofthetheoremtogetthedesiredconclusion.Toprovethefourthpartofthetheoremsinceistheindexofgby[ 30 ,Theorem6]weknowthat,moduloone,is)]TJ /F6 11.955 Tf 9.3 0 Td[(1 2logqZ)]TJ /F13 5.978 Tf 4.82 -1.18 Td[(0logjgjd0)]TJ /F10 11.955 Tf 11.96 16.27 Td[(Z)]TJ /F13 5.978 Tf 4.82 -1.18 Td[(1logjgjd1. 21

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Applyingthefactthatjgj=clogjq)]TJ /F13 5.978 Tf 5.75 0 Td[(1=2j 1=2theaboveexpressionsimpliesto1 4logqZ)]TJ /F13 5.978 Tf 4.82 -1.18 Td[(0logjjd0)]TJ /F10 11.955 Tf 11.96 16.27 Td[(Z)]TJ /F13 5.978 Tf 4.82 -1.17 Td[(1logjjd1.Finally,supposethatTg=0andalsoTh=0.Fromwhathasalreadybeenprovedgandhareouterandthereexistsnonzeroc,d2Rsuchthatjgj2=clogq)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2andjhj2=dlogq)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2on)]TJ /F1 11.955 Tf 6.77 0 Td[(.Sinceisalmosteverywherenonzerowehavethatjgj2=c djhj2andbecausegandhareouter,theyareequaluptoacomplexscalarmultiple,see[ 30 ,Theorem7.9]. 2.2.3ProofsoftheCorollaries 2.2.3.1ProofofCorollary 2.2.5 Observethatesssupf(z):z2)]TJ /F9 7.97 Tf 6.78 -1.79 Td[(0g=m<0
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2.2.3.2ProofofCorollary 2.2.6 Let2L1(T)berealvalued.Ifisaneigenvalueassociatedto,thenTheorem 2.3.3 impliesthatthereexistsac2Candag2H2(D)suchthat()]TJ /F8 11.955 Tf 11.95 0 Td[()jgj2=clogq)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2.Thereforewithoutlossofgeneralitywemayassumethatesssupf(z):z2)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(1g=mM=essinff(z):z2)]TJ /F9 7.97 Tf 6.78 -1.79 Td[(0g.ByCorollary 2.2.5 ifmMorM.Since>M=essinff(z):z2)]TJ /F9 7.97 Tf 6.78 -1.79 Td[(0gwehavethat(f(z))]TJ /F8 11.955 Tf 11.96 0 Td[(<0:z2)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(0g)>0.Ontheotherhand,ifisaneigenvalue,thenthereisanonzerod2R,an2[0,1),andanouterfunctionh2H2(A;)suchthat()]TJ /F8 11.955 Tf 12.28 0 Td[()jhj2=dlogq)]TJ /F9 7.97 Tf 6.58 0 Td[(1=2whichimpliesthateither()]TJ /F8 11.955 Tf 11.95 0 Td[()j)]TJ /F13 5.978 Tf 4.82 -1.18 Td[(1ispositivealmosteverywhereor()]TJ /F8 11.955 Tf 11.95 0 Td[()j)]TJ /F13 5.978 Tf 4.82 -1.18 Td[(0ispositivealmosteverywhere.Thisisacontradictionsinceweknowthat()]TJ /F8 11.955 Tf 11.05 0 Td[()isnegativeontheinnerboundaryoftheannulusandnotpositivealmosteverywhereontheouterboundary.ThisprovesCorollary 2.2.6 for>M.Theproofforthecase
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NoticethatasapproachesMon)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(1wehavethatR)]TJ /F13 5.978 Tf 4.82 -1.17 Td[(1logj)]TJ /F8 11.955 Tf 11.95 0 Td[(j<1since2L1(\andesssupf(z):z2)]TJ /F9 7.97 Tf 6.77 -1.8 Td[(1g
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Lemma2.3.1. LetanonzeroproperVCCzbegiven.IfgisouterandinH2(D;V),thenthesetfag:a2AgisdenseinH2(D;V).Let(z)=zandA=fa:a2Ag.ThefollowingpropositioncharacterizesthemeasureswhichannihilateAA. Proposition2.3.2. AmeasureabsolutelycontinuoustoswhoseRadon-NikodymderivativewithrespecttosisinL1(T)annihilatesAAifandonlyifthereexistsac2Csuchthatd ds=c+(c).ThefollowingtheoremisananalogofTheorem 2.2.4 andgivesnecessaryandsucientconditionfortheexistenceofeigenvaluesforToeplitzoperatorsonH2(D;V). Theorem2.3.3. Fixareal-valued2L1andletVCCzandnonzerog2H2(D;V)begiven.IfTVg=0,thengisouterandmoreoverthereisac2Csuchthat,onT, jgj2=c+(c).(2{6)Conversely,ifthereisac2C,gisouterinH2(D),andEquation( 2{6 )holds,thenthereexistsanonzeroproperVC+CzsuchthatTVg=0andVisdeterminedbythevaluesg(0)andg0(0).Inparticular,thereexistsatmostoneVsuchthatTVhaseigenvalue0andthedimensionofthiseigenspaceisatmostone.Byanalogywiththecaseoftheannulus,wesaythatisaneigenvalueassociatedtorelativetoAifthereexistsaVCCzandnontrivialsolutiontoTVg=g.Nowwecanstatethefollowinganalogsofcorollaries 2.2.5 ,and 2.2.6 .Butrstforz2Dandtreal,letH(z,t)=eit+z eit)]TJ /F3 11.955 Tf 11.96 0 Td[(z. 25

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Corollary2.3.4. Ifthereisac2Csuchthatesssupf(z):cz+(cz)<0g=m<00g,theneach2(m,M)isaneigenvalueassociatedtorelativetoA.Further,foreachsuchthereisanessentiallyuniqueouterfunctionfsuchthat()]TJ /F8 11.955 Tf 11.95 0 Td[()jfj2=c+(c).Additionally,lethc(z)=exp1 2Z)]TJ /F14 7.97 Tf 6.59 0 Td[(H(z,)logjc+c j1=2ds,g(z)=exp1 2Z)]TJ /F14 7.97 Tf 6.59 0 Td[(H(z,)logj)]TJ /F8 11.955 Tf 11.95 0 Td[(j)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2ds,andVbethenontrivialsubspaceofCCzorthogonaltohc(0)g(0)+(hc(0)g0(0)+h0c(0)g(0))z.Thenf2H2(D;V)andTVf=f.Further,sincehcandgareouterfunctionsneitherhc(0)norg(0)arezero.Moreover,M(resp.m)isaneigenvalueifandonlyifc+(c) )]TJ /F12 7.97 Tf 6.59 0 Td[(M(resp.c+(c) )]TJ /F12 7.97 Tf 6.58 0 Td[(m)isinL1(T)andislogintegrable.Additionallywegetthefollowingeasycorollary. Corollary2.3.5. ThesetofeigenvaluesassociatedtorelativetoAiseitherempty,apoint,oraninterval.ThereisnoanalogofCorollary 2.2.7 orofthemainresultof[ 6 ]fortheNeileparabola.InfactCorollary 2.3.4 impliesthatifeCCzisspannedby1,thennoToeplitzoperatoronH2(D;e)haseigenvalues.NotethataneasierwaytoseethatnoToeplitzoperatoronH2(D;e)haseigenvaluesistonotethatH2(D;e)=zH2.Moreover,forsimilarreasonsnoToeplitzoperatoronH2(D;V)haseigenvaluesifV=f0gorV=CCz.InfactthefollowingcorollarysaysthattherearemanynonzeroproperV,notjuste,suchthatTVhas 26

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noeigenvalues.Toshowthisweneedamapbetweenthegap(m,M)andthesetofspacesfH2(D;V):VC+Czisnonzeroandproperg.First,identifyeachnonzeroproperVwithanelementofthecomplexprojectiveline,P1(C),whichisX=C2nf0gmodulotheequivalencerelationvwifandonlyifthereisacomplexnumbersuchthatv=worw=v.Let:X!P1(C)denotethequotientmapping.ThespaceP1(C)canberealizedasaRiemannsurfacebythechartsj:C!P1(C)denedby0()=1Tand1()=1T.Indeed,thetransitionmappingsbetweenthesechartsare=1 and=1 .AmapF:R!P1(C)isdierentiableifthemaps)]TJ /F9 7.97 Tf 6.58 0 Td[(10Fand)]TJ /F9 7.97 Tf 6.59 0 Td[(11Faredierentiablewheredened.Finally,ifIisanintervalinRandg:I!Xistwicedierentiable,thensoisgandinthiscasetheHausdordimensionoftherangeofgisatmostoneandinthissensetherangeisarelativelysmallsubsetofP1(C).ForadiscussionofpropertiesoftheHausdordimensionsee[ 32 ]. Corollary2.3.6. Thefunction,,from(m,M)toP1(C)denedby:7!0B@0B@hc(0)g(0)h0c(0)g(0)+hc(0)g0(0)1CA1CAislocallyLipschitzwithrespecttoon(m,M).ThusinadditiontoV=spanf1gthereexistnonzeroproperVCCzsuchthatTVhasnoeigenvalues. 2.3.1ProofofLemma 2.3.1 Letf2H2Vbegiven.SincegisouterthereexistsasequenceoffunctionsfangH1(D)suchthatangconvergestofinH2(D).Letbn=an)]TJ /F3 11.955 Tf 12.23 0 Td[(a0n(0),soeachbn2A.Toshowthatbngalsoconvergestofitsucestoshowthata0n(0)convergestozero.TodothisnotethatforeachnonzeroproperVthereexistsapair(,)2C2notbothzerosuchthatifh2H2V,thenthereexistsa2Candq2H2(D)suchthath=+z+z2q.Sinceg(0)6=0thismeansthatf(0) g(0)g0(0)=f0(0). 27

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Becausean(0)g(0)convergestof(0)and(ang)0(0)convergestof0(0)wehavethatlimn!1an(0)=f(0) g(0)limn!1a0n(0)=f0(0))]TJ /F12 7.97 Tf 13.28 5.48 Td[(f(0) g(0)g0(0) g(0)=0Whichcompletestheproofthelemma. 2.3.2ProofofProposition 2.3.2 AssumethatannihilatesAA,thenforeachn2Znf1,)]TJ /F6 11.955 Tf 9.29 0 Td[(1gZTnd=ZTnd dsds=0.Henceforanyndierentfrom1thecorrespondingFouriercoecientofd dsiszero.Thusthereexistsc2CsuchthatalltheFouriercoecientsofd ds)]TJ /F3 11.955 Tf 10.95 0 Td[(c+(c)are0,whichimpliesthatd ds=c+(c)almosteverywhereonT.Nowletbeameasuresuchthatd ds=c+(c)almosteverywhereonT.Sincethespanofthesetfznjn2Znf1,)]TJ /F6 11.955 Tf 9.3 0 Td[(1ggisdenseinAAitwillsucetoshowthatforn2Znf1,)]TJ /F6 11.955 Tf 9.29 0 Td[(1gZTnd=ZTn(c+(c))ds=0toprovethatannihilatesAA.Butforn2Znf1,)]TJ /F6 11.955 Tf 9.29 0 Td[(1gZTn(c+(c))ds=ZTcn+1+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(cn)]TJ /F9 7.97 Tf 6.58 0 Td[(1ds=0,whichprovestheproposition. 2.3.3ProofofTheorem 2.3.3 Fixareal-valued2L1andletVCCzandnonzerog2H2(D;V)begiven.SupposeTVg=0.Usingthefactthat,ifa2A,thenag2H2(D;V)itfollowsthat0=TVg,ag=ZTjgj2 ads. 28

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Sincejgj2isreal-valueditisalsothecasethatZTjgj2ads=0.Thusjgj2annihilatesAA.ThusbyProposition 2.3.2 thereexistssomec2Csuchthatjgj2=c+(c).Thenextobjectiveistoshowgisouter.Tothisend,letg=Fdenotetheinner-outerfactorizationofgasanH2(D)function.Observethat,znF2H2(D;V)forintegersn2.Thus,forsuchn,0=Tvg,Fzn=ZTjgj2 znds=ZT(cz+(cz)) znds.Itfollows,writing=P1k=0kzk,thatcn)]TJ /F9 7.97 Tf 6.59 0 Td[(1+cn+1=0forn2.Inparticular,2k+1=()]TJ /F3 11.955 Tf 12.86 8.09 Td[(c c)k1fork1andlikewise,2k+2=()]TJ /F3 11.955 Tf 12.86 8.09 Td[(c c)k2.Because2H2(D)theselasttwoequationsimplythatk=0fork1;i.e.,isaunimodularconstantandthusgisouter,andtherstpartofthetheoremisestablished.Toprovetheconverse,supposethatgisouterandthereisac2CsuchthatEquation( 2{6 )holds.LetVbethenonzerosubspaceofCCzsuchthatg(0)+g0(0)z2V?.In 29

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particular,g2H2(D;V).ByLemma 2.3.1 wehavethat,foranya2A,TVg,ag=ZTjgj2 ads=ZT(c+(c))a=0andthusTVg=0.Finally,supposethatTVg=0andTWh=0.Fromwhathasalreadybeenprovedweknowthatgandhareouterandthereexistsc,d2Csuchthatjgj2=c+(c)andjhj2=d+(d)onalmosteverywhereonT.Itfollowsthatispositivealmosteverywherebothwherec+(c)and(d)+(d)arepositive.Hencec=tdforsomepositiverealnumbert.Butthen,tjgj=jhjandbecausegandhareouter,theyareequaluptoa(complex)scalarmultiple. 2.3.4ProofsoftheCorollaries 2.3.4.1ProofofCorollary 2.3.4 Observethattheexistenceofac2Csuchthatesssupf(z):cz+(cz)<0g=m<00g,implies,form<(2{7) 30

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becauseRTlogjc+(c)jds>,isessentiallyboundedandsgn(c+(c))=sgn().Hencethereisanouterfunctionf2H2(D)suchthat()]TJ /F8 11.955 Tf 11.96 0 Td[()jfj2=c+(c).Toprovethesecondpartuse( 2{6 )andthefactthatfisoutertoseethatf(z)=exp ZTH(z,)logc+(c) )]TJ /F8 11.955 Tf 11.96 0 Td[(1=2ds!=expZTH(z,)logjc+(c)j1=2dsexpZTH(z,)logj)]TJ /F8 11.955 Tf 11.95 0 Td[(j)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2ds=hc(z)g(z).Thusf(0)=hc(0)g(0)andf0(0)=hc(0)g0(0)+h0c(0)g(0)soifVisthenontrivialsubspaceofCCzorthogonaltohc(0)g(0)+(hc(0)g0(0)+h0c(0)g(0))z,thenf2H2(D;V).FromTheorem 2.3.3 weknowthatTVf=f.Thecase=M(resp.=m)aresimilar,withtheonlyissuebeingthatahypothesisisneededtoguaranteethat ,asdenedabove,isintegrableandlogintegrable. 2.3.4.2ProofofCorollary 2.3.5 Let2L1(T)bereal-valued.Ifisaneigenvalueassociatedto,thenTheorem 2.3.3 impliesthatthereexistsac2Candag2H2(D)suchthat()]TJ /F8 11.955 Tf 11.95 0 Td[()jgj2=c+(c).Nowletesssupf(z):cz+(cz)<0,z2Tg=mM=essinff(z):cz+(cz)>0,z2Tg.IfmMor
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thenisnotaneigenvalue.Accordinglysuppose>M.InthiscasethemeasureofthesetS=fz2T:(z)>gislessthan 2sincesgn()]TJ /F3 11.955 Tf 11.51 0 Td[(M)=c+(c)almosteverywhereandM=essinff(z):cz+cz>0,z2Tg.Ontheotherhand,sinceisaneigenvaluethereisanon-zerodandouterfunctionh2H2suchthat()]TJ /F8 11.955 Tf 11.96 0 Td[()jhj2=d+(d)ButthenthemeasureofthesetSis 2,acontradiction.Whichprovesthecorollarywhen>M.Theproofofthecase0suchthat)]TJ /F8 11.955 Tf 12.27 0 Td[(isessentiallyboundedaboveandawayfromzeroforj)]TJ /F8 11.955 Tf 12.28 0 Td[(0j<.Itfollowsthatfor2(0)]TJ /F8 11.955 Tf 12.04 0 Td[(,0+),thefunctionsH(0,t)log(j(t))]TJ /F8 11.955 Tf 11.95 0 Td[(j)andH0(0)log(j(t))]TJ /F8 11.955 Tf 11.96 0 Td[(j)aswellas((t))]TJ /F8 11.955 Tf 12.27 0 Td[())]TJ /F9 7.97 Tf 6.58 0 Td[(1areallboundedaboveandbelow.Thusastandardapplicationofthedominatedconvergencetheoremestablishesthedesireddierentiability.Asimilarargumentshowsthatinfactbothfunctionsareinnitelydierentiable.Since1istwicedierentiableon(m,M)itislocallyLipschitz.Because(m,M)canbewrittenasacountableunionofintervalswith1LipschitzoneachintervaltheHausdordimensionof1((m,M)) 32

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isatmost1.So((m,M))cannotbeallofP1(C)nf((01)Tgsince1isinjectiveonitsrange.LetL=fVjVisanonzeropropersubspaceofC+Czg.FinallyforeachnonzeroproperVchooseaf2H2Vwithf(0)andf0(0)notbothzeroandlet:L!P1(C)bedenedbymap:V7!((f(0)f0(0))T).ThemapisabijectionbetweenLandP1(C),thusif(V)62((m,M)),thenby 2.3.5 and 2.3.4 wehavethatTVhasnoeigenvalues.Additionallyifm(resp.M)isaneigenvalueassociatedtorelativetoAthenweneedtoaddtheconditionthat(V)6=(m)(resp.(V)6=(M))fortheaboveconclusiontohold. 33

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CHAPTER3GENERALTHEOREM 3.1IntroductionLetRbeamultiplyconnecteddomainwhoseboundaryisn+1disjointanalyticsimpleclosedcurves.LetU0betheunboundedcomponentofCn R,and,for1jn,letUjeachbeadierentboundedcomponentofCn R.Let)-355(=@Rand,for1jn,let)]TJ /F12 7.97 Tf 6.77 -1.8 Td[(j=@Uj.Notethat)-277(=Snj=0)]TJ /F12 7.97 Tf 6.77 -1.79 Td[(j.TodenetheHardy-HilbertspaceonRletsbearclengthmeasureon)]TJ /F1 11.955 Tf 6.77 0 Td[(,andletL2(\=f:)]TJ /F2 11.955 Tf 16.9 0 Td[(!C:fismeasurableandkfk2=Z)]TJ /F2 11.955 Tf 7.31 10.8 Td[(jfj2ds<1L1(\=ff:)]TJ /F2 11.955 Tf 16.91 0 Td[(!C:fismeasurableandkfk1=esssupjfj<1g.Further,letA(R)=f: R!C:fiscontinuousin RandanalyticinRandH2(R)= A(R)L2(\.WecandeneacollectionofspacessimilartoH2(R)calledtheSarason-Hardy-Hilbertspaces. 3.2Sarason-Hardy-HilbertSpacesonMultiplyConnectedDomainsFollowing[ 2 ]letDdenotetheunitcircleandletbeacoveringmapfromDtoRthatrealizesDastheuniversalcoveringspaceofR.LetGbethegroupofhomeomorphisms,A,ofthedisksuchthatA=.NotethegroupGisisomorphictothefundamentalgroupofRandthattheelementsofGarelinearfractionaltransformationsthatalsox@D.WesayafunctionfonDisG-automorphiciffA=fforallA2G.FurtheritiseasytoseethatifgisafunctiononR,thengisG-automorphic.AdditionallyitcanbeshownthatinducesabijectionfromtheorbitsofpointsofDunderactionbyGtoR,soalloftheG-automorphicfunctionsareoftheformgforsomefunctiongonR. 34

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NowletHbeaseparableHilbertspace.Following[ 33 ]letL2H(T)bethefunctionsfromTtoHwhicharemeasurableandwhichsatisfykfk2L2H(T)=ZTkfk2Hds<1.WecandeneH2H(D)byH2H(D)=(f=1Xj=0ajzj:fajg2Hand1Xj=1kajk2<1).LiketheclassicalHardy-HilbertspaceontheunitdiskNagy-Foias[ 33 ]showedthefunctionsinH2H(D)aredenedinsidethediskandhaveboundaryvalueswhichareinL2H(T).Moreover,theidenticationoff2H2H(D)withitsboundaryvaluesisanisometricisomorphismbetweenH2H(D)andthesubspaceofL2H(T)offunctionswithzeronegativeFouriercoecients.ForamultiplyconnecteddomainRwhichhasn+1smoothdisjointboundarycomponentsletL2H(\bethemeasurablefunctionsfrom)]TJ /F1 11.955 Tf 10.68 0 Td[(toHwhichsatisfykfk2L2H(R)=Z)]TJ /F2 11.955 Tf 5.32 10.79 Td[(kfk2Hds<1anddeneH2H(R)byH2H(R)= A(R)HL2H(\.IfH=C,thenL2H(\=L2(\andH2H(R)=H2(R).ItwasshownbyAbrahamseandDouglas[ 2 ]thatthefunctionsinH2H(R)aredenedinsideRandhaveboundaryvaluesalmosteverywhereon)]TJ /F1 11.955 Tf 10.68 0 Td[(whichareinL2H(\.Similartothecaseforthedisc,theidenticationoff2H2H(R)withitsboundaryvaluesisanisometricisomorphismbetweenH2H(R)andasubspaceofL2H(\.Next,letU(H)bethegroupofunitaryoperatorsonHand=(R,H)=Hom(G,U(H))bethegroupofrepresentationsofGintoU(H).TheSarason-Hardy-HilbertspacesonR,isthesetofspacesfH2H(R;):2gwhereH2H(R;)=f2H2H(D):f(A(z))=(A)f(z)forallA2G. 35

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Notethatforf,g2H2(R;),A2G,andz2Dhf(A(z)),g(A(z))iH=h(A)f(z),(A)g(z)iH=hf(z),g(z)iHsohf(z),g(z)iHandkf(z)kHareG-automorphicfunctions.TodeneanormonH2H(R;)weneedthenotionofaharmonicmeasureonR.LetG(,z)betheGreen'sfunctionforRwithpoleatz,andlet@ @nbethederivativewithrespecttotheoutwardnormal.IffisharmonicinRandcontinuouson R,thenby[ 16 ],foranyz2Rf(z)=)]TJ /F6 11.955 Tf 14.03 8.09 Td[(1 2Z)]TJ /F3 11.955 Tf 7.31 10.79 Td[(f@G @n(;z)ds.Themeasured(z)=)]TJ /F9 7.97 Tf 13.07 4.71 Td[(1 2@G @n(;z)dsiscalledharmonicmeasureatz.AdditionallyifweletmdenotearclengthmeasureonTthenforanyfharmoniconRweknowthatZTfdm=f((0)).Infact,mpushesforwardto((0)),harmonicmeasureat(0),[ 16 ].Letds dbetheRadon-Nikodymderivativeofs,arclengthmeasureonR,withrespectto((0)).Weknowthatds disboundedandabsolutelycontinuous.Moreover,givenaG-automorphicfunctionf2L1(T)wehavethatf=FforsomeFon)]TJ /F1 11.955 Tf 10.67 0 Td[(andZTfds ddm=ZTFds ddm=Z)]TJ /F3 11.955 Tf 7.31 10.79 Td[(Fds dd((0))=Z)]TJ /F3 11.955 Tf 7.31 10.79 Td[(Fds.For2[0,1)andf,g2H2H(R;)denethenormonH2H(R;)bykfk22=ZTkf()k2Hds ddm.andtheinnerproductbyhf,gi=ZThf(),g()iHds ddm. 36

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Since,kf(z)kHandhf(z),g(z)iHareG-automorphicthereexistsfunctionNfandPf,gsuchthatkf(z)kH=Nf(z)andhf(z),g(z)i=Pf,g(z).Thisimpliesthatkfk22=Z)]TJ /F3 11.955 Tf 7.31 10.79 Td[(Nfdsandhf,gi=Z)]TJ /F3 11.955 Tf 7.32 10.79 Td[(Pf,gds.Fromnowonwewillwritekf()kHandhf(),g()iHevenwhenwemeanNf()andPf,g().Lastly,aswiththeHardy-HilbertspacesoverD,ifH=CwewillwriteH2(R;).TheSarason-Hardy-HilbertspacesareageneralizationofthestandardHardy-Hilbertspaces.Indeed,if=idH,thenf(A(Z))=f(z)forallA2GsofisG-automorphicwhichwasshowntoimplythatf=gforsomeg2H2H(R).Hence,thereisabijectionbetweenH2H(R;idH)andH2H(R).MoreoverifRisanannulus,A,andH=ClikeinSection 2.2 ,thenG=ZandU(C)=T.ThusHom(G,U(C))=[0,1),where2[0,1)isidentiedwithexp(2i)2T.SothedenitionsH2(A;)giveninthissectionandthedenitionsgiveninSection 2.2 agree.Wecallafunctionf:D!HmodulusautomorphiconRifkfkHisG-automorphic.Given2andf2H2H(R;),byconstructionweknowthatforanyA2Gandz2Dkf(A(z))kH=k(A)f(z)kH=kf(z)kHwhichimpliesthatthefunctionsinH2H(R;)aremodulusautomorphic.IfH=C,andRisdomainofconnectivityn+1,thenGisisomorphictothefreegrouponnvariablesandisisomorphicto[0,1)nunderadditionmodulo1.Theisomorphismidenties=idCwith~0.LikewiththeannulusthefunctionsinH2(R;)haveperiodsabouteachboundarybuttondtheperiodsweneedtodevelopthenotionofanouterfunction.Iffislocallyanalytic,modulusautomorphic,jfj2majorizedbyanintegrableharmonicfunction,thenjfjhasboundaryvaluesalmosteverywhereon)]TJ /F1 11.955 Tf 6.78 0 Td[(.Wewillcallsuchanfouterifforsome 37

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z2Rlogjf(z)j=)]TJ /F6 11.955 Tf 14.03 8.08 Td[(1 2Z)]TJ /F6 11.955 Tf 7.31 10.8 Td[(logjfj@G @n(;z)ds.Aneasyadaptationof[ 16 ,Proposition5.4.6]showsthatfor2iff2H2(R;)isouter,thenA(R)fisdenseinH2(R;).Therefore,thisnotionofanouterfunctioninH2(R;)isageneralizationofouterfunctionsinH2(D).Next,lethjbethefunctionharmonicinRsuchthat hj=8>><>>:1on)]TJ /F12 7.97 Tf 6.77 -1.79 Td[(j0on)]TJ /F12 7.97 Tf 6.77 -1.79 Td[(kfork6=j.(3{1)Iff2H2(R;)isouter,thenisidentiedwith(j)nj=1=1 2Z)]TJ /F6 11.955 Tf 7.31 10.79 Td[(logjfj@hj @nds(mod1)nj=1.Conversely,iffislocallyanalytic,modulusautomorphic,jfj2ismajorizedbyanintegrableharmonicfunction,thenthereexistsan2suchthatf2H2(R;).Iff2H2(R;)isalsoouter,thenwecanusetheequationabovetoidentify2. 3.3ExistenceofEigenvaluesTheoremLetAA(R)beaunitalsubalgebra,andletA=fa:a2Ag.WewillcallD=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(L2(\(A_A)thedefectofA.WewillsayAhasnitedimensionaldefectifDisanitedimensionalsubspaceofL2(\.FromnowonwewillassumethatDL1(\andthatitisanitedimensionalsubspaceofL2(\.Notethatspan@hj @n:1jnDwherethefunctionshjweredenedin( 3{1 ).Wewillcallameasurablecomplexvaluedfunctionfon)]TJ /F4 11.955 Tf 10.68 0 Td[(logintegrableifZ)]TJ /F6 11.955 Tf 7.31 10.8 Td[(logjfjds<1.Lastly,wewillassumethatA_D_Aisweak-*denseinL1(\,i.e.,foranyf2L1(\thereexistsasequenceoffunctionffng1j=0A_D_Asuchthatfnconvergesboundedlypointwisealmosteverywheretof.SeeFisher[ 16 ]foraproofthatA_D_Aisweak-*denseinL1(\ 38

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inthecaseoftheAnnulus,infactheprovesitformultiplyconnecteddomains.ThefactthatifAistheNeilealgebra,thenA_D_Aisweak-*denseinL1(T)followsfromFourierseries. Theorem3.3.1. Letarealvalued2L1(\,2,anAinvariantNH2H(R;),andag2Nbegiven.IfTNg=0,thenthereexistsa2<(D)suchthat kgk2=(3{2)almosteverywhere.Converselysupposethat12A,h2L1(\ispositiveandlogintegrable,and2<(D).Ifh=,thenthereexistsan2,anAinvariantNH2H(R;),andg2Nsuchthatkgk2=handTNg=0.Infact,givenan2andg2H2H(R;)suchthatkgk2=thereexistsanAinvariantNgH2H(R;)suchthatTNgg=0.Additionally,givenanyKHthereexistsanAinvariantNKH2H(R;)suchthatdimker(TNK)dim(K).WewillsaythatisaneigenvalueassociatedtorelativetoAifthereexistsan2andsomeAinvariantNH2H(R;)suchthatTNf=f.ToshowthattherecanbemanyeigenvaluesrelativetoAwewillneedtheconceptofagapintherangeof.Letbethedistributionfunctionof,i.e.,(x)=m(ft:(t)xg).Wewillsaythathasagaparoundaifessinf()
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Corollary3.3.2. Ifhasagaparound0,andtheconstantfunction1isinA,thenthereexistsalogintegrable2<(D)suchthatsgn()=sgn()almosteverywhereifandonlyifthereexistsan2,anAinvariantNH2H(R;),andanonzerofunctiong2NsuchthatTNg=0.LetF=f2<(D):sgn()=sgn()almosteverywhereandislogintegrableg. Corollary3.3.3. Ifhasagaparound0,O0,andtheconstantfunction1isinA,thenforeach2O0and2Fthereexistsan2,anAinvariantNH2H(R;)andanonzerofunctiong,2Nsuchthat()]TJ /F8 11.955 Tf 11.95 0 Td[()kg,k2=andhenceTNg,=g,.WewilldiscusssomeimplicationsandusesofthiscorollaryinSection 3.4 3.3.1ProofofTheorem 3.3.1 ToprovetherstpartofTheorem 3.3.1 xarealvalued2L1(\,2,NH2H(R;)andg2NsuchthatTNg=0.Alsoletc=dim(D)andfkgck=1DbeaorthonormalbasisforD.SinceTNg=0wehavethatforanya2A0=TNg,ag=hg,agi=Z)]TJ /F8 11.955 Tf 7.32 10.79 Td[(kgk2 ads.Moreoversinceisrealvaluedwealsoknow0=Z)]TJ /F8 11.955 Tf 7.31 10.8 Td[(kgk2ads. 40

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Nowletdk=Z)]TJ /F8 11.955 Tf 7.31 10.8 Td[(kgk2 kds.Becausek2L1(\andkgk22L1(\weknoweachdkisnite.Therefore,sinceDisnitedimensionalwehavethat=Pck=1dkk2L1(\.Soforanya2AwehavethatZ)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(kgk2)]TJ /F6 11.955 Tf 11.96 0 Td[( ads=Z)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(kgk2)]TJ /F6 11.955 Tf 11.95 0 Td[(ads=0,sincekgk2and2DannihilateAandA.Additionallysincethetheprojectionofkgk2ontoDwehavethatforanya2DZ)]TJ /F10 11.955 Tf 7.31 20.47 Td[()]TJ /F8 11.955 Tf 5.48 -9.68 Td[(kgk2)]TJ /F6 11.955 Tf 11.96 0 Td[(ads=0.BecauseAADisboundedlypointwisedenseinL1(\,byassumption,and2L1(\weknowthatkgk2=almosteverywhere.Since2Dandisrealvaluedwehavethat2<(D)whichprovestherstpartofthetheorem.Toprovethesecondpartofthetheoremassumethatthereexistsah2L1(\and2<(D)suchthath=.SincewecansolvetheDirichletproblemonRandhislogintegrablewecanndareal-valuedharmonicfunction,g2L1(\,suchthattheboundaryvaluesofgequal1 2log(h)almosteverywhere.LetgbetheharmonicconjugateofginDandletf=exp(g+ig).So,jfj2=exp(2g)andbecauseandexponentiationarecontinuoustheboundaryvaluesofjfj2areequaltoexp(log(h))=halmosteverywhere.Sofismodulusautomorphicandafteridentifyingjfj2withthefunctionitinduceson)]TJ /F1 11.955 Tf 10.68 0 Td[(itsatisesjfj2=.Additionallynotethatlogjfj=gandsotheboundaryvaluesoflogjfj=1 2log(h)almosteverywhere.SincegisharmonicinRwehavethatlogjf(z)j=g(z)=)]TJ /F6 11.955 Tf 14.03 8.08 Td[(1 2Z)]TJ /F6 11.955 Tf 8.51 18.88 Td[(1 2logjhj@G @n(;z)ds=)]TJ /F6 11.955 Tf 14.03 8.08 Td[(1 2Z)]TJ /F6 11.955 Tf 7.31 10.8 Td[(logjfj@G @n(;z)ds.Sofisouter.Lastly,sincefismodulusautomorphicisinaSarason-Hardy-Hilbertspacei.e.,thereexistsan2(R,H)suchthatf2H2(R;). 41

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ToconstructtherequiredAinvariantspaceandfunctiong,letKHbegiven,chooseav2Ksuchthatkvk=1andletg(z)=f(z)v.Sincekgk2(z)=jfj2(z)=h(z)2L1(\itfollowsthatg2L2H(\.TondtheSarason-Hardy-HilbertspacewhichcontainsgnotethatbecauseeachelementofU(C)isleftmultiplicationbyaunitaryelementofCeach2(R;C)canbeidentiedwithanb2(R;H).SoforanyA2Gwehavethatg(Az)=f(Az)v=((A)f)v=b(A)(fv)=b(A)gthereforeg2H2H(R;b).FinallyletN=_a2A[k2Kafk.Wecanchoosef2H2H(R;b)tobeouter,andN=H2H(R;b).Sinceforeachk2H,thefunctionfkisinH2H(R;b),H2H(R;b)isinvariantundermultiplicationbyfunctionsfromH1(R)[ 2 ],andtheconstantfunction1isinA,itfollowsthatNH2H(R;b)isanAinvariantsubspacewhichcontainsg.ToseethatTNg=0itsucesproveTNg,afk=0fora2AsincethespanfunctionsofthatformisdenseN.Thusbecause2D=L2(\(AA)wehavethatTNg,afk=Z)]TJ /F8 11.955 Tf 7.31 10.79 Td[(jfj2 ahv,kids=hv,kiZ)]TJ /F6 11.955 Tf 7.31 10.8 Td[( ads=0whichprovesthenalpartofTheorem 3.3.1 .NotethatifA=A(R),thenD=spanf@hj @n:j2f1,...,ngg. 42

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3.3.2ProofofCorollaries 3.3.2.1ProofofCorollary 3.3.2 .Assumethathasagaparound0,and12A.Furtherassumethatthereexistsalogintegrable2<(D)suchthatsgn()=sgn()almosteverywhere.Sincehasagaparound0and2L1(\wehavethath= 2L1(\Becausesgn()=sgn()wehavethath0almosteverywhere.Also,sincehasagaparound0,andislogintegrableweknowthathislogintegrable.SobyTheorem 3.3.1 thereexistsan2,anAinvariantNH2H(R;),andafunctiong2NsuchthatTNg=0.Nowassumethereexistsan2,anAinvariantNH2H(R;),andafunctiong2NsuchthatTNg=0,thenbyTheorem 3.3.1 thereexitsa2<(D)suchthatkgk2=butkgk20sosgn()=sgn()almosteverywhere.Finally,sincebothkgk2andarelogintegrablemustalsobelogintegrable. 3.3.2.2ProofofCorollary 3.3.3 .Assumethathasagaparound0,12A.Let2O0and2Fbegiven.Notethat)]TJ /F8 11.955 Tf 12.49 0 Td[(hasagaparoundzero,theconstantfunction1isinAandbythedenitionofFweknowthatislogintegrableandsgn()=sgn().Further,because2O0sgn()=sgn()]TJ /F8 11.955 Tf 11.95 0 Td[().NowapplyCorollary 3.3.2 togetan2,anAinvariantNH2H(R;),andafunctiong,2Nsuchthat()]TJ /F8 11.955 Tf 11.96 0 Td[()kg,k=andTNg,=g,. 3.4ModelswithEigenvaluesFortopologicalspaceTandSTletdimHSdenotetheHausdordimensionofSandforavectorspaceVletdimRVdenotedimensionofVasarealvectorspace.AsetWV 43

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iscalledaconeifforalla2R+andw2W,awisinW.Ifweaddtheconditionthatforallv,w2Wv+wisinW,thenWiscalledaconvexcone.Let2L1(\berealvaluedandhaveagaparound0,O.Further,letF=f 2<(D):sgn()=sgn( )a.e.andislogintegrableg.NoteCorollary 3.3.2 andCorollary 3.3.3 implythat2O0isaneigenvalueassociatedtorelativetoAifandonlyifFisnonempty.IfH=C,2O0,and2F,thenbyCorollary 3.3.3 thereexistsan2,anAinvariantNH2(R;)andg,2Nsuchthat()]TJ /F8 11.955 Tf 12.37 0 Td[()jg,j=andTNg,=g,.The2,AinvariantsubspaceNH2(R;),andfunctiong,2Narenotunique.Butifwecanchoose2,NH2(R;)andg,2Ninauniquefashion,saybychoosingg,tobetheessentiallyuniqueouterfunctionsuchthat()]TJ /F8 11.955 Tf 12.46 0 Td[()jg,j2=,2suchthatg,2H2(R;)andN=_a2Aag,,thenCorollary 3.3.3 inducesamape:O0F!fNH2(R;):2andNisAinvariantgdenedbye(,)=N,whereg,2N,.Asanexample,ifR=AandA=A(A)likeinSection 2.2 ,thenchoosingg,tobeouterforcesN,=H2(A;)forsome2[0,1).IfinsteadweletR=DandAbetheNeilealgebralikeinSection 2.3 ,thenchoosingg,tobeouterforcesN,=H2(D;V)forV=(g(0)+g0(0)z)?.ThesetFisclearlyaconeandifitisaconvexconeandweletFdenotetheinteriorofFwhenconsideredasasubsetofspanRFwiththerelativetopology,thenFcanbeidentiedwithaopensubsetofRmwherem=dimR(spanRF)dimR(D).Moreover,if2F,c>0,andg,2Nandh,c2Maresolutionsto()]TJ /F8 11.955 Tf 12.68 0 Td[()jg,j2=and()]TJ /F8 11.955 Tf 12.13 0 Td[()jh,j2=c,thenp cg,=h,c.Soifweletbetheequivalencerelationwhichidentiestwofunctionsifoneisapositivescalarmultipleoftheother,thenletting2Oand[]2P+F=F=themapdenedby(,[])=e(,)=N,iswelldened.Infact,ifFisnonempty,thenthespaceP+Fisalsoadierentiablemanifold. 44

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Toseethis,rstobservethatFisanopensubsetofspanRFwhichisavectorspaceofdimensionm,soFishomeomorphictoaopensubsetofRm.Indeed,sinceP+Rmisasmoothmanifold,P+Fisasmoothsubmanifold.Indeed,for1jmletUj=[a]=[a1,a2,...,am]2P+Rm:aj>0andU)]TJ /F12 7.97 Tf 6.59 0 Td[(j=[a]=[a1,a2,...,am]2P+Rm:aj<0.Additionallydenek:Uk!Rm)]TJ /F9 7.97 Tf 6.58 .01 Td[(1and)]TJ /F12 7.97 Tf 6.59 0 Td[(k:U)]TJ /F12 7.97 Tf 6.59 0 Td[(k!Rm)]TJ /F9 7.97 Tf 6.58 .01 Td[(1byk([a])=a1 ak,...,ak)]TJ /F9 7.97 Tf 6.59 0 Td[(1 ak,ak+1 ak,...,an akand)]TJ /F12 7.97 Tf 6.59 0 Td[(k([a])=a1 ak,...,ak)]TJ /F9 7.97 Tf 6.59 0 Td[(1 ak,ak+1 ak,...,an ak.TheproofthatthesechartsmakeP+Rmintoasmoothmanifoldaresimilartotheproofthattheprojectivespacesaresmoothmanifolds.NotethatdimH(P+F)=dimR(spanRF))]TJ /F6 11.955 Tf 12.42 0 Td[(1.Additionally,wecouldalsoconsiderFasasubmanifoldofthem)]TJ /F6 11.955 Tf 11.76 0 Td[(1spherebutthechartsforP+FreectthevectorspacestructureofspanRF.Ifwecanequip(P+FO)withaLipschitzmanifoldstructurethenwecansaymoreaboutthesizeof(P+FO). Proposition3.4.1. If(P+FO)isaLipschitzmanifoldandislocallyLipschitz,thendimH((P+FO))dimHFdimR<(D).Also,ifweconsiderasafunctionontoitsimageand)]TJ /F9 7.97 Tf 6.58 0 Td[(1existsandisLipschitzinaneighborhoodofapointthendimH((P+FO))dimHF Proof. LipschitzmapspreserveorreducetheHausdordimensionofthedomainandHausdordimensionispreservedundercountableunions[ 32 ].So,ifthemapislocallyLipschitz,thenbecauseP+FOis-compactwehavethatdimH)]TJ /F6 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 5.48 -9.68 Td[(P+FOdimH)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(P+FO=dimH(F). 45

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CHAPTER4NOVELEXAMPLES 4.1PreviousWorkToeplitzoperatorsonn-holeddomainswererststudiedbyAbrahamsein[ 1 ].AryanaandClancystudiedself-adjointToeplitzoperatorsonasymmetric2-holeddomainandwereabletondresultssimilartoCorollary 2.2.7 [ 6 ]. 4.2ToeplitzOperatorsonaMultiplyConnectedDomainLetRbeamultiplyconnecteddomainofconnectivityn+1.FollowingthenotationofChapter 3 ,let)-282(=@R,theboundaryoftheunboundedcomponentofCnRbedenotedby)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(0,andtheboundariesoftheboundedcomponentsbedenotedby)]TJ /F12 7.97 Tf 6.78 -1.79 Td[(jfor1jn.Usedstodenotearclengthmeasureon)]TJ /F1 11.955 Tf 6.77 0 Td[(,anddenehiasin( 3{1 ).Lastly,letA=A(R)andH=C.Observethatif@f @nisthederivativeoffwithrespecttotheoutwardnormal,thenby[ 16 ]weknowthat@hj @nisstrictlypositiveonthejthboundaryandstrictlynegativeontheothersandweknowthat@hj @nnj=1isalinearlyindependentsetthatspansD,infacttheyalsospan<(D).In[ 2 ]AbrahamseandDouglasshowedthattheSarason-Hardy-HilbertspacesofRpermitthefollowingversionofBeurling'sTheorem. Theorem4.2.1(AbrahamseandDouglas). Let2=Hom(G,U(C))andNbeanonzerosubspaceofH2(R;).IfNisanAinvariant,thenthereexists,2andaninnerfunction2H2(R;)suchthatN=H2(R;)=f:f2H2(R;).Thisallowsustheprovethefollowinglemma. Lemma4.2.2. Let2,anAinvariantNH2(R;),andg2Nbegiven.IfTNg=0,thenthereexistsa2andanouterfunctioneg2H2(R;)suchthatTH2(R;)eg=0. 46

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SoweonlyneedtoconsiderToeplitzoperatorsforwhichN=H2(R;)forsome2andwhichhaveanoutereigenfunction.NotethatasidefrominthecasesdiscussedinChapter 2 wedonotknowifeigenfunctionsmustbeouter.Givenafunction2L1(\,recallthatthegaparounda,Oa,isthelargestconnectedopensetcontainingaonwhichthedistributionofisconstant.Additionally,recallthatF=f2<(D):sgn()=sgn(),andislogintegrableg. Lemma4.2.3. If2L1(\isrealvaluedandhasagaparoundzero,and2F,thenR)]TJ /F6 11.955 Tf 7.31 4.15 Td[(logjj@hj @ndsisniteforeachj2f1,...,ng.InSection 3.2 wenotedthatthereisanisomorphismbetween(R,C)and[0,1)nunderadditionmodulo1.Infact,byadaptingtheworkofFisherin[ 16 ]onfunctionswithperiodswecanrealizethisisomorphismbyidentifying2and(j(mod1))nj=1whenthefunctionsinH2(R;)haveperiods(j(mod1))nj=1.Equip[0,1)nwiththequotienttopologyandgivethetopologyinducedbytheisomorphismbetweenitand[0,1)n. Lemma4.2.4. Suppose2L1(\isrealvaluedandhasagaparoundzero,O0.Foreach2Fand2O0thereexistsan(,)2andanouterg,2H2(R;(,))suchthatTH2(R;)g,=0.Moreover,the(,)2isdeterminedby (j(,))nj=1=1 4Z)]TJ /F6 11.955 Tf 7.31 10.63 Td[((logjj)]TJ /F6 11.955 Tf 17.94 0 Td[(logj)]TJ /F8 11.955 Tf 11.95 0 Td[(j)@hj @ndsnj=1(4{1)andtheisomorphismbetweenand[0,1)n.Whenthedependenceof(,)and(j(,))nj=1onandisunimportantwewillsuppressthefunctionnotation.Letmbeontheedgeofthegaparoundzeroand Jm=j2f0,...,ngjm2essrange(j)]TJ /F18 5.978 Tf 4.82 -1.4 Td[(j).(4{2) 47

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Fork2f0,...,ngand2O0letLk,m()=1 4Xj2JmZ)]TJ /F18 5.978 Tf 4.82 -1.41 Td[(jlogj)]TJ /F8 11.955 Tf 11.96 0 Td[(j@hk @nds.Notethatifk=2Jm,thenlim!mLk,m()=1ifandonlyifR)]TJ /F6 11.955 Tf 7.31 4.15 Td[(logj)]TJ /F3 11.955 Tf 11.95 0 Td[(mjds=. Proposition4.2.5. Let2L1(\berealvaluedwithagaparoundzero,O0,and2Fbegiven.IfmisontheedgeofO0,Jmischosenasin( 4{2 ),andk2f1,...,ngischosensuchthatlim!mLk,m()=,thenforanyxed2Flim!mk(,)+Lk,m()existsandisnite.Theconditionthatlim!mLk,m()=,forsomek2f1,...,ngholdsforalargeclassoffunctions. Lemma4.2.6. Let2L1(\berealvalued,haveconstantsignoneachboundarycomponent,andhaveagaparoundzero,O0,withmontheedgeofthegap.If Z)]TJ /F6 11.955 Tf 7.31 10.79 Td[(logj)]TJ /F3 11.955 Tf 11.95 0 Td[(mjds=,(4{3)thenthereexistsak02f1,...,ngsuchthatlim!mLk0,m()=.Additionally,ifthereisauniquel02f0,...,ngsuchthatm2essrange(j)]TJ /F18 5.978 Tf 4.82 -1.41 Td[(l0)and( 4{3 )holds,thenlim!mLk,m()=foreveryk2f1,...,ng.FortheremainderofthesectionthesetJmwillbechosenasin( 4{2 )andk02f1,...,ngwillbechosensuchthatlim!mLk0,m()=.Lemma 4.2.4 andProposition 4.2.5 bothneedF=f2<(D):sgn()=sgn(),andfislogintegrablegtobenonempty.ThefollowinglemmaprovidesaneasytocheckconditionwhichguarantiesFisnonempty. 48

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Lemma4.2.7. Givenarealvalued2L1(\withagaparoundzero,ifhasconstantsignoneachboundarycomponent,thenthereexistsac2Rnsuchthat=c=nXj=1cj@hj @n2F.Moreover,thesetC=fc2Rn:c2Fghasnonemptyinterior,denotedbyC.LetBr(([],m))=P+Br()(m)]TJ /F3 11.955 Tf 12.67 0 Td[(r,m+r)whereBr()istheballofradiusrcenteredat.Withtheselemmasinhandwecangiveaconditionwhichimpliesthatthesetof2suchthatTH2(R;)hasaneigenvalueislarge. Theorem4.2.8. Let2L1(\berealvalued,haveconstantsignoneachboundarycomponent,andhaveagap,O0,aroundzero.Ifa)misontheedgeofthegapofaroundzero;b)Jmischosenasin( 4{2 );c)k02f1,...,mgandlim!mLk0,m()=;d)foreachk2f1,...,ngthereexistsak2Rsuchthat lim!mLk,m())]TJ /F8 11.955 Tf 11.95 0 Td[(kLk0,m()(4{4)exists;ande)thesetf1,...,ngislinearlyindependentoverQ,thenthesetof2suchthatTH2(R;)hasaneigenvalueisdensein.Moreover,ifthereexistsa2FsuchthatasdenedinSection 3.4 isinvertibleinBr(([],m))\(P+FO0)forsomer>0,thentheimageofonthatneighborhoodisanopendensesetin.Wecansimplifytheconditionsfortheabovetheoremforsomefunctions. Corollary4.2.9. Let2L1(\berealvalued,haveconstantsignoneachboundarycomponent,andhaveagaparoundzero.Ifa)misontheedgeofthegapofaroundzero;b)Jmischosenasin( 4{2 ); 49

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c)k02f1,...,ngandlim!mLk0,m()=;andd)j2Jmimpliesthatj)]TJ /F18 5.978 Tf 4.83 -1.41 Td[(j=m,thenforeachk2f1,...,ngk=Pj2JmR)]TJ /F18 5.978 Tf 4.82 -1.41 Td[(j@hk @nds Pj2JmR)]TJ /F18 5.978 Tf 4.82 -1.4 Td[(j@hk0 @ndsexists.Ifinadditionthesetf1,...,ngislinearlyindependentoverQ,thenthesetof2suchthatTH2(R;)hasaneigenvalueisdensein.Moreover,ifthereexistsa2FsuchthatasdenedinSection 3.4 isinvertibleinBr(([],m))\(P+FO0)forsomer>0,thentheimageofonthatneighborhoodisanopendensesetin. 4.2.1ProofsofLemmas 4.2.1.1ProofofLemma 4.2.2 .Let2,NH2(R;),andg2NsuchthatTNg=0begiven.LetM=_a2Aag.ItisclearthatMisanonzeroAinvariantsubspaceofH2(R;)sobyTheorem 4.2.1 thereexists,2andaninnerfunction2H2(R;)suchthatM=H2(R;).SinceM=_a2AagandM=H2(R;)thefunction g2H2(R;)mustbeouter.Soleteg= g.ByTheorem 3.3.1 weknowthatthereexistsa2<(D)suchthatjgj2=.So,sincejegj= g=jgjwehavethatjegj2=andTheorem 3.3.1 saysthatundertheseconditionsthereexistsanAinvarianteNH2(R;)containing~gsuchthatTeNeg=0.ButbecauseegisoutertheonlyclosedAinvariantsubspaceofH2(R;)containingegisH2(R;)itself.Thusthelemmahasbeenshown. 50

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4.2.1.2ProofofLemma 4.2.4 .Since2L1(\hasagaparoundzero,theconstantfunction1isinA,2FthehypothesesofCorollary 3.3.3 aresatised.Therefore,weknowthatforeach2O0thereexistsan2,anAinvariantsubspaceNH2(R;)andafunctiong,2NsatisfyingTNg,=g,.ButbyLemma 4.2.2 wecanassumethatN=H2(R;)andg,2H2(R;)isouter.Additionally,weknowthatjg,j2= )]TJ /F8 11.955 Tf 11.96 0 Td[(.By[ 16 ]wehavethattheperiod,j,ofanoutermodulusautomorphicfunctiongaround)]TJ /F12 7.97 Tf 6.77 -1.79 Td[(jisj=1 2Z)]TJ /F6 11.955 Tf 7.32 10.8 Td[(logjgj@hj @nds.Thusforg=g,j(,)=1 4Z)]TJ /F6 11.955 Tf 7.32 10.63 Td[((logjj)]TJ /F6 11.955 Tf 17.93 0 Td[(logj)]TJ /F8 11.955 Tf 11.96 0 Td[(j)@hj @ndswhichbytheisomorphismbetweenand(j(,)(mod1))ndetermines(,).Lastly,since2O0weknowthat)]TJ /F8 11.955 Tf 12.44 0 Td[(isboundedawayfromzeroandthusforeachj2f1,...,ngZ)]TJ /F6 11.955 Tf 7.31 10.8 Td[(logj)]TJ /F8 11.955 Tf 11.96 0 Td[(j@hj @ndsisnite.Therefore,Z)]TJ /F6 11.955 Tf 7.31 10.8 Td[(logjj=j(,)+Z)]TJ /F6 11.955 Tf 7.31 10.8 Td[(logj)]TJ /F8 11.955 Tf 11.96 0 Td[(j@hj @ndsisnite. 4.2.1.3ProofofLemma 4.2.6 .Let)]TJ /F12 7.97 Tf 6.77 4.33 Td[(m=Sj2Jm)]TJ /F12 7.97 Tf 6.78 -1.8 Td[(j.SincemisontheedgeofthegapO0andhasconstantsignconstantoneachboundarycomponentthereexistsaboundary`suchthatm=2essrange(j)]TJ /F17 5.978 Tf 4.82 -1.4 Td[(`)and)]TJ /F2 11.955 Tf 10.02 0 Td[(n)]TJ /F12 7.97 Tf 6.77 4.34 Td[(misnonempty.Additionally,sincem=2essrange(j)]TJ /F7 7.97 Tf 4.82 0 Td[(n)]TJ /F18 5.978 Tf 4.82 2.27 Td[(m)weknowthat)]TJ /F3 11.955 Tf 12.55 0 Td[(mis 51

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boundedawayfromzeroon)]TJ /F2 11.955 Tf 9.44 0 Td[(n)]TJ /F12 7.97 Tf 6.77 4.34 Td[(mandZ)]TJ /F7 7.97 Tf 4.82 0 Td[(n)]TJ /F18 5.978 Tf 4.82 2.27 Td[(mlogj)]TJ /F3 11.955 Tf 11.96 0 Td[(mjdsisnite.Therefore,since@h` @nisboundedon)]TJ /F1 11.955 Tf 10.68 0 Td[(andnegativeon)]TJ /F12 7.97 Tf 6.78 4.34 Td[(mandR)]TJ /F6 11.955 Tf 7.31 4.15 Td[(logj)]TJ /F3 11.955 Tf 11.96 0 Td[(mjds=wehavethatL`,m(m)+1 4Z)]TJ /F7 7.97 Tf 4.82 0 Td[(n)]TJ /F18 5.978 Tf 4.82 2.27 Td[(mlogj)]TJ /F3 11.955 Tf 11.96 0 Td[(mj@h` @nds=1 4Z)]TJ /F6 11.955 Tf 7.31 10.79 Td[(logj)]TJ /F3 11.955 Tf 11.95 0 Td[(mj@h` @nds=1.Forclosetoandapproachingm,R)]TJ /F18 5.978 Tf 4.82 2.27 Td[(mlogj)]TJ /F8 11.955 Tf 11.95 0 Td[(jdsisdecreasing,thereforebytheMonotoneConvergenceTheoremlim!mL`,m()=.Nowwecanchoosek0.If`6=0,thenletk0=`.Ontheotherhandif`=0,then@h0 @n=)]TJ /F10 11.955 Tf 11.29 8.96 Td[(Pni=1@hi @n.Solim!mZ)]TJ /F6 11.955 Tf 7.31 10.79 Td[(logj)]TJ /F8 11.955 Tf 11.95 0 Td[(j@h0 @nds=lim!m)]TJ /F12 7.97 Tf 17.64 14.95 Td[(nXj=1Z)]TJ /F6 11.955 Tf 7.31 10.79 Td[(logj)]TJ /F8 11.955 Tf 11.96 0 Td[(j@hj @nds=1thustheremustbesomek02f1,...,ngsuchthatlim!mLk0,m()=lim!m1 4Z)]TJ /F6 11.955 Tf 7.31 10.79 Td[(logj)]TJ /F8 11.955 Tf 11.96 0 Td[(j@hk0 @nds=.Finally,ifthereexistsaunique`0suchthatm2essrange(j)]TJ /F17 5.978 Tf 4.82 -1.41 Td[(`),thentheaboveproofshowsthatforany`2f1,...,ngsuchthatfor`6=`0wehavethatlim!mL`,m()=1.Ontheotherhandif`=`06=0,then)]TJ /F3 11.955 Tf 11.96 0 Td[(misboundedawayfromzeroon)]TJ /F2 11.955 Tf 9.44 0 Td[(n)]TJ /F14 7.97 Tf 6.77 -1.8 Td[(`0soZ)]TJ /F7 7.97 Tf 4.82 0 Td[(n)]TJ /F17 5.978 Tf 4.82 -1.41 Td[(`0logj)]TJ /F3 11.955 Tf 11.95 0 Td[(mjdsisnite.Similartobeforeweknow@h`0 @nisboundedon)]TJ /F1 11.955 Tf 10.68 0 Td[(andpositiveon)]TJ /F14 7.97 Tf 6.77 -1.79 Td[(`0soZ)]TJ /F6 11.955 Tf 7.31 10.8 Td[(logj)]TJ /F3 11.955 Tf 11.96 0 Td[(mj@h`0 @nds=.AnotherapplicationoftheMonotoneConvergenceTheoremsayslim!mL`0,m()=.Thatcompletesthiscaseandtheproof. 52

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4.2.1.4ProofofLemma 4.2.7 .Letarealvalued2L1(\withagaparoundzeroandconstantsignoneachboundarycomponentbegiven.LetJ+=j2f0,...,ng:j)]TJ /F18 5.978 Tf 4.82 -1.41 Td[(j>0.Thenthereexistsatleastoneboundary)]TJ /F12 7.97 Tf 6.77 -1.8 Td[(jwherej)]TJ /F18 5.978 Tf 4.82 -1.4 Td[(jispositiveandanotherboundary)]TJ /F12 7.97 Tf 6.77 -1.79 Td[(jwherej)]TJ /F18 5.978 Tf 4.83 -1.41 Td[(jisnegative.ThusJ+isanonemptypropersubsetoff1,...,ng.Leth=Pj2J+hj.Thushistheharmonicfunctionwhichis1on)]TJ /F9 7.97 Tf 6.77 4.34 Td[(+=Sj2J+)]TJ /F12 7.97 Tf 6.78 -1.79 Td[(jandzeroontheothercomponentsof)]TJ /F1 11.955 Tf 10.67 0 Td[(and@h @nisstrictlypositiveon)]TJ /F9 7.97 Tf 6.78 4.34 Td[(+andstrictlynegativeontheothercomponentsof)]TJ /F1 11.955 Tf 10.67 0 Td[(byHopf'sLemma,[ 15 ].FurtherbecausePni=0@hi @n=0wehavethat@h0 @n=)]TJ /F10 11.955 Tf 11.29 8.97 Td[(Pnj=1@hj @nsowecanwritec=@h @n=nXj=1cj@hj @nwherecj2f)]TJ /F6 11.955 Tf 27.02 0 Td[(1,0,1g.Thereforebecauseofthecontinuityof@hj @nforeachj2f1,...,ngandthecontinuityofadditionwecanperturbthecoecients,cj,andcwillstillbeinF. 4.2.2ProofofProposition 4.2.5 Let)]TJ /F12 7.97 Tf 6.77 4.34 Td[(m=Sj2Jm)]TJ /F12 7.97 Tf 6.78 -1.79 Td[(j.ByLemma 4.2.4 weknowthatk=1 4Z)]TJ /F6 11.955 Tf 7.31 10.62 Td[((logjj)]TJ /F6 11.955 Tf 17.93 0 Td[(logj)]TJ /F8 11.955 Tf 11.96 0 Td[(j)@hk @nds.Firstnotethat,1 4R)]TJ /F6 11.955 Tf 7.32 4.15 Td[(logjj@hk @ndsisniteandindependentof.Further,m=2essrange(j)]TJ /F7 7.97 Tf 4.82 0 Td[(n)]TJ /F18 5.978 Tf 4.82 2.27 Td[(m)sologj)]TJ /F3 11.955 Tf 11.96 0 Td[(mjisboundedon)]TJ /F2 11.955 Tf 9.71 0 Td[(n)]TJ /F12 7.97 Tf 6.77 4.34 Td[(m.Thuswecanboundlogj)]TJ /F8 11.955 Tf 11.96 0 Td[(j@hk @non)]TJ /F2 11.955 Tf 9.71 0 Td[(n)]TJ /F12 7.97 Tf 6.77 4.34 Td[(m.SobytheDominatedConvergenceTheoremlim!mZ)]TJ /F7 7.97 Tf 4.82 0 Td[(n)]TJ /F18 5.978 Tf 4.82 2.27 Td[(mlogj)]TJ /F8 11.955 Tf 11.96 0 Td[(j@hk @ndsexistsandisnite.Nowitiseasytoseethatlim!mk(,)+Lk,m()=lim!mZ)]TJ /F6 11.955 Tf 7.31 10.79 Td[(logjj@hk @nds)]TJ /F10 11.955 Tf 11.95 16.28 Td[(Z)]TJ /F7 7.97 Tf 4.82 0 Td[(n)]TJ /F18 5.978 Tf 4.82 2.27 Td[(mlogj)]TJ /F8 11.955 Tf 11.96 0 Td[(j@hk @nds 53

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existsandisnite. 4.2.3ProofofTheorem 4.2.8 Let2L1(\berealvalued,haveagaparoundzero,O0,andhaveconstantsignoneachboundary.Bythehypothesisweknowthatmisontheedgeofthegaparoundzero,Jmischosenasin( 4{2 ),forsomek02f1,...,mglim!mLk0,m()=,andthereexistsasetf1,...,ngsuchthatlim!mLk,m())]TJ /F8 11.955 Tf 11.96 0 Td[(kLk0,m()existsforeachk2f1,...,ngandislinearlyindependentoverQ.Assumethatlim!mLk0,m()=1,thecasethatlim!mLk0,m()=issimilar.Lastly,let(j(mod1))nj=12[0,1)nand>0begiven.Foreachj2f1,...,ngletdj=lim!mLj,m())]TJ /F8 11.955 Tf 11.95 0 Td[(jLk0,m(),andej=lim!mj(,)+Lj,m().ByhypothesiseachdjexistsandbyLemma 4.2.7 andProposition 4.2.5 thereexistsasuchthatejexistsforallj2f1,...,ng.Fixsome0inO0andforeachp2Z+letp2O0bechosensuchthatjm)]TJ /F8 11.955 Tf 11.96 0 Td[(pj
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isdensein[0,1)nunderadditionmodulo1,sothereexistsasubsequence()]TJ /F8 11.955 Tf 9.3 0 Td[(jpq(mod1))nj=11q=0whichconvergesto(j+dj+jLk0,m(0))]TJ /F3 11.955 Tf 11.96 0 Td[(ej(mod1))nj=1.Ifwexarepresentative(j)nj=1of(j(mod1))nj=1,thenthereexistsasequenceofintegersf(sj,q)nj=1g1q=1suchthatf()]TJ /F8 11.955 Tf 9.3 0 Td[(jpq)]TJ /F3 11.955 Tf 11.95 0 Td[(sj,q)nj=1g1q=1convergesto(j+dj+jLk0,m(0))]TJ /F3 11.955 Tf 11.95 0 Td[(ej)nj=1inRn.Therefore,forlargeqinRnweknowthatd)]TJ /F8 11.955 Tf 5.47 -9.68 Td[(j(pq,))]TJ /F3 11.955 Tf 11.95 0 Td[(sj,qnj=1,(j)nj=1Rn< 3+d)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(Lj,m(pq)+ej)]TJ /F3 11.955 Tf 11.96 0 Td[(sj,qnj=1,(j)nj=1Rn<2 3+d)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F8 11.955 Tf 9.3 0 Td[(jLk0,m(pq)+ej)]TJ /F3 11.955 Tf 11.95 0 Td[(dj)]TJ /F3 11.955 Tf 11.96 0 Td[(sj,qnj=1,(j)nj=1Rn=2 3+d)]TJ /F6 11.955 Tf 5.48 -9.85 Td[(()]TJ /F8 11.955 Tf 9.3 0 Td[(jpq)]TJ /F3 11.955 Tf 11.96 0 Td[(sj,q)]TJ /F8 11.955 Tf 11.96 0 Td[(jLk0,m(0)+ej)]TJ /F3 11.955 Tf 11.96 0 Td[(dj)nj=1,(j)nj=1Rn=2 3+d)]TJ /F6 11.955 Tf 5.48 -9.85 Td[(()]TJ /F8 11.955 Tf 9.3 0 Td[(jpq)]TJ /F3 11.955 Tf 11.96 0 Td[(sj,q)nj=1,(j+dj+jLk0,m(0))]TJ /F3 11.955 Tf 11.95 0 Td[(ej)nj=1Rn<.So)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(j(,pq)]TJ /F3 11.955 Tf 11.96 0 Td[(sj,q)nj=1convergesto(j)nj=1inRn.Sincefsj,qgZ,wehavethatf)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(j(,pq)(mod1)nj=1g1q=1convergesto(j)nj=1.Thusforadensesubsetof[0,1)ntheToeplitzoperatorTH2(R;)hasaneigenvalue2O0.Finally,ifthemapdenedinSection 3.4 isinvertibleinBr(([],m))\(P+FO0)forsomer>0then(Br(([],m))\(P+FO0))isopenandcontainstheimageof[]((m)]TJ /F3 11.955 Tf 11.97 0 Td[(r,m+r)\O0)whichsincethechoiceof02O0wasarbitraryisadensesubsetof. 4.2.4ProofofCorollary 4.2.9 Choosek0tobeanelementofJm.Sincej)]TJ /F18 5.978 Tf 4.82 -1.4 Td[(j=mwhenj2JmwehavethatLk,m()=logjm)]TJ /F8 11.955 Tf 11.95 0 Td[(jXj2JmZ)]TJ /F18 5.978 Tf 4.82 -1.41 Td[(j@hk @nds. 55

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Becauselim!mLk0,m()=itfollowsthatthekasdenedinitem(d)existandmoreover,kLk0,m()=Lk,m().AnapplicationofTheorem 4.2.8 completestheproof. 4.3SomeToeplitzOperatorsonaConstrainedDiskAlgebraInadditiontoallowingustoproveresultsforToeplitzoperatorsonmultiplyconnecteddomainstheresultsinChapter 3 allowustoeasilyobtainresultsforalgebrassimilartotheNeilealgebra.LetA=ff2H1(D):f0(0)=f00(0)=0g.ItisclearthatD=L2(T)(A_A)=spanfz2,z, z, z2gL1(T).Additionally,notethatsincethefundamentalgroup,G,ofD,istrivial,thegroupofrepresentationsofGinU(C)isalsotrivial.SowewillonlybeworkingwiththeAinvariantsubspacesofH2(D).SimilartotheNeilealgebracaseinChapter 2 ,foraVCCzCz2letH2(D;V)=H2(D)V.NowwecanproveaversionofBurling'sTheoremforAinvariantsubspacesofH2(D). Lemma4.3.1. IfNisanAinvariantsubspaceofH2(D),thenthereexistsaVCCzCz2withtheconstantfunction1=2VandaninnerfunctionsuchthatN=H2(D;V)whereH2(D;V)=H2(D)V.Moreover,givenanf2H2(D)wecandetermineaVsuchthatf2H2(D;V). Lemma4.3.2. Iff2H2(D)andVf(0)+f0(0)z+1 2f00(0)z2?,thenf2H2(D;V).Notethat,iff(0)6=0,then1=2V.Theselemmasallowustoprovethefollowing. 56

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Lemma4.3.3. LetNanAinvariantsubspaceofH2(D),andg2Nbegiven.IfTNg=0,thenthereexistsaeVCCzCz2andanouterfunctioneg2H2(D;eV)suchthatTH2(D;eV)eg=0.Moreover,eVcanbechosentobe)]TJ 5.54 -9.18 Td[(eg(0)+eg0(0)z+eg00(0)z2?and1=2eV.SoweonlyneedtoconsiderToeplitzoperatorsforwhichN=H2(D;V)forsomeVCCzCz2andwhichhaveanoutereigenfunction.Thefunctionsin<(D)areoftheformc2z2+c1z+ c1 z+ c2 z2=<(c2z2)+<(c1z)andarelogintegrablebybasiccalculus.Whilethedimensionof<(D)asarealvectorspaceisfour,thenextlemmademonstratesthatforaspecicreal-valued2L1(T)theHausdordimensionofF=f 2<(D):sgn()=sgn( )andislogintegrableg<(D)maybemuchsmaller. Lemma4.3.4. Givenareal-valued2L1(T),ifthereexistsab2Csuchthatsgn()=<(bz),thenbisuniqueuptopositivescaling.Moreover,P+Fisasmoothmanifoldofdimensionatmosttwo.Forgeneralreal-valued2L1(T),Corollary 3.3.2 andLemma 4.3.1 sayifhasagaparoundzero,thenthereexistsa2FifandonlyifthereexistsaV,anouterfunctiong2H2(D;V)suchthatTH2(D;V)g=0.Moreover,ifFisnonempty,theneachinthegaparoundzero,O0,isaneigenvalueassociatedtorelativetoA.Further,ifweletF=f2D:sgn()=sgn()gand=fH2(D;V):V=(1+d1z+d2z2)?,d1,d22Cg,thenbySection 3.4 andLemma 4.3.3 wehaveamap:P+FO0!denedby(([],))mapstotheuniqueH2(D;V)2suchthatthereexistsanouterfunctionf2H2(D;V)satisfyingjfj2=whereisarepresentativeof[].InfactwecanequipwithamanifoldstructurebylettingbethemapfromC2todenedby((d1,d2))=H2(D;V)whereV=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1+d1z+d2z2?.Then,isabijectionandinheritsthestructureonC2through. 57

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BySection 3.4 ifwecanshowthatisLipschitzwecanndtheHausdordimensionofthespaceofmodelsoftheToeplitzoperatorwithsymbolwhichhaveeigenvalues.IfweknowthatFistheconegeneratedbyasinglefunctionthenwehavethefollowingtheorem. Theorem4.3.5. IfP+F=f[ ]g,thenthemapistwicedierentiable.Thustherearemanyspacesinwhichhavenoeigenvalues.Unfortunatelyweareleftwiththefollowingquestion. Question4.3.6. Underwhatconditionsisdierentiablewithrespectto[]? 4.3.1ProofsofLemmas 4.3.1.1ProofofLemma 4.3.1 LetNH2(D)beanAinvariantsubspaceandM=_n2NznN.ItisclearthatMisinvariantundermultiplicationbyH1(D).So,thereexistsaninnerfunctionsuchthatM=H2(D).Inotherwords M=f= g:g2M=H2(D).Thus,eN= NH2(D)andbecauseWn2N+znN$Mweknowthat2Nandhence12eN.Further,ifn>2,thenzn2eN.Therefore,z3H2(D)eN.Thus,V=H2(D)eNC+Cz+Cz2and1=2V.Whichprovesthelemma. 4.3.1.2ProofofLemma 4.3.2 Givenf2H2(D),leth=f(0)+f0(0)z+f00(0) 2z2.Thereexistsag2H2(D)suchthatf=d0+d1z+d2z2+z3g.Thus,f2H2(D;V)ifandonlyifV(h)?. 4.3.1.3ProofofLemma 4.3.3 LetNanAinvariantsubspace,andg2NsuchthatTNg=0begiven.LetM=_a2Aag. 58

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ItisclearthatMisanonzeroAinvariantsubspaceofH2(D)sobyTheorem 4.3.1 thereexistseVandaninnerfunction2H2(D)suchthatM=H2(R;eV).SinceM=_a2AagandM=H2(R;eV)thefunction g2H2(R;eV)mustbeouter.Soleteg= g.ByTheorem 3.3.1 weknowthatthereexistsa2<(D)suchthatjgj2=.So,sincejegj= g=jgjwehavethatjegj2=andTheorem 3.3.1 saysthatundertheseconditionsthereexistsanAinvarianteNH2(R;eV)suchthatTeNeg=0.ButbecauseegisoutertheonlyclosedAinvariantsubspaceofH2(R;eV)containingegisH2(R;eV)itself.Thusthelemmahasbeenshown. 4.3.1.4ProofofLemma 4.3.4 Assumethatsgn()=sgn(<(bz))forsomeb2C,andchooserb,b2Rsuchthatb=rbeib.Additionally,assumethatsgn()=sgn(<(cz))forsomec=rceic2C.Notethatsincez2Twecanwrite<(bei)=rbcos(+b)and<(cei)=rccos(+c).Sincebotharecontinuousfunctionsofandsgn(<(bz))=sgn(<(cz))wehavethatrbcos(+b)=0ifandonlyifrccos(+c)=0.Fromthereitiseasytoseethatc=b+nforsomeintegern.Ifnisodd,thencos(+c)=)]TJ /F6 11.955 Tf 11.29 0 Td[(cos(+b).Thusc=b+2nandcisapositivescalarmultipleofb.Similarly,ifd=rdeid2Candsgn(<(bei))=sgn(<(cei))+<(dei2),thenifrbcos(+b)=0,thenrccos(+c)+rdcos(2+d)=0.Usingthesumtoproductformulastowritethesecondequationintermsof+b,c)]TJ /F8 11.955 Tf 11.01 0 Td[(b,andd)]TJ /F8 11.955 Tf 11.01 0 Td[(bwecanshowthatc=b+nforsomeintegernandd=2b+(2n+1) 2.ThusrcandrdaretheonlypossiblefreecontinuousvariablessodimH(spanR(F))2andthelemmahasbeenshown. 59

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4.3.2ProofofTheorem 4.3.5 Toshowthatistwicedierentiableweneedtoshowthatthemap)]TJ /F9 7.97 Tf 6.58 0 Td[(1(([],))istwicedierentiableonf[]gO0.ByCorollary 3.3.3 weknowthereisafunctiongthatsatisestheequation()]TJ /F8 11.955 Tf 11.95 0 Td[()jgj2=,thereforejgj2= )]TJ /F8 11.955 Tf 11.95 0 Td[(.Further,sincebyLemma 4.3.3 wemayassumethatgisouterwehavethatifH(z,t)=eit+z eit)]TJ /F3 11.955 Tf 11.96 0 Td[(z,theng(z)=exp1 2ZTH(z,)(logjj)]TJ /F6 11.955 Tf 17.94 0 Td[(logj)]TJ /F8 11.955 Tf 11.95 0 Td[(j)ds.So,ifweleth(z)=1 2ZTH(z,)logjcjdsandf(z)=1 2ZTH(z,)logj)]TJ /F8 11.955 Tf 11.95 0 Td[(jds,theng(0)=exp(h(0))]TJ /F3 11.955 Tf 11.96 0 Td[(f(0)),g0(0)=(h0(0))]TJ /F3 11.955 Tf 11.95 0 Td[(f0(0))exp(h(0))]TJ /F3 11.955 Tf 11.95 0 Td[(f(0)),andg00(0)=h00(0))]TJ /F3 11.955 Tf 11.96 0 Td[(f00(0)+(h0(0))]TJ /F3 11.955 Tf 11.95 0 Td[(f0(0))2exp(h(0))]TJ /F3 11.955 Tf 11.95 0 Td[(f(0)).Thus,toshowthat)]TJ /F9 7.97 Tf 6.59 0 Td[(1([],)istwicedierentiableweneedtoshowthatthefunctionsf(0)=RTH(0,)logj)]TJ /F8 11.955 Tf 11.95 0 Td[(jds,f0(0)=RTH0(0,)logj)]TJ /F8 11.955 Tf 11.96 0 Td[(jds,f00(0)=RTH00(0,)logj)]TJ /F8 11.955 Tf 11.95 0 Td[(jdsaretwicedierentiablewithrespecttoinO0.Infactthatf(0)andf0(0)aredierentiablewasshowninSubsection 2.3.4 andthesamereasoningappliestof00(0).ThustherangeofhasHausdordimensionatmostone.Sinceweconcludedwasamanifoldofdimensionfour,weknowthatmanypointsincannotbeintherangeof. 60

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CHAPTER5BUNDLESHIFTS 5.1PreviousWorkItisnaturaltoaskwhatdistinguishesthefamiliesofrepresentationsf:0<1gandfV:VCCzgofthealgebrasA(A)andAasmultiplicationoperatorsonthespacesfH2:0<1gandfH2V:VCCzgrespectively.Fortheannulus,anansweristhatSarasonrecognizedthatthecollectionofrepresentations(,H2(A;))playedthesameroleontheannulusasthesinglerepresentationdeterminedbytheshiftoperatorSgivenby A(D)3f!f(S)(5{1)playsforthediscalgebraA(D).ForAtherepresentationsVgenerateafamilyofpositivityconditionssucientforPickinterpolationinA[ 12 ].Likelyitisaminimalsetofconditionstoo.Corollary 5.3.6 belowcanbeinterpretedassayingthattherepresentations(V,H2(D;V);VCCz)shouldplaytheroleoftherankonebundleshiftsfortheNeilealgebraA.Inadualdirection[ 14 ]foundaminimalsetoftestfunctionsforA.Forsimilarresultsonmultiplyconnecteddomainssee[ 8 ]and[ 9 ].Theresultspresentedinthischapteroriginallyappearedin[ 10 ]. 5.2RepresentationsLetHbeaHilbertspace.Forpositiveintegersn,thealgebraMn(B(H))ofnnmatriceswithentriesfromB(H)isnaturallyidentiedwithB(CnH),theoperatorsontheHilbertspaceCnHn1H.Inparticular,itisthennaturaltogiveanelementX2Mn(B(H))thenormkXknitinheritsasanoperatoronCnH.IfAisasubalgebraofB(H),thenthenorms Theresultsinthischapteroriginallyappearedin[ 10 ]andareusedwithpermissionfromSpringer. 61

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kknofcourserestricttoMn(A),thennmatriceswithentriesfromA,andAtogetherwiththissequenceofnormsisaconcreteoperatoralgebra.Turningtothediscalgebra,anelementF2Mn(A(D))takestheformF=(Fj,k)nj,k=1forFj,k2A(D).Inparticular,Mn(A(D))isitselfanalgebraandcomesnaturallyequippedwiththenorm,kFkn=supfkF(z)k:z2Dg.wherekF(z)kistheusualoperatornormofthennmatrixF(z).Therepresentation:A(D)7!B(H2(D))givenby(a)=MaextendsnaturallytoMn(A(D))as1n:Mn(A(D))!B(nH2(D))by1n(F)=(Fj,k)nj,k=1.Moreover,themaps1nareisometric.ThusthealgebraA(D)canbeviewedasanoperatoralgebrabyidentifyingA(D)togetherwiththesequenceofnorms(kkn)withitsimageinB(H2(D))underthemappings1n.Ofcourse,anysubalgebraofA(D)canalsothenbeviewedasanoperatoralgebrabyinclusion.GivenanoperatoralgebraA,arepresentation:A!B(H)iscompletelycontractiveifk1n(F)knkFknforeachnandF2Mn(A).IfAandBareunital,thenisaunitalrepresentationif(1)=1.TherepresentationonB(H)ispureif\a2A(a)H=(0).ItisimmediatethattherepresentationsofA(D)determinedbySaswellastherepresentationsofA(A)andVofAareunital,completelycontractive,andpure.FollowingAgler[ 3 ],acompletelycontractive(unital)representation:A!B(H)ofAontheHilbertspaceHisextremalifwhenever:A!B(K)isacompletelycontractiverepresentationontheHilbertspaceKandV:H!Kisanisometrysuchthat(a)=V(a)V 62

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theninfactV(a)=(a)V.LetH2H(D)denotetheHilbertHardyspaceofH-valuedanalyticfunctionsonthediscwithsquareintegrableboundaryvalues.AssociatedtoHistherepresentation:A(D)!B(H2H(D))denedby(')f='f.Thus,(')ismultiplicationbythescalar-valued'onthevector-valuedHardy-HilbertspaceH2H(D).Ofcourse,H2H(D)isnaturallyidentiedwithHH2(D)andtherepresentationisthentheidentityonHtensoredwiththerepresentationofA(D)inEquation( 5{1 ).IfisacompletelycontractiveunitalpureextremalrepresentationofA(D),thenthereexistsaHilbertspaceHsothat,uptounitaryequivalence,:A(D)!B(H2H(D))isgivenby(')f='f. 5.3RepresentationsontheNeileAlgebraForAitturnsoutthatthesubspacesofH2H(A),whichareoftheformH2H(D;V)=H2H(D)V,giverisetotheextremalrepresentations[ 24 ].Indeed,givenaHilbertspaceHandasubspaceVofthesubspaceNzNofH2H(D),themappingV:A!B(H2H(D;V)denedbyV(a)f=af,iseasilyseentobeaunitalpurecompletelycontractiverepresentationofA.WhileitiseasytoseethattherepresentationsV:A!B(H2(D;V))areunital,pure,andcompletelycontractiveshowingthattheyarealsoextremalisabitharder.Toprovetheyareextremalwewillrstproveapropositionwhichgivesusaneasytoverifysucientconditionforarepresentationtobeextremal.TherstlemmaweneedisawellknowngeneralizationofSarason'sLemma[ 31 ].Givenarepresentation:A!B(K),asubspaceMofKisinvariantforif(a)MMforalla2A.AsubspaceHofKissemi-invariantforifthereexistinvariantsubspacesMand 63

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NsuchthatH=NM.Notethat,lettingV:H!Kdenotetheinclusion,themappingA3a7!V(a)VisalsoarepresentationofA. Lemma5.3.1. Let:A!B(H)bearepresentationofAinB(H)and:A!B(K)bearepresentationofAinB(K)andV:H!Kanisometry.If(a)=V(a)Vforalla2A,thenVHissemi-invariantfor.Thenextlemmaallowsustoimprovesemi-invariancetoinvariance. Lemma5.3.2. IfHKisasemi-invariantsubspaceforacontractionTandS=PHTjHisanisometry,thenHisaninvariantsubspaceforT.CombiningLemmas 5.3.1 and 5.3.2 yieldsthefollowingproposition. Proposition5.3.3. Let:A!B(H)beacontractiverepresentationofAinB(H)andfaigi2JAbeasetthatgeneratesadensesubalgebraofAwithkaik=1foralli2J.If(ai)isanisometryforalli2J,thenisextremal.NowitiseasytoshowthatalloftheV'sareextremalrepresentationsofA. Corollary5.3.4. TherepresentationVisanextremalrepresentationofA.Infactwealsohaveaconverse. Theorem5.3.5. TherepresentationsVareunitalpurecompletelycontractiveextremalrepresentations.Moreover,ifisaunitalpureextremalcompletelycontractiverepresentationofA,thenisunitarilyequivalenttoVforsomeHilbertspaceHandVHHz.Finallywewillsayarepresentationhasrankoneiftheredoesnotexistanontrivialorthogonalpairofsubspacesinvariantfortherepresentation. Corollary5.3.6. TherepresentationsVforVCCzhaverankone.MoreoveriftherepresentationisaunitalpureextremalcompletelycontractiverankonerepresentationofA,thenthereisaVCCzsuchthatisunitarilyequivalenttoV. 5.3.1ProofofLemmas 5.3.1.1ProofofLemma 5.3.1 LetN=_a2A(a)VH, 64

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thesmallest(closed)subspaceofKcontainingallofthespaces(a)VH.NoticethattheelementsoftheformNXi=0(ai)VhiwherefhigH,faigA,andN>0formadensesubsetofN.Sinceisarepresentation,foranya2A,fhigH,faigA,andN>0wehavethat(a) NXi=0(ai)Vhi!=NXi=0(aai)Vhi2NandthusNis(a)invariant.TocompletethisdirectionoftheproofweonlyneedtoshowthatM=NVHisalsoinvariantfor(a).Noticethat(a)=V(a)VimpliesV(a) NXi=0(ai)Vhi!=NXi=0V(aai)Vhi=NXi=0(aaj)hj=NXi=0(a)(ai)hi=(a)NXi=0V(ai)Vhi.ThusV(a)jN=(a)VjN.Ifm2M,thenm2NandbytheFredholmalternativeVm=0.Thus,ifa2A,thenV(a)m=(a)Vm=0,which,againbytheFredholmalternative,implies(a)m2M.Whichprovesthelemma. 5.3.1.2ProofofLemma 5.3.2 SinceHissemi-invariantforTandS=PHTjHweknowthatthereexiststwoTinvariantspacesNKandMKsuchthatN=MHandT=266664ABC0SF00K377775 65

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WhereA:M!M,B:M!H,C:M!N?,F:H!N?,andK:N?!N?.SinceTisacontractionwehaveI)]TJ /F3 11.955 Tf 11.96 0 Td[(TT0thus,forallh2H,0*(I)]TJ /F3 11.955 Tf 11.96 0 Td[(TT)2666640h0377775,2666640h0377775+=*266664)]TJ /F3 11.955 Tf 9.3 0 Td[(ABh(I)]TJ /F3 11.955 Tf 11.96 0 Td[(BB)]TJ /F3 11.955 Tf 11.96 0 Td[(SS)h)]TJ /F6 11.955 Tf 11.29 -.16 Td[((CB+FS)h377775,2666640h0377775+=khk2kBhk2kShk2=kBhk2soBh=0forallh2H.ThusHisaTinvariantsubspace. 5.3.2ProofofProposition 5.3.3 Let:A!B(K)beacontractiverepresentationofAinB(K)andV:H!Kbeanisometrysuchthat(a)=V(a)Vforalla2A.ByLemma 5.3.1 ,VHisasemi-invariantsubspaceofKfor.ByLemma 5.3.2 ,VHisinvariantfor(ai)foreachi2J.BecauseisarepresentationwehavethatVHisinvariantfor(a)foreachainthealgebrageneratedbythesetfaigi2J.ThusVHisinvariantfor.NowsinceVHis(a)invariantand(a)=V(a)Vforalla2AwehavethatV(a)=VV(a)V=PVH(a)V=(a)Vforalla2A.Thusisextremal. 5.3.3ProofofTheorem 5.3.5 Let:A!B(H)beapureextremalrepresentationofAonsomeseparableHilbertspaceH.By[ 23 ,Corollary7.7],therepresentationhasanC(T)-dilation;i.e.,thereexistsacompletelycontractiverepresentation:L1(D)!B(K)andanisometryV:H!Ksuchthat(a)=V(a)Vforalla2A.MoreoversinceisextremalV(a)=(a)Vforalla2AandVHisinvariantfor.FinallyletE=1_i=0(zi)VHK. 66

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Sincezi2Aforalli2Nandi6=1,1_i=0i6=1(zi)VH=1_i=0i6=1V(zi)H=VH.Inparticular,E=(z)VH_VH.FirstwewillshowthatS=(z)jEisapureisometryonE;iff,g2K,thenh(z)f,(z)gi=h(z)(z)f,gi=h( zz)f,gi=hf,gi.SinceSistherestrictionofanisometrytoaninvariantsubspaceSisanisometry.ToshowthatSispurenotethat(z2)E=(z2)((z)VH_VH)=(z3)VH_(z2)VHVH.Sinceispurewehave\b2A(D)(b)E\b=z2aa2A(a)(z2)E\a2A(a)VH=\a2AV(a)H=f0g.ThusSisapureshiftonE.SinceSisapureshiftthereisaHilbertspaceNandaunitarymapW:E!H2N(D)suchthatWS=MzW.SincethesubspaceS2EliesinVHandVHisasubspaceofE,thereexistsasubspaceVofNzNsuchthatWVH=H2N(D;V)=H2N(D)V.LetU:H!H2N(D;V)bedenedbyU=WV,thisisaunitarymapsuchthatUV(a)Uh=UMaUh=(a)hforallh2H,i.e.Visunitarilyequivalentto. 67

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5.3.4ProofofCorollaries 5.3.4.1ProofofCorollary 5.3.4 SinceV(1),V(z2),andV(z3)areisometriesand1,z2,andz3generateA,byProposition 5.3.3 weknowthatVisextremal. 5.3.4.2ProofofCorollary 5.3.6 SupposeVCCzandMandNareorthogonalsubspacesofH2VinvariantforA.Choosingnon-zero'and fromMandNrespectively,itfollowsthathzm',zn i=0fornaturalnumbersm6=16=n.HenceifisnormalizedarclengthmeasureonTand(z)=z,0=ZT' jdforalljandtherefore' =0.Sinceboth'and areinH2,eachisnon-zeroalmosteverywherewheneveritisnotthezerofunction.Thusatleastonemustbezero,whichisacontradiction.SoifVCCz,thenVisrankone.ByTheorem 5.3.5 itsucestocheckthesecondpartofthecorollaryforVwhereVNNz.IfNisonedimensionalthenNisunitarilyequivalenttoCandwearedone.IfNisnotonedimensional,thenchooseapairofnon-zerovectorseandfinNsuchthathe,fi=0andletE=z2H2eandF=z2H2f.BothEandFarenon-trivialsubspacesofH2VforanyVandareAinvariant.Theyarealsoorthogonalbyconstruction.HenceVisnotrankone. 68

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CHAPTER6CONCLUSIONLetRbeamultiplyconnecteddomain,2L1(R)bereal-valued,and2[0,1)n.Additionally,letAA(R)whichhasnitedefectandNH2(R,)beAinvariant.WhileTheorem 3.3.1 givesnecessaryandsucientconditionsfortheexistenceofeigenvaluesforTNtherearemanyadditionalquestionsregardingthespectrumofTNlefttobeanswered.Forexample,dotheeigenvectorsofTNhavetobeouterfunctions?DoesthespectralmeasureofTNhaveasingularcontinuouscomponent?Ifhasagaparound0,O0,andsgn()6=sgn()forall2D=L2(\(A_A),theniseach2O0inthespectrumofTN?Given2andanAinvariantMH2(R;)doTNandTMhavethesamecontinuousspectrum?CanthespectrumofTNhaveembeddedeigenvalues?Thatis,canthecontinuousspectrumofTNcontainandopenneighborhoodof0and0beaneigenvalueofTN?UnderwhatconditionsdoesTNhaveeigenvaluesforany2[0,1)nandAinvariantNH2(R,)?Whatcanweproveifwedroptheconditionthatisrealvalued?Whatcanwesayifisunimodular?CanweproveaversionofRosenblum'sTheorem[ 25 ]fortheNeilealgebra?Lastly,canweextendtheseresultstoothertypesofspacesofanalyticfunctions?ForexampletheFockspace.TheresultsinChapter 5 alsosuggestfutureresearch.Forexample,canwecharacterizetheextremalrepresentationsforanyAA(D)ofnitedefect? 69

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[32] DierkSchleicher.Hausdordimension,itsproperties,anditssurprises.Amer.Math.Monthly,114(6):509{528,2007. [33] BelaSz.-Nagy,CiprianFoias,HariBercovici,andLaszloKerchy.HarmonicanalysisofoperatorsonHilbertspace.Universitext.Springer,NewYork,secondedition,2010. 72

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BIOGRAPHICALSKETCH AdamBroschinskiwasborninWaynesboroVirginiaandlivedinthatareauntilhegraduatedWaynesboroHigh-Schoolin2002.HethentraveledtotheUniversityofWestVirginiaInstituteofTechnologyinMontgomeryWestVirginatostudymathematicsandchemistry.HereceivedaBachelorsofScienceinbotheldsin2007.HeenteredgraduateschoolattheUniversityofFloridatostudymathematicsin2008andgraduatedwithadoctorateinmathematicsin2014. 73