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Composition Operators on Hardy Spaces of the Disk and Half-Plane

Permanent Link: http://ufdc.ufl.edu/UFE0045410/00001

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Title: Composition Operators on Hardy Spaces of the Disk and Half-Plane
Physical Description: 1 online resource (74 p.)
Language: english
Creator: Luery, Kristin E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: composition -- operator
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: In his work, J. H. Shapiro provided an integral formula for the Nevanlinna counting function and used it to prove many results for composition operators on the Hardy space of the disk.  We derive an integral formula for a counting function in the upper half-plane and use it to provide a function theoretic proof that composition operators are bounded above on the Hardy space of the upper half-plane.  We also derive a new tool, the reproducing kernel thesis, to show that composition operators have closed range on the Hardy space of the disk.  With it we are able to provide a direct geometric equivalence between the criterion of Cima, Thomson, and Wogen and the one of Zorboska for composition operators to have closed range on the Hardy space of the disk.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kristin E Luery.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Jury, Michael Thomas.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045410:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045410/00001

Material Information

Title: Composition Operators on Hardy Spaces of the Disk and Half-Plane
Physical Description: 1 online resource (74 p.)
Language: english
Creator: Luery, Kristin E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: composition -- operator
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In his work, J. H. Shapiro provided an integral formula for the Nevanlinna counting function and used it to prove many results for composition operators on the Hardy space of the disk.  We derive an integral formula for a counting function in the upper half-plane and use it to provide a function theoretic proof that composition operators are bounded above on the Hardy space of the upper half-plane.  We also derive a new tool, the reproducing kernel thesis, to show that composition operators have closed range on the Hardy space of the disk.  With it we are able to provide a direct geometric equivalence between the criterion of Cima, Thomson, and Wogen and the one of Zorboska for composition operators to have closed range on the Hardy space of the disk.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kristin E Luery.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Jury, Michael Thomas.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045410:00001


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COMPOSITIONOPERATORSONHARDYSPACESOFTHEDISKANDHALF-PLANEByKRISTINLUERYADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013KristinLuery 2

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Iwouldliketodedicatethisdissertationtomyfamily. 3

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ACKNOWLEDGMENTS First,Iwouldliketothankmyadviser,Dr.MichaelJury,forallofhishelpandsupportthroughoutmygraduatecareer.Hehasbeenanexcellentadviser.Second,Iwouldliketothankmycommittee:Dr.LouisBlock,Dr.ScottMcCullough,Dr.LiShen,andDr.ChrisStanton. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 9 1.1HardyspaceoftheDisk ............................ 9 1.2HardyspaceoftheHalf-Plane ........................ 11 1.3CompositionOperators ............................ 12 2PRELIMINARIES ................................... 13 2.1Inner-OuterFactorization ........................... 13 2.1.1DiskFactorization ............................ 13 2.1.2Half-PlaneFactorization ........................ 16 2.2MobiusTransformations ............................ 17 2.3Frostman'sTheorem .............................. 18 2.4ReproducingKernelHilbertSpaces ..................... 19 2.5NevanlinnaCountingFunction ........................ 22 2.6ChangeofVariablesFormula ......................... 23 2.7AleksandrovClarkMeasures ......................... 25 2.8CarlesonRegions ............................... 28 2.9Non-tangentialLimitsandAngularDerivatives ................ 30 3ORIGINALPROOFSOFBOUNDEDABOVE ................... 33 3.1BoundedAboveontheDisk .......................... 33 3.2BoundedAboveontheHalf-Plane ...................... 37 3.3Remarks .................................... 39 4BOUNDEDABOVEANDTHECOUNTINGFUNCTION ............. 41 4.1NevanlinnaCountingFunctionontheDisk .................. 41 4.2CountingFunctionintheHalf-Plane ..................... 45 5BOUNDEDBELOW ................................. 53 5.1ClosedRange ................................. 53 5.2TheReproducingKernelThesis ........................ 55 5.3ANewProofofZorboska'sCondition ..................... 58 5.4FutureWork ................................... 71 5

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REFERENCES ....................................... 72 BIOGRAPHICALSKETCH ................................ 74 6

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LISTOFFIGURES Figure page 2-1Carlesonwindow ................................... 29 2-2Carlesoncircle .................................... 29 7

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCOMPOSITIONOPERATORSONHARDYSPACESOFTHEDISKANDHALF-PLANEByKristinLueryMay2013Chair:Dr.MichaelJuryMajor:Mathematics Inhiswork,J.H.ShapiroprovidedanintegralformulafortheNevanlinnacountingfunctionandusedittoprovemanyresultsforcompositionoperatorsontheHardyspaceofthedisk.Wederiveanintegralformulaforacountingfunctionintheupperhalf-planeanduseittoprovideafunctiontheoreticproofthatcompositionoperatorsareboundedaboveontheHardyspaceoftheupperhalf-plane.Wealsoderiveanewtool,thereproducingkernelthesis,toshowthatcompositionoperatorshaveclosedrangeontheHardyspaceofthedisk.WithitweareabletoprovideadirectgeometricequivalencebetweenthecriterionofCima,Thomson,andWogenandtheoneofZorboskaforcompositionoperatorstohaveclosedrangeontheHardyspaceofthedisk. 8

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CHAPTER1INTRODUCTION Thestudyofcompositionoperatorsisabeautifulsubjectthatutilizestechniquesfromallpartsofanalysistoproveresults:operatortheory,functionalanalysis,measuretheory,andanalyticfunctiontheorytonameafew.Theconceptofacompositionoperatorisrelativelysimpleandnaturaltostudy;givenafunction'wedenethecompositionoperatorC'onaspaceofanalyticfunctionsbyC'f=f'wherefisinthefunctionspace.Studyofcompositionoperatorsassuchtrulybeganinthe1960swithNordgren,andhassteadilygrownsincethen.Theyhavebeenstudiedonseveraldifferentfunctionspacesofvariousdomainsinthecomplexplaneaswellasinhigherdimensions.IthasbeenfoundthatmanypropertiesoftheoperatorC'canbeextractedfromfunctiontheoreticpropertiesofthefunction'.Inthesimplestandmostintuitivecase,theHardyspaceH2oftheunitdiskinthecomplexplane,manyproblemshavealreadybeensolvedsuchasconditionsforC'tobecompact[ 22 ],haveclosedrange[ 5 11 26 ],beanisometry[ 16 ],orsimilartoanisometry[ 2 ].Anotherinterestingsetting,andoneofalotofrecentactivity,istheHardyspaceH2oftheupperhalfofthecomplexplane.ThoughthetwoHardyspacesareisomorphic,compositionoperatorsonthetwospacesbehaveverydifferently.Forexample,therearenocompactcompositionoperatorsonH2oftheupperhalfplane[ 14 ],andthougheverycompositionoperatorisboundedonH2ofthedisknotallareboundedonH2oftheupperhalf-plane.HerewewillbefocusingoncompositionoperatorsontheHardyspaceofthediskandoftheupperhalfofthecomplexplane. 1.1HardyspaceoftheDisk First,wedenesomenotation.WedenotetheopenunitdiskofthecomplexplanebyD=fz2C:jzj<1g,anditsboundary,theunitcirclebyT=fz2C:jzj=1g.ThereareseveralwaystodenetheHardyspaceH2ofthedisk.Initsmostbasicsense,H2(D)isdenedtobethespaceofallanalyticfunctionsonthediskwhoseTaylorseries 9

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coefcientsaresquare-summable,thatis H2(D)=(f(z)=1Xn=0anzn:1Xn=0janj2<1).(1) ThisleadstothedenitionoftheinnerproductonH2(D);letf,g2H2(D)wheref(z)=Panznandg(z)=Pbnzn,wedenehf,giH2(D)=1Xn=0an bn. Thenormisthen kfk2H2(D)=hf,fiH2(D)=1Xn=0janj2.(1) ThesedenitionsgiveadirectcorrespondencebetweentheHilbertspaceH2(D)andtheHilbertspaceofsquare-summableinnitesequencesl2(N).AnotherdenitionforH2(D)relatesittotheHilbertspaceoffunctionsL2(T).InthiscasewedeneH2(D)=fanalyticonD:sup0
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Severaloftheresultsgiven,especiallyinChapter 2 ,applymoregenerallytoHpspacesofthedisk.ThespaceH2(D)isactuallyaspecialcaseofanHp(D)spacewherep=2.For00g,whichhasboundarytherealline,R=fz2C:=z=0g.AverysimilarsituationholdsfortheHardyspaceoftheupperhalf-planeH2(H),thoughitisusuallydenedonlybyintegralmeans.Letz=x+iy2C,thenwedeneH2(H)=fanalyticonH:supy>0ZRjf(x+iy)j2dm(x)<1, withnormkfk2H2(H)=supy>0ZRjf(x+iy)j2dm(x) wheremdenotesLebesguemeasureonthereallineR.ThespaceH1(H)isdenedsimilarlytothatofthedisk;itisthespaceofboundedanalyticfunctionsonHandhasnormkfkH1(H)=supz2Hjf(z)j. WecandeneasimilargeneralizationoftheH2(H)spacetoanHp(H)space.For00ZRjf(x+iy)jpdm(x)<1, 11

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withnormkfkpHp(H)=supy>0ZRjf(x+iy)jpdm(x) wheremdenotesLebesguemeasureonthereallineR. Throughoutthispaper,wewillbesuppressingthesubscriptsonthenorms,asitshouldbeeasilyunderstoodfromcontextwhichfunctionspacethegivenfunctionisin. 1.3CompositionOperators AtthistimewegiveaformaldenitionofacompositionoperatoronaHardyspace. Denition1.3.1. Let':D!Dbeanalytic.WedenethecompositionoperatorC'onHp(D)for0
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CHAPTER2PRELIMINARIES Thepurposeofthischapteristoreproducesomewell-knownresultsintheeld,sothattheywillbeonhandtoreferenceinlaterchapters. 2.1Inner-OuterFactorization NowthatwehavedenedtheHardyspacesHp(D)andHp(H)for0
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Nowthatwehavedeterminedthatboundaryvaluesexistalmosteverywhere,wecandenepreciselywhatitmeansforafunctiontobeinnerorouter. Denition2.1.2. Aninnerfunctionisananalyticfunctiong:D!Csuchthatjg(z)j1andg(ei)=1m-a.e.2T. AnouterfunctionisananalyticfunctionF:D!CoftheformF(z)=expZTei+z ei)]TJ /F4 11.955 Tf 11.95 0 Td[(zk()dm() wherekisareal-valuedintegrablefunctiononthecircleand2Chasmodulus1. WhenFisanouterfunctioninHp(D)for0
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wheremisanonnegativeintegerandfangisanonzerosequencethatsatises( 2 ). Theconditiononthesequencefang,( 2 ),isusuallycalledtheBlaschkecondition.Forconvenience,wewillalsocallafunctionthatisaunimodularconstanttimesB(z)aBlaschkeproduct. AcoupleofobservationsabouttheBlaschkeproduct: B(z)isinH1(D); thezerosofB(z)arepreciselyfangifm=0orf0g[fangifm>0; jB(z)j1andB(ei)=1m-almosteverywhereonT,i.e.B(z)isaninnerfunction. ThefollowingconditionisequivalenttoafunctionbeingaBlaschkeproductandwillbeveryusefulwhenweencounterFrostman'stheoreminSection 2.3 Theorem2.1.6. [ 9 ,p.54]LetfbeanalyticinD,thenthefollowingareequivalent: (a) fisaBlaschkeproduct,i.e.f(z)=B(z),whereisaunimodularconstantandB(z)isaBlaschkeproduct. (b) limr!1)]TJ /F10 11.955 Tf 8.25 22.09 Td[(ZTlogf(rei)dm()=0. Finally,wedenethesingularinnerfactor.Itencodesalltheinformationaboutthezerosthataccumulateontheboundaryofthedisk. Denition2.1.7. AsingularfunctionisafunctionoftheformS(z)=exp)]TJ /F10 11.955 Tf 11.29 16.27 Td[(ZTei+z ei)]TJ /F4 11.955 Tf 11.95 0 Td[(zd() wherethemeasureispositiveandsingulartoLebesguemeasure. Notethatthesingularfunctionisaninnerfunctionthathasnozerosinsideofthedisk.Nowthatwehavethesedenitions,wecanwriteanyfunctioninHpofthediskasaproductofeachtypeoffunction,aBlaschkeproduct,asingularinnerfunction,andanouterfunction. 15

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Theorem2.1.8. Letf2Hp(D)for0
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Finally,theouterfunctionisgivenbyF(z)=exp1 ZRlogjf(t)jitz+1 z)]TJ /F6 11.955 Tf 11.95 0 Td[(1dt 1+t2. Therefore,wecanfactoranyfunctionfinHp(H)for0
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theworkbeingdoneonH2(H)reliesontransportingquantitiestothedisk,exploitingpropertiesalreadyknownthere,andthentransportingbacktothehalf-plane.AthoroughdiscussionofMobiustransformationscanbefoundinanycomplexanalysisbook,forexamplein[ 6 ]or[ 20 ]. 2.3Frostman'sTheorem BeforewegettoFrostman'stheorem,weneedtounderstandalittlebitaboutlogarithmiccapacity.ThisbriefdiscussionisfromGarnett[ 9 ,SectionII.6],andamorerobustaccountcanbefoundinRansford[ 18 ,Chapter5]. Denition2.3.1. [ 9 ,p.75]AcompactsetKinChaspositivelogarithmiccapacityifthereisapositivemeasure6=0onKsuchthatthelogarithmicpotentialU(z)=ZKlog1 jw)]TJ /F4 11.955 Tf 11.96 0 Td[(zjd(w) isboundedonsomeneighborhoodofK. IfKisasubsetofD,thenKhaspositivecapacityifandonlyifKsupportsapositivemeasureforwhichGreen'spotentialG(z)=ZKlog1)]TJ ET q .478 w 258.16 -367.75 m 267.8 -367.75 l S Q BT /F4 11.955 Tf 258.16 -375.07 Td[(wz w)]TJ /F4 11.955 Tf 11.95 0 Td[(zd(w) isboundedonD. AnarbitrarysetEissaidtohavepositivecapacityifthereisacompactsubsetofEthathaspositivecapacity. Theorem2.3.2(Frostman). [ 9 ,p.75]Letf(z)beanonconstantinnerfunctionontheunitdisk,thenforallw2D,exceptpossiblyforasetofcapacityzero,thefunctionfw(z)=w)]TJ /F4 11.955 Tf 11.96 0 Td[(f(z) 1)]TJ ET q .478 w 249.91 -551.39 m 259.55 -551.39 l S Q BT /F4 11.955 Tf 249.91 -558.71 Td[(wf(z) isaBlaschkeproduct. ForaproofofFrostman'stheorem,seeGarnett[ 9 ,p.76]. 18

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Noticethatthefunctionfwissimplyequaltowf,henceitisfrequentlycalledtheFrostmantransformoff.ThewonderfulthingabouttheFrostmantransformoffisthatf)]TJ /F7 7.97 Tf 6.58 0 Td[(1(w)=f)]TJ /F7 7.97 Tf 6.58 0 Td[(1w(0).ThisprovesincrediblyusefulinourdiscussionoftheNevanlinnacountingfunctioninChapter 4 Theconditionforallw2D,exceptpossiblyforasetofcapacityzerousedinFrostman'stheoremisfrequentlywordedasforquasieveryw2Dorq.e.w2Dsimilartotheconditionalmosteveryoralmosteverywhereusedinmeasuretheory.Infact,quasieverywhereimpliesalmosteverywherewithrespecttoLebesgueareameasure. In[ 19 ],WalterRudingeneralizedFrostman'stheoremtoapplytoallHpfunctions,ratherthanjustnonconstantinnerfunctions.Inthiscase,theconclusionissimplythatfwhasnosingularinnerfactorquasieverywhere,orinotherwordsthatfw=BFforq.e.w2D. Theorem2.3.3(Rudin). Iff2Hp(D)for0
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(iii) foreveryy2X,thelinearevaluationfunctional,Ey:H!C,denedbyEy(f)=f(y)isbounded. SinceeveryboundedlinearfunctionalinareproducingkernelHilbertspaceHisgivenbytheinnerproductwithauniquevectorinH,wehavethatforeveryy2X,thereisauniquevectorky2H,suchthatf(y)=hf,kyiforeveryf2H. Denition2.4.2. Thefunctionkyiscalledthereproducingkernelatthepointy.ThetwovariablefunctiondenedbyK(x,y)=ky(x) iscalledthereproducingkernelforH. Alsonotethat K(x,y)=ky(x)=hky,kxiforx,y2X,and kEyk2=kkyk2=hky,kyi=K(y,y)fory2X. Therstpropertyaboveiscalledthereproducingpropertyofthekernelfunctions. OneofthereasonsreproducingkernelfunctionsaresousefulisbecausetheyhavedenselinearspanintheirassociatedHilbertspaces. Proposition2.4.3. [ 17 ]LetHbeareproducingkernelHilbertspaceonthesetXwithkernelK.Thenthelinearspanofthefunctionsky()=K(,y)isdenseinH. Werequireonemorefactaboutkernelfunctions,itisusedintheprooftheacompositionoperatorisboundedaboveonH2(H).ThispropositionstatesthatiftherearetwopositivekernelsoverthesamesetX,thentheirproductisalsoapositivekernel. Proposition2.4.4. [ 17 ]LetXbeasetandletKi:XX!C,i=1,2betwopositivedenitekernels,thentheirproduct,P:XX!CgivenbyP(x,y)=K1(x,y)K2(x,y) isapositivedenitekernel. InthispaperwewillbeconcernedwithtworeproducingkernelHilbertspacesinparticular,H2(D)andH2(H).ThereproducingkernelsfortheHardyspaceonthedisk 20

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H2(D)aretheSzegokernelskw(z)=1 1)]TJ ET q .478 w 226.68 -16.81 m 236.32 -16.81 l S Q BT /F4 11.955 Tf 226.68 -24.13 Td[(wzforw,z2D.ThenormoftheSzegokernelsisgivenbykkwk2=K(w,w)=1 1)-222(jwj2. Becauseofthis,ratherthanusingthetraditionalSzegokernels,wewillbeusingthenormalizedSzegokernels,~kw(z)=q 1)-222(jwj2 1)]TJ ET q .478 w 220.33 -125.32 m 229.98 -125.32 l S Q BT /F4 11.955 Tf 220.33 -132.64 Td[(wz,whicharenormalizedtohavenormone.AsforH2(H),thereproducingkernelsarethefunctionskw(z)=1 z+ wforw,z2H. ThereproducingkernelsofareproducingkernelHilbertspaceinteractverynicelywiththeadjointofacompositionoperator,ascanbeseenin[ 7 ].ThispropertywillbeusedintheoriginalproofthatC'isboundedaboveonH2(H)inChapter 3 .Recallthedenitionofanadjoint: Denition2.4.5. LetHbeaHilbertspacewithinnerproducth,i,andconsideraboundedlinearoperatorA:H!H,thenthereexistsauniqueboundedlinearoperatorA:H!HsuchthathAx,yi=hx,Ayiforallx,y2H. TheoperatorAiscalledtheadjointofA. WewillsimplydenotetheadjointofacompositionoperatorC'byC'.NowletHbeareproducingkernelHilbertspaceonXoverCwithreproducingkernelsfkxgx2X. Theorem2.4.6. [ 7 ,p.4]IfAisaboundedlinearoperatormappingareproducingkernelHilbertspaceHtoitself,thenAisacompositionoperatorifandonlyifthesetfkxgx2XisinvariantunderA.Inthiscase,A=C'where'andAarerelatedbyA(kx)=k'(x). Proof. [ 7 ,p.4]IfA=C',thenforeachfunctionf(A(kx))(f)=kx(Af)=kx(f')=f('(x))=k'(x)f soA(kx)=k'(x),andthesetofpointevaluationlinearfunctionalsisinvariantunderA. 21

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Conversely,ifthesetofpointevaluationlinearfunctionalsisinvariantunderA,thendenethemap'onXbyA(kx)=k'(x).Thisdenes'sincethevectorsinaHilbertspacesseparatethepointsofX.Then(Af)(x)=kx(Af)=(A(kx))(f)=k'(x)(f)=f('(x)), soA=C'. 2.5NevanlinnaCountingFunction TheNevanlinnacountingfunctionisatoolthatisintimatelylinkedwithcompositionoperators.Forexample,in[ 22 ]JoelShapirousedthecountingfunctioninacharacterizationoftheessentialnormofacompositionoperatorandacriterionforcompositionoperatorstobecompact. Denition2.5.1. Foraholomorphicmap':D!D,denetheNevanlinnacountingfunctionof'byN'(w)=Xz2')]TJ /F13 5.978 Tf 5.76 0 Td[(1(w)log1 jzj forallw2'(D)nf'(0)g,where')]TJ /F7 7.97 Tf 6.58 0 Td[(1(w)denotesthesetof'-preimagesofwcountedaccordingtotheirmultiplicity,andN'(w)=0ifw=2'(D). Sincelog1 jzj1)-258(jzjforjzjcloseenoughto1,theNevanlinnacountingfunctionprovidesameasureoftheafnitythat'hasforthevaluewbyweightingeachpreimageofwbytheproductofitsmultiplicityandaweightthatisessentiallyitsdistancefromtheboundaryofthecircleT. Wewishtodeneasimilarfunctionontheupperhalf-plane.InthiscasetheboundaryisthereallineR,soourgoalistoconstructafunctionthatweightseach'-preimagebyitsdistancetoR.Thisisrelativelysimpleas=zgivespreciselythedistancefromz2HtoR. 22

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Denition2.5.2. Foraholomorphicmap':H!H,denethecountingfunctionof'onHbyN'(w)=Xz2')]TJ /F13 5.978 Tf 5.76 0 Td[(1(w)=z forallw2'(H),where')]TJ /F7 7.97 Tf 6.58 0 Td[(1(w)denotesthesetof'-preimagesofwcountedaccordingtotheirmultiplicity,andN'(w)=0ifw=2'(H). Thiscountingfunctionweightseach'-preimagebytheproductofitsmultiplicityanditsdistancetotheboundaryoftheupperhalf-planeR. 2.6ChangeofVariablesFormula InChapter 1 ,wediscussedthemainformulationsofthenormonHpspaces,specicallyH2(D)andH2(H).Thereisonemoreformulationofthenormthatweproveexceedinglyusefultous,itistheLittlewood-Paleyidentity. Proposition2.6.1(Littlewood-PaleyIdentity). IffisholomorphiconD,then kfk2=2ZDjf0(z)j2log1 jzjdA(z)+jf(0)j2(2) wherekkdenotestheH2(D)norm,andkfk=1meansthatf=2H2(D). ForaproofseeGarnett[ 9 ,ChapterIV.3]. Aswewillbefocusingourstudyoncompositionoperators,wewouldlikeasimpleformulaforthenormofC'actingonafunctionf2H2(D).WhenweusetheLittlewood-PaleyidentitytondkC'fkaninterestingthinghappens,theNevanlinnacountingfunctionappearsintheintegral. Proposition2.6.2(ChangeofVariablesFormula). Suppose'isholomorphiconD,then kC'fk2=2ZDjf0(z)j2N'(z)dA(z)+jf('(0))j2.(2) ThereasonwhytheNevanlinnacountingfunctionappearsisbecausewedonotrequire'tobeunivalent.Typicallyachangeofvariablesrequiresthetransformationfunctiontobeunivalent,howeverthatwouldlimitthestudyofcompositionoperatorssignicantly.So,wedivide')]TJ /F7 7.97 Tf 6.59 0 Td[(1(D)intoregionswhere'isone-to-one,thenapply 23

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theusualchangeofvariablesoneachoftheseregionsindividually.TheNevanlinnacountingfunctionappearswhenwesumoveralloftheseregions.ItfollowsthatthemainstepinprovingtheChangeofVariablesFormulaisprovingthechangesofvariableswithintheintegralof( 2 )[ 23 ,Section10.3]. Lemma2.6.3. [ 23 ,p.186]LetgbeanonnegativemeasurablefunctiononDand':D!Dbeholomorphic,thenZDg('(z))j'0(z)j2log1 jzjdA(z)=ZDg(z)N'(z)dA(z). Proof. [ 23 ,p.186]Herewemakeexplicituseoftheunderlyingassumptionthat'isnotaconstantfunction.Assuch,itsderivative'0vanishesonanatmostcountablesubsetofDthathasnolimitpointinD;letuscallthissetZ.AroundeachpointinDnZthereisanopensetonwhich'isahomeomorphism,sowecanrewriteDnZasanatmostcountabledisjointunionofsemi-closedpolarrectanglesRnsuchthat'isunivalentoneachone.NowwecanapplytheusualchangeofvariablesformulaoneachRnindividually;foreaseofnotation,let ndenotetheinverseof'Rn.Therefore,ZRng('(z))j'(z)j2log1 jzjdA(z)=ZDg(w)'(Rn)(w)log1 j n(w)jdA(w). Summingbothsidesovernyields ZDg('(z))j'(z)j2log1 jzjdA(z)=ZDg(w)"Xn'(Rn)(w)log1 j n(w)j#dA(w).(2) Now,ifw2'(D)n'(Z),everypointof')]TJ /F7 7.97 Tf 6.58 0 Td[(1(fwg)hasmultiplicityone,soXn'(Rn)(w)log1 j n(w)j=N'(w) foralmosteveryw2'(D).Ontheotherhand,ifw=2'(D),thenbydenitionN'(w)=0andthebracketedtermin( 2 )iszero. 24

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WehaveasimilarformulationfortheH2normintheupperhalf-plane.Herethetermlog1 jzjisreplacedby=z,whichrepresentsthedistancetotheboundaryintheupperhalf-plane. Proposition2.6.4(Littlewood-PaleyIdentityfortheHalf-Plane). IffisholomorphiconD,then kfk2=2ZDjf0(z)j2=zdA(z)(2) wherekkdenotestheH2(H)norm,andkfk=1meansthatf=2H2(H). 2.7AleksandrovClarkMeasures AnothersetoftoolsthatareintimatelytiedwithcompositionoperatorsaretheAleksandrov-Clarkmeasures.SimilartohowtheNevanlinnacountingfunctionN'(w)measurestheafnity'haswiththevaluew2D,thesingularpartofanAleksandrovmeasuremeasurestheafnitythat'hasfortheboundaryvalue2T.Infact,NieminenandSaksmanshowedthatinacertainsensetheNevanlinnacountingfunctionactuallyconvergesweaktoanAleksandrov-Clarkmeasure[ 15 ,Theorem1.1].Consequently,wendthattheAleksandrov-Clarkmeasuresprovidemuchvaluableinformationabouttheboundarybehavior'.SeeCima,Matheson,andRoss[ 4 ,Chapter9]foramoreindepthstudy. ThedenitionofAleksandrov-ClarkmeasuresisbasedoffofHerglotz'stheorem: Theorem2.7.1(Herglotz). Ifuisanon-negativeharmonicfunctiononD,thenthereisauniquepositiveBorelmeasuresuchthatu(z)=ZT1)-222(jzj2 j)]TJ /F4 11.955 Tf 11.96 0 Td[(zj2d(), thePoissonintegralof. Denition2.7.2. [ 4 ,p.201]Forananalyticfunction':D!Dandapoint2T,thefunction<+'(z) )]TJ /F3 11.955 Tf 11.96 0 Td[('(z)=1)-222(j'(z)j2 j)]TJ /F3 11.955 Tf 11.96 0 Td[('(z)j2 25

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ispositiveandharmoniconD.ByHerglotz'stheorem1)-222(j'(z)j2 j)]TJ /F3 11.955 Tf 11.95 0 Td[('(z)j2=ZT1)-221(jzj2 j)]TJ /F4 11.955 Tf 11.95 0 Td[(zj2d() forsomeuniquepositiveBorelmeasureonT.Thesetfg2TisthesetofAleksan-drovmeasures,orClarkmeasuresifthefunction'isinner. Now,wewillenumerateseveralimportantquantitiesassociatedwithAleksandrov-Clarkmeaures:thetotalvariation,absolutelycontinuouspart,andcarriers. Proposition2.7.3. [ 4 ,p.204]IfisanAleksandrovmeasure,thenitstotalvariationisgivenby kk=1)-222(j'(0)j2 j)]TJ /F3 11.955 Tf 11.96 0 Td[('(0)j2.(2) Denition2.7.4. [ 4 ,p.205]ForanAleksandrovmeasure,,let d=hdm+ds,h2L1,s?m,(2) betheLebesguedecompositionofwithrespecttom.Observethath=d dmm-a.e. Proposition2.7.5. [ 4 ,p.205]Form-a.e.2T, h()=1)-222(j'()j2 j)]TJ /F3 11.955 Tf 11.95 0 Td[('()j2.(2) Itfollowsfromthepreviouspropositionthatif'isinner,thentheresultingClarkmeasureissingulartoLebesguemeasure. Corollary2.7.6. [ 4 ,p.205]If'isaninnerfunction,then?mforevery2T. Acarrierofameasureisageneralizationofthesupportofameasure.Thesupportforameasureisalwaysacarrier,howeveracarrierdoesnotneedtobeclosed. Denition2.7.7. [ 4 ,p.16]ForaBorelmeasure,aBorelsetHTforwhich(H\A)=(A)forallBorelsubsetsATiscalledacarrierof. ThecarrierfortheabsolutelycontinuouspartofanAleksandrovmeasureisapparentfrom( 2 ),itissimplythesetwherej'()j<1.ThecarrierforthesingularpartofanAleksandrovmeasureisdenedbelow. 26

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Proposition2.7.8. [ 4 ,p.207] (1) LetGbetheBorelsetforwhichlimr!1)]TJ /F3 11.955 Tf 8.24 5.81 Td[('(r)existsanddene'()=limr!1)]TJ /F3 11.955 Tf 8.25 5.81 Td[(G()'(r).ThenforanAleksandrovmeasuretheset('))]TJ /F7 7.97 Tf 6.58 0 Td[(1(fg)isacarrierfors. (2) Fors-a.e.2T,'()=. AnimportanttheoremconcerningAleksandrovmeasuresisAleksandrov'sdisintegrationtheorem. Theorem2.7.9(Aleksandrov'sdisintegrationtheorem). [ 4 ,p.216]Forf2L1ZTf()dm()=ZTZTf()ddm(). ThemostimportantresultaboutAleksandrovmeasuresthatwewillbeusingistheirrelationshiptothenormofC'onthereproducingkernelsofH2(D).In[ 3 ],CimaandMathesonfoundthattheradiallimitofC'~krisequaltothetotalvariationofthesingularpartoftheAleksandrovmeasure. Proposition2.7.10. [ 3 ]Let':D!Dbeholomorphicand2T,thenksk=limr!1)]TJ /F10 11.955 Tf 8.24 19.56 Td[(C'~kr2. Proof. [ 3 ]BytheLebesguedecompositionoftheAleksandrovmeasure,ksk=kk)]TJ /F10 11.955 Tf 20.59 16.27 Td[(ZTh()dm()=<+'(0) )]TJ /F3 11.955 Tf 11.95 0 Td[('(0))]TJ /F10 11.955 Tf 11.96 16.27 Td[(ZT1)-222(j'()j2 j)]TJ /F3 11.955 Tf 11.95 0 Td[('()j2dm(). Ontheotherhand,simplycomputingthenormofC'~kryieldsC'~kr2=ZT1)]TJ /F4 11.955 Tf 11.96 0 Td[(r2 j)]TJ /F4 11.955 Tf 11.95 0 Td[(r'()j2dm()=ZT1)]TJ /F4 11.955 Tf 11.96 0 Td[(r2j'()j2 j)]TJ /F4 11.955 Tf 11.95 0 Td[(r'()j2dm())]TJ /F4 11.955 Tf 11.95 0 Td[(r2ZT1)-221(j'()j2 j)]TJ /F4 11.955 Tf 11.96 0 Td[(r'()j2dm()=<+r'(0) )]TJ /F4 11.955 Tf 11.95 0 Td[(r'(0))]TJ /F4 11.955 Tf 11.96 0 Td[(r2ZT1)-222(j'()j2 j)]TJ /F4 11.955 Tf 11.96 0 Td[(r'()j2dm(). 27

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Sincej'(0)j<1,<+r'(0) )]TJ /F4 11.955 Tf 11.95 0 Td[(r'(0)convergesto<+'(0) )]TJ /F5 7.97 Tf 6.59 0 Td[('(0)uniformlyinasrincreasesto1.Nowif0
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Figure2-1. Carlesonwindow Figure2-2. Carlesoncircle (1) thereisaconstantK<1suchthat(W(,h))
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MoreinformationaboutCarlesonregionsandtheoremscanbefoundinGarnett[ 9 ];CowenandMacCluer[ 7 ]provideamoreindepthdiscussionofhowtheyrelatetocompositionoperators. 2.9Non-tangentialLimitsandAngularDerivatives Theideasofradiallimitsandderivativesarewell-known.Inthissection,webrieydiscussageneralizationofthoseideastonon-tangentiallimitsandangularderivatives. Denition2.9.1. For2Tand>1wedeneanon-tangentialapproachregionatby\(,)=fz2D:jz)]TJ /F3 11.955 Tf 11.95 0 Td[(j<(1)-222(jzj)g. Notethat\(,)isateardropshapedregionwithitsvertexat,hencethenamenon-tangential:ifz!afromwithin\(,),thenzcannotapproachalongacurvethatistangenttothecircleat.Anon-tangentiallimitissimplyalimitwherezapproachesfromwithinanon-tangentialapproachregion,ormoreformally: Denition2.9.2. [ 7 ,p.50]Afunctionfissaidtohaveanon-tangentiallimitAatiflimz!f(z)=Aineachnon-tangentialregion\(,),written\limz!f(z)=A. Nowwecandenetheangularderivativeaszapproachesapoint2T.Theonlydifferencebetweentheangularderivativeandthetraditionalderivativeisthattheareaofapproachisnowanon-tangentialapproachregionratherthanaradius. Denition2.9.3. Forananalyticfunction':D!Dandapoint2T,wesaythat'hasanangularderivativeat2Tifforsome2T,\limz!'(z))]TJ /F3 11.955 Tf 11.96 0 Td[( z)]TJ /F3 11.955 Tf 11.95 0 Td[( existsandisnite.Wedenotetheabovelimitwheneveritexistsby'0(). OneofthemostimportanttheoremsconcerningangularderivativesistheJulia-Caratheodorytheorem,itprovidesconditionsfortheirexistenceanduniqueness.A 30

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proofcanbefoundinCowen,MacCluer[ 7 ]orShapiro[ 23 ],andareformulationutilizingreproducingkernelscanbefoundinSarason[ 21 ]. Theorem2.9.4(Julia-Caratheodory). [ 4 ,p.28]Forananalyticfunction':D!Dand2Tthefollowingstatementsareequivalent: (1) liminfz!1)-221(j'(z)j 1)-222(jzj=<1, (2) \limz!'(z))]TJ /F3 11.955 Tf 11.95 0 Td[( z)]TJ /F3 11.955 Tf 11.95 0 Td[(='0()existsforsome2T, (3) \limz!'0()existsand\limz!'(z)=2T. Furthermore, (a) >0in(1); (b) thepointsin(2)and(3)arethesame; (c) '0()= and\limz!'0(z)='0(); (d) ifanyoftheaboveequationshold,then=\limz!1)-222(j'(z)j 1)-222(jzj. Inthecaseoftheupperhalf-plane,thedenitionsandtheoremsworkasexpectedfornon-tangentiallimitsandangularderivativesonR,howeversomecareneedstobetakenwiththeboundarypointat1.ThedenitionsandtheorembelowareduetoElliottandJuryin[ 8 ]. Denition2.9.5. [ 8 ]Asequenceofpointszn=xn+iyninHapproaches1non-tangentiallyifxn!1andtheratiosjynj xnareuniformlybounded. Wesayamap':H!Hxesinnitynon-tangentiallyif'(zn)!1wheneverzn!1non-tangentiallyandwewrite'(1)=1. If'(1)=1,wesaythat'hasniteangularderivativeat1ifthenon-tangentiallimit limz!1z '(z)z2H(2) existsandisnite,andwewrite'0(1). 31

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In( 2 ),thequotientmayappeartobeupside-down,usuallywhenwetakederivativestheterminvolvingthefunctionisinthenumerator.However,inthiscaserememberthatz!1ratherthanzero,sothisquotientisexactlywhatwewant. If :D!Disthemapthatisconjugateto'viatheCayleytransform,thatis =J)]TJ /F7 7.97 Tf 6.59 0 Td[(1'J,thentheexistenceofthelimitin( 2 )isequivalenttotheexistenceofthelimit\limz!11)]TJ /F3 11.955 Tf 11.96 0 Td[( (z) 1)]TJ /F4 11.955 Tf 11.96 0 Td[(zz2D. BytheJulia-CaratheodoryTheorem 2.9.4 ,thislimitisequalto\limz!1 0(z).WehavethefollowingJulia-Caratheodorytheoremforangularderivativesat1inthehalf-plane. Theorem2.9.6(Julia-CaratheodoryinH). [ 8 ]Forananalyticfunction':H!H,thefollowingareequivalent: (1) '(1)=1and'0(1)exists, (2) supz2H
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CHAPTER3ORIGINALPROOFSOFBOUNDEDABOVE InthischapterwewillreviewtheoriginalproofsthatcompositionoperatorsareboundedaboveonH2(D)andH2(H).ThoughLittlewoodoriginallyprovedtheupperboundforcompositionoperatorsH2(D),theproofhereistakenfromShapiroin[ 23 ,Section1.3].Theessenceoftheproofisthesame,howeverShapirousesthemodernnotationforobjects.TheproofthatcompositionoperatorsareboundedaboveonH2(H)isveryrecent,from2010byElliottandJuryin[ 8 ];theproofinthischapteristakendirectlyfromtheirpaper. 3.1BoundedAboveontheDisk IthasbeenknownforalongtimethatcompositionoperatorsareboundedaboveonH2(D).Thoughcompositionoperatorswerenotstudiedassuchuntilaboutthe1960s,theproofoftheirboundednessharkensbacktoLittlewood'stheoremfrom1925[ 12 ]. Theorem3.1.1(Littlewood'sTheorem). [ 23 ,p.16]Suppose':D!Disholomorphic,thenC'isboundedonH2(D)andkC'ks 1+j'(0)j 1)-222(j'(0)j. Toprovethistheorem,weneedtostartwiththesimplercasewhere'(0)=0.Thisisanassumptionwewillmakeseveraltimesthroughoutthispaper,andisnotfatal.Inmostcaseswecanlifttothemoregeneralcasewhere'(0)=aforsomevaluea2Dbyconjugatingwiththeconformalautomorphismofthediska,asmentionedinSection 2.2 Theorem3.1.2(Littlewood'sSubordinationPrinciple). [ 23 ,p.13]Let':D!Dbeaholomorphicfunctionwith'(0)=0,thenforeachf2H2(D),C'f2H2(D)andkC'fkkfk. 33

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ThisproofofLittlewood'ssubordinationprincipleistakenfromShapiro[ 23 ].Inadditiontothecompositionoperator,thisproofwillrequiretwoadditionaloperatorsonH2(D).Firstisthemultiplicationoperator: Denition3.1.3. Forananalyticfunction':D!D,wedenethemultiplicationoperatorM'onH2(D)byM'f='ff2H2(D). Theonlypropertywewillneedofmultiplicationoperatorswithsymbol':D!Disthattheyarecontractive,inotherwordsthat kM'fkkfkforf2H2(D).(3) Additionally,wewillneedthebackwardshiftoperatoronH2(D).Recallfrom( 1 )thatforf2H2(D),wecanwritefasapowerseries1Xn=0anznwhere1Xn=0janj2<1. Denition3.1.4. Letf(z)=1Xn=0anznbeafunctioninH2(D).ThebackwardshiftoperatorBonH2(D)isdenedby(Bf)(z)=1Xn=0an+1znf2H2(D). ThenamecomesfromthefactthatBshiftsthepowerseriescoefcientsofftotheleftoneunitanddropstheconstantterm.Infact,wecanrewritethefunctionfas f(z)=f(0)+zBf(z)z2D.(3) FromthedenitionitisobviousthatBisacontractiononH2(D),kBfkkfkforeveryf2H2(D).Wewouldexpectthistobeimportantintheproof,sinceweareprovingthatC'isacontractionwhen'(0)=0,howeverthisisnotthecase. ProofofLittlewood'sSubordinationPrinciple. [ 23 ,p.13]Themaintoolinthisproofisthebackwardshift. 34

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Webeginwiththesimplifyingassumptionthatfisapolynomial.Thenf'isboundedonD,soitisinH2(D);thisleavesustoprovethenormestimate.Tothisend,substitute'(z)forzin( 3 )toobtainf('(z))=f(0)+'(z)(Bf)('(z))forz2D. WecanrewritethisequationintermsoftheoperatorsC',M',andBtoget C'f=f(0)+M'C'Bf.(3) Ifwewrite'asapowerseries,allthetermsmusthaveacommonfactorofzsince'(0)=0,thereforeallthetermsinM'C'Bfdoaswell.ItfollowsthatM'C'Bfisorthogonaltotheconstantfunctionf(0)inH2(D).Thus,computingthenormofC'fgiveskC'fk2=jf(0)j2+kM'C'Bfk2. SinceM'iscontractive( 3 ),kM'C'BfkkC'Bfkandwehave kC'fkjf(0)j2+kC'Bfk.(3) Next,successivelyapply( 3 )toBf,B2f,B3f,...toobtainthefollowingstringofinequalitieskC'Bfk2jBf(0)j2+C'B2f2C'B2f2B2f(0)2+C'B3f2...kC'Bnfk2jBnf(0)j2+C'Bn+1f2. Puttingallthesetogetherwith( 3 ),weget kC'fk2nXk=0(Bkf)(0)2+C'Bn+1f2(3) 35

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foreverynonnegativeintegern. Byassumptionfisapolynomial,soletndenotethedegreeoff.Ifweshiftfn+1timeswewillgetzero,henceBn+1f=0.Therefore( 3 )reducestokC'fk2nXk=0(Bkf)(0)2. Notethat(Bkf)(0)=ak,sofromthepreviousinequalitywehavethatkC'fk2nXk=0jakj2=kfk2. ThisshowsthatC'isacontractiononthevectorsubspaceofpolynomialsofH2(D).Thefactthatthisholdsforallfunctionsf2H2(D)followsfromthefactthatconvergenceintheH2(D)normimpliesuniformconvergenceoncompactsubsetsofD,see[ 23 ,pg.15]fordetails. TheproofofLittlewood'stheoremfollowseasilyfromLittlewood'ssubordinationprinciple.Totakecareoftheassumption'(0)=0,wewillrstlookatthenormofacompositionoperatorwithsymbolw,oneofourautomorphismsofthedisk.Thentheself-inversepropertyofwplussomebasicoperatortheoryresultswillyieldtheconclusion. ProofofLittlewood'sTheorem. [ 23 ,p.16]Supposethat'(0)=wforsomew2D.FirstweneedtondthenormofCwwherew(z)=w)]TJ /F4 11.955 Tf 11.95 0 Td[(z 1)]TJ ET q .478 w 277.27 -459.35 m 286.91 -459.35 l S Q BT /F4 11.955 Tf 277.27 -466.67 Td[(wz.Iff2H2(D)isholomorphicinaneighborhoodoftheclosedunitdisk,sayr0Dforsomer0>1,thenkfk2=limr!1)]TJ /F10 11.955 Tf 8.24 22.08 Td[(ZTjf(r)j2dm()=ZTjf()j2dm(). ApplyingthecompositionoperatorCwtofwehavekCwfk2=kfwk2=ZTjf(w())j2dm(), 36

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andusingthechangeofvariables=w()yields kCwfk2=ZTjf()j2j0w()jdm()=ZTjf()j21)-222(jwj2 j1)]TJ ET q .478 w 269.87 -80.62 m 279.51 -80.62 l S Q BT /F4 11.955 Tf 269.87 -87.94 Td[(wj2dm()1)-222(jwj2 (1)-222(jwj)2ZTjf()j2dm()=1+jwj 1)-222(jwjkfk2.(3) Therefore,thedesiredresultholdsforallfunctionsanalyticinr0D,andhenceforpolynomials.Thisholdsforallfunctionsf2H2(D)bythesameargumentasintheproofofLittlewood'ssubordinationprinciple. Finally,considerthefunction =w'.Sincewisaself-inverseautomorphismofthedisk,wealsohavethat'=w ,henceC'=CwC .Underthisdenition (0)=0,sobyLittlewood'ssubordinationprinciple 3.1.2 ,kC k1.Combiningthiswithequation( 3 )yieldskC'kkCwkkC ks 1+jwj 1)-222(jwj. 3.2BoundedAboveontheHalf-Plane ResultsaboutcompositionoperatorsonH2(H)arerelativelyrecent,duetoElliottandJuryin2010[ 8 ].Inthehalf-planecase,boundednessofthecompositionoperatorC'ischaracterizedbytheangularderivativeof'atthepoint1.ThisdiscussionisfromElliottandJury'spaper[ 8 ]. Theorem3.2.1(Elliott,Jury). [ 8 ]Let':H!Hbeholomorphic.ThecompositionoperatorC'isboundedonH2(H)ifandonlyif'hasniteangularderivative0<<1atinnity,inwhichcasekC'k=p TheproofofElliottandJuryhingesonrestatingtheexistenceoftheangularderivativeintermsofthepositivityofakernel.RecallfromtheJulia-Caratheodoryin 37

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thehalf-plane 2.9.6 that'0(1)existsifandonlyifsupz2H
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'(w)))]TJ /F7 7.97 Tf 6.58 0 Td[(1ispositivesinceitfactorsashk'(w),k'(z)i.Thekernel('(z))]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1z)+ ('(w))]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1w) z+ w ispositiveby( 3 ).Thushkw,kzi)]TJ /F10 11.955 Tf 19.26 9.68 Td[(C'kw,C'kzisapositivekernel,asdesired. Fortheotherdirection,supposethatC'isboundedabovebyM.Thenforeachz2H, kC'kzk2M2kkzk2(3) However,kkzk2=1 2
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Asimilarobservationcanbemadeabouttheproofofboundednessfromaboveintheupperhalf-plane,Theorem 3.2.1 .Though,ElliottandJurydoutilizetheangularderivativeof'at1,whichisapurelyfunctiontheoreticproperty,theygoaboutprovingthatitisboundedusingHilbertspacetechniques.Theactionoftheadjointofacompositionoperatoronkernelfunctionshassurprisingimportance,andprovingthattheangularderivativeisboundedreducestoshowingakernelispositive. OurgoalinthenextchapteristoprovidepurelyfunctiontheoreticproofsofbothoftheseresultsbyutilizingtheNevanlinnacountingfunction.Thekeyisexpressingthecountingfunctionasaintegralthatiseasytobound.MuchoftheworkonthediskhasalreadybeendonebyJoelShapiroin[ 22 ]and[ 25 ],howevertheintegralformulaforthehalf-planecountingfunctionanditsresultsarenew. 40

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CHAPTER4BOUNDEDABOVEANDTHECOUNTINGFUNCTION InthischapterwewilldevelopanintegralformulafortheNevanlinnacountingfunctiononthehalf-plane( 2.5.2 )followingasimilarmethodtoShapiro'sfrom[ 24 ].WewillusethisnewformulatoprovideapurelyfunctiontheoreticproofthatcompositionoperatorsareboundedaboveonH2(H). 4.1NevanlinnaCountingFunctionontheDisk First,wewillreviewShapiro'sconstructionofanintegralformulafortheNevanlinnacountingfunctiononthediskfrom[ 22 24 ].Hisformulaprovedtobeincrediblyuseful;itenabledhimtoprovideafunctiontheoreticproofthatcompositionoperatorsareboundedaboveonH2(D)andtocharacterizewhenequalityoccursinLittlewood'sinequality. RecallfromSection 2.5 theNevanlinnacountingfunction: Denition4.1.1. Foraholomorphicmap':D!D,denetheNevanlinnacountingfunctionof'byN'(w)=Xz2')]TJ /F13 5.978 Tf 5.76 0 Td[(1(w)log1 jzj forallw2'(D)nf'(0)g,where')]TJ /F7 7.97 Tf 6.58 0 Td[(1(w)denotesthesetof'-preimagesofwcountedaccordingtotheirmultiplicity,andN'(w)=0ifw=2'(D). AnimportantauxiliaryfunctionwewillneedistheFrostmantransformof'fromSection 2.3 ,'w=w'forw2D.ThenicethingabouttheFrostmantransformisthat')]TJ /F7 7.97 Tf 6.58 0 Td[(1w(0)=')]TJ /F7 7.97 Tf 6.58 0 Td[(1(w).AswesawinChapter 3 ,sometimesitiseasiertoprovepropertiesundertheassumptionthat'(0)=0rstandthenliftthemtothemoregeneralcase.TheFrostmantransformeasesthistransitionsinceif'(0)=wforsomew2D,then'w(0)=0. Denition4.1.2. Givenaholomorphicmap':D!D,foreveryw2DdeneitsFrostmantransformby'w(z)=w)]TJ /F3 11.955 Tf 11.95 0 Td[('(z) 1)]TJ ET q .478 w 251.23 -636.46 m 260.87 -636.46 l S Q BT /F4 11.955 Tf 251.23 -643.78 Td[(w'(z) 41

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forz2D. NowwecanstateandproveShapiro'sintegralformula;itoriginallyappearedin[ 22 ]. Proposition4.1.3(Shapiro,1987). Let':D!Dbeaholomorphicfunction,thenN'(w)=limr!1)]TJ /F10 11.955 Tf 8.25 22.09 Td[(ZTlogj'w(r)jdm())]TJ /F6 11.955 Tf 11.95 0 Td[(logj'w(0)j wheremdenotesnormalizedLebesgueonthecircleT. ThisversionofShapiro'sproofistakenfromhisunpublishedmanuscript[ 24 ]. Proof. [ 24 ]First,supposethat'(0)6=0.Arrangethe'-preimagesof0inasequenceinorderofincreasingmoduliandrepeatedaccordingtotheirmultiplicity.Wedenotethissequenceby')]TJ /F7 7.97 Tf 6.58 0 Td[(1(0)=(z1,z2,z3,...).Thesequence')]TJ /F7 7.97 Tf 6.59 0 Td[(1(0)maybeinnite,sorstwewillconsiderpreimagesinacircleofradiusr<1.Tothisend,letn(r)bethenumberofpreimageswithmultiplicitycountedthathavemodulusr.UsingJensen'sformula[ 20 ,p.307],wehavethatn(r)Xj=1logr jzjj=ZTlogj'(r)jdm())]TJ /F6 11.955 Tf 11.95 0 Td[(logj'(0)j. Wetakethelimitasr%1ofthisequation.Sincelogj'jissubharmoniconthedisk,theintegralontherighthandsideincreasesasrincreasesto1,butstaysnitebecause'isboundedonD.Thesumonthelefthandsideincreasesto1Xj=1log1 jzjjasrincreasestoonebymonotoneconvergence.Therefore,wehavethat 1Xj=1log1 jzjj=limr!1)]TJ /F10 11.955 Tf 8.25 22.08 Td[(ZTlogj'(r)jdm())]TJ /F6 11.955 Tf 11.96 0 Td[(logj'(0)j.(4) If0=2'(D)wedenetheemptysumonthelefthandsideof( 4 )tobezero,asinthedenitionofN'.Inthiscasetheintegralontherighthandsideisharmonic,soisequaltozerobythemeanvaluetheorem.Hence,equalityholdsif0=2'(D).If'(0)=0,boththeleftandrighthandsidescanbeinterpretedasbeing+1;thereforeequalityalso 42

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holdsif'(0)=0.ThisgivesusthatN'(0)=limr!1)]TJ /F10 11.955 Tf 8.25 22.08 Td[(ZTlogj'(r)jdm())]TJ /F6 11.955 Tf 11.95 0 Td[(logj'(0)j. Finally,togettheintegralformulaforthepreimageofanypointwinthedisk,simplyapplytheintegralformulato'wratherthan'.Notethat')]TJ /F7 7.97 Tf 6.59 0 Td[(1w(0)=')]TJ /F7 7.97 Tf 6.59 0 Td[(1(w),soN'(w)=N'w(0).Hence,N'(w)=N'w(0)=limr!1)]TJ /F10 11.955 Tf 8.25 22.08 Td[(ZTlogj'w(r)jdm())]TJ /F6 11.955 Tf 11.96 0 Td[(logj'w(0)j. Littlewood'sinequalityisaneasyconsequenceofProposition 4.1.3 Theorem4.1.4(Littlewood'sInequality). Foraholomorphicfunction':D!D, N'(w)log1 j'w(0)j=log1)]TJ ET q .478 w 297.8 -277.47 m 307.44 -277.47 l S Q BT /F4 11.955 Tf 297.8 -284.79 Td[(w'(0) w)]TJ /F3 11.955 Tf 11.95 0 Td[('(0)(4) forallw2D. Proof. InProposition 4.1.3 ,notethattheintegrandontherighthandsideisnonpositive,thereforeN'(w)log1 j'w(0)j=log1)]TJ ET q .478 w 295.06 -391.04 m 304.7 -391.04 l S Q BT /F4 11.955 Tf 295.06 -398.36 Td[(w'(0) w)]TJ /F3 11.955 Tf 11.95 0 Td[('(0). Shapiro'sresultalsoprovidesapurelyfunctiontheoreticproofofLittlewood'sTheorem 3.1.1 .AsinChapter 3 ,webeginwithLittlewood'ssubordinationprinciple,andthenuseautomorphismsofthedisktogetthegeneralresult. Theorem4.1.5(Littlewood'sSubordinationPrinciple). Let':D!Dbeaholomorphicfunction,with'(0)=0,thenforeachf2H2(D),C'f2H2(D)andkC'fkkfk. 43

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Proof. Foreveryf2H2(D)bythechangeofvariablesformulawith'(0)=0,wehavethatkC'fk2=kf'k2=2ZDjf0(z)j2N'(z)dA(z)+jf(0)j2 whereAdenotesnormalizedLebesgueareameasureonthediskD.ApplyingLittlewood'sinequality( 4 )andtheLittlewood-Paleyidentity( 2 )yieldskC'fk22ZDjf0(z)j2log1 jzjdA(z)+jf(0)j2=kfk2. ThiscaneasilybeextendedtoLittlewood'stheorem: Theorem4.1.6(Littlewood'sTheorem). Suppose':D!Disholomorphic,thenC'isboundedonH2(D)andkC'ks 1+j'(0)j 1)-222(j'(0)j. Proof. ThisproofworksidenticallytothatinSection 3.1 Asaconsequence,in[ 25 ]ShapirowasabletoprovideacharacterizationinnerfunctionsbasedonpropertiesoftheNevanlinnacountingfunctionN'. Theorem4.1.7(Shapiro,1999). Foraholomorphicself-map'ofD,thefollowingareequivalent: (a) 'isinner. (b) N'(w)=log1 j'w(0)jforsomew2D. (c) N'(w)=log1 j'w(0)jforquasi-everyw2D. Proof. [ 24 ]WebeginwiththeintegralformulafromProposition 4.1.3 ,N'(w)=limr!1)]TJ /F10 11.955 Tf 8.25 22.08 Td[(ZTlogj'w(r)jdm())]TJ /F6 11.955 Tf 11.95 0 Td[(logj'w(0)j. 44

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Theintegrandontherighthandsideisnonpositive,sobyFatou'slemma, N'(w)ZTlogj'w()jdm())]TJ /F6 11.955 Tf 11.95 0 Td[(logj'w(0)j(4) where'w()indicatestheradiallimitof'wat. SupposethatN'(w)=log1 j'w(0)jforsomew2D.Thenforthatw,ZTlogj'w()jdm()=0. Sincetheintegrandisnonpositive,itmustvanishalmosteverywhereonD,thatislog1 j'w()j=0foralmostevery2T.Thisoccurswhenj'w()j=1foralmostevery2T.Therefore,'wmustbeaninnerfunction,andwecanconcludethat'isalsoaninnerfunction.Thisprovestheimplication(b))(a). Toprove(a))(c),supposethat'isinner.ByFrostman'sTheorem 2.3.2 ,'wisaBlaschkeproductforquasi-everyw2D.Hence,limr!1)]TJ /F10 11.955 Tf 8.25 22.08 Td[(ZTlogj'w(r)jdm()=0 forquasi-everyw2DbyTheorem 2.1.6 .Therefore,byProposition 4.1.3 ,N'(w)=log1 j'w(0)j forquasi-everyw2D. Finally,theimplication(c))(b)isobvious;ifequalityholdsforquasi-everyw2D,thenitcertainlyholdsforonew2D. 4.2CountingFunctionintheHalf-Plane WewishtoemployasimilarmethodasShapiro'stoobtainanintegralformulaforacountingfunctionontheupperhalf-plane.Wewillusethecountingfunctionforthehalf-planedenedinSection 2.5 .AswiththeNevanlinnacountingfunctionforthedisk,thiscountingfunctionweightseach'-preimagebytheproductofitsmultiplicityanditsdistancetotheboundaryoftheupperhalf-planeR. 45

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Denition4.2.1. Foraholomorphicmap':H!H,denethecountingfunctionof'onHbyN'(w)=Xz2')]TJ /F13 5.978 Tf 5.76 0 Td[(1(w)=z forallw2'(H),where')]TJ /F7 7.97 Tf 6.58 0 Td[(1(w)denotesthesetof'-preimagesofwcountedaccordingtotheirmultiplicity,andN'(w)=0ifw=2'(H). Next,weneedtodeneaFrostmantransformofthehalf-plane.Inthediskcase,theFrostmantransformofafunctionfwassimplyfw=wf,anautomorphismofthediskcomposedwithf.Wehopetoachievesomethingsimilarhere.Mobiustransformationsthatareautomorphismsoftheupperhalf-planecanonlydilateavaluez2Hbyapositiverealnumberaorshiftitrightorleftbyaddingarealconstantb.Thatis,automorphismsoftheupperhalf-planelooklikeaz+bwherea>0andb2R. Denition4.2.2. Givenaholomorphicmap':H!H,foreveryw2HdeneitsFrostmanTransformby'w(z)=1 =w'(z))]TJ 13.15 8.08 Td[(
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Theorem4.2.3(Ahern,Clark,1974). [ 1 ]LetfbeaholomorphicfunctiononDboundedinmodulusby1,thenforall2T,jf0()j=Xn1)-222(janj2 j)]TJ /F4 11.955 Tf 11.95 0 Td[(anj2+2ZT1 j)]TJ /F3 11.955 Tf 11.96 0 Td[(j2d(), wherefangistheBlaschkesequenceforfandisdenedasabove. WhenwetransportAhernandClark'sconditiontotheupperhalf-plane,somethinginterestinghappens.Theangularderivativetermreducestotheimaginarypartofanumberw2H,andtheBlaschkeproducttermreducestoPz2')]TJ /F13 5.978 Tf 5.76 0 Td[(1(w)=z,theNevalinnacountingfunctionforthehalf-plane. Theorem4.2.4. Let':H!Hbeholomorphicsuchthat'(1)=1and'0(1)=1,thenforq.e.w2HN'(w)=Xz2')]TJ /F13 5.978 Tf 5.75 0 Td[(1(w)=z==w+1 2ZRlogj'w(t)jdt. Notethatweareassuming'xes1and'0(1)=1.RecallthattheseweretheconditionsnecessaryforacompositionoperatortobeboundedaboveonH2(H)by 3.2.1 .Toprovethistheorem,wewillmakeextensiveuseoftheCayleytranformfromSection 2.2 :J(z)=i1+z 1)]TJ /F4 11.955 Tf 11.95 0 Td[(z, whichconformallymapstheunitdisktotheupperhalfplane,anditsinverseJ)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z)=z)]TJ /F4 11.955 Tf 11.95 0 Td[(i z+i, 47

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whichconformallymapstheupperhalf-planetotheunitdisk.Webeginwithasimplecomputation;letb2D,then =J(b)=1 2i(J(b))]TJ ET q .478 w 208.38 -57.4 m 231.32 -57.4 l S Q BT /F4 11.955 Tf 208.38 -68.04 Td[(J(b))=1 2ii1+b 1)]TJ /F4 11.955 Tf 11.96 0 Td[(b)-222()]TJ /F4 11.955 Tf 21.26 0 Td[(i1+ b 1)]TJ ET q .478 w 290.64 -98.41 m 297.1 -98.41 l S Q BT /F4 11.955 Tf 290.64 -108.39 Td[(b=1 21+b 1)]TJ /F4 11.955 Tf 11.96 0 Td[(b+1+ b 1)]TJ ET q .478 w 239.29 -132.53 m 245.75 -132.53 l S Q BT /F4 11.955 Tf 239.29 -142.51 Td[(b=1 2(1+b)(1)]TJ ET q .478 w 236.93 -149.19 m 243.39 -149.19 l S Q BT /F4 11.955 Tf 236.93 -159.17 Td[(b)+(1+ b)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(b) j1)]TJ /F4 11.955 Tf 11.96 0 Td[(bj2=1 2 1)]TJ ET q .478 w 195.6 -186.81 m 202.06 -186.81 l S Q BT /F4 11.955 Tf 195.6 -196.79 Td[(b+b)-222(jbj2+1)]TJ /F4 11.955 Tf 11.96 0 Td[(b+ b)-222(jbj2 j1)]TJ /F4 11.955 Tf 11.95 0 Td[(bj2!=1)-222(jbj2 j1)]TJ /F4 11.955 Tf 11.96 0 Td[(bj2.(4) InthisproofwewilluseamodiedversionoftheFrostmantransformfromthedisk.Weinsertanadditionalunimodularfactor1)]TJ ET q .478 w 221.54 -283.18 m 228 -283.18 l S Q BT /F4 11.955 Tf 221.54 -293.16 Td[(b b)]TJ /F6 11.955 Tf 11.96 0 Td[(1toensurethat b(z)=1whenever (z)=1. Denition4.2.5. Givenaholomorphicmap :D!D,foreveryb2DdeneitsmodiedFrostmantransformby b(z)=1)]TJ ET q .478 w 236.63 -366.86 m 243.09 -366.86 l S Q BT /F4 11.955 Tf 236.63 -376.84 Td[(b b)]TJ /F6 11.955 Tf 11.96 0 Td[(1b)]TJ /F3 11.955 Tf 11.96 0 Td[( (z) 1)]TJ ET q .478 w 266.37 -384.33 m 272.84 -384.33 l S Q BT /F4 11.955 Tf 266.37 -394.3 Td[(b (z) forz2D. Lemma4.2.6. Supposethat':H!Hisholomorphic,andlet denotethefunctioninthediskthatisconjugatetoHviatheCayleytransform,'=J J)]TJ /F7 7.97 Tf 6.59 0 Td[(1,then'w=J bJ)]TJ /F7 7.97 Tf 6.59 0 Td[(1whereb=J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(w). 48

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Proof. Letz2H,then(J bJ)]TJ /F7 7.97 Tf 6.58 0 Td[(1)(z)=J1)]TJ ET q .478 w 119.25 -27.38 m 125.71 -27.38 l S Q BT /F4 11.955 Tf 119.25 -37.36 Td[(b b)]TJ /F6 11.955 Tf 11.95 0 Td[(1b)]TJ /F3 11.955 Tf 11.95 0 Td[( (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z)) 1)]TJ ET q .478 w 148.99 -44.85 m 155.45 -44.85 l S Q BT /F4 11.955 Tf 148.99 -54.82 Td[(b (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z))=i0@1+1)]TJ ET q .359 w 130.3 -62.99 m 134.87 -62.99 l S Q BT /F12 7.97 Tf 130.3 -69.78 Td[(b b)]TJ /F7 7.97 Tf 6.59 0 Td[(1b)]TJ /F5 7.97 Tf 6.59 0 Td[( (J)]TJ /F13 5.978 Tf 5.75 0 Td[(1(z)) 1)]TJ ET q .359 w 148.28 -72.75 m 152.85 -72.75 l S Q BT /F12 7.97 Tf 148.28 -79.54 Td[(b (J)]TJ /F13 5.978 Tf 5.76 0 Td[(1(z)) 1)]TJ /F7 7.97 Tf 13.15 4.7 Td[(1)]TJ ET q .359 w 130.3 -85.2 m 134.87 -85.2 l S Q BT /F12 7.97 Tf 130.3 -91.99 Td[(b b)]TJ /F7 7.97 Tf 6.59 0 Td[(1b)]TJ /F5 7.97 Tf 6.59 0 Td[( (J)]TJ /F13 5.978 Tf 5.75 0 Td[(1(z)) 1)]TJ ET q .359 w 148.28 -94.96 m 152.85 -94.96 l S Q BT /F12 7.97 Tf 148.28 -101.75 Td[(b (J)]TJ /F13 5.978 Tf 5.76 0 Td[(1(z))1A(b)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(1)]TJ ET q .478 w 270.24 -67.91 m 276.7 -67.91 l S Q BT /F4 11.955 Tf 270.24 -77.89 Td[(b (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z))) (b)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(1)]TJ ET q .478 w 270.24 -85.38 m 276.7 -85.38 l S Q BT /F4 11.955 Tf 270.24 -95.35 Td[(b (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z)))=i(b)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(1)]TJ ET q .478 w 158.42 -109.21 m 164.88 -109.21 l S Q BT /F4 11.955 Tf 158.42 -119.18 Td[(b (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z)))+(1)]TJ ET q .478 w 262.51 -109.21 m 268.97 -109.21 l S Q BT /F4 11.955 Tf 262.51 -119.18 Td[(b)(b)]TJ /F3 11.955 Tf 11.96 0 Td[( (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z))) (b)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(1)]TJ ET q .478 w 158.42 -126.67 m 164.88 -126.67 l S Q BT /F4 11.955 Tf 158.42 -136.65 Td[(b (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z))))]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ ET q .478 w 262.51 -126.67 m 268.97 -126.67 l S Q BT /F4 11.955 Tf 262.51 -136.65 Td[(b)(b)]TJ /F3 11.955 Tf 11.96 0 Td[( (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)))=i b)-222(jbj2 (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z)))]TJ /F6 11.955 Tf 11.95 0 Td[(1+ b (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z))+b)]TJ /F3 11.955 Tf 11.95 0 Td[( (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)))-222(jbj2+ b (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)) b)-222(jbj2 (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)))]TJ /F6 11.955 Tf 11.96 0 Td[(1+ b (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)))]TJ /F4 11.955 Tf 11.96 0 Td[(b+ J)]TJ /F7 7.97 Tf 6.59 0 Td[(1+jbj2)]TJ ET q .478 w 406.26 -163.96 m 412.72 -163.96 l S Q BT /F4 11.955 Tf 406.26 -173.94 Td[(b (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z))!=i )-166(jbj2+b)]TJ /F6 11.955 Tf 11.95 0 Td[(1+ b)]TJ ET q .478 w 203.17 -186 m 209.63 -186 l S Q BT /F4 11.955 Tf 203.17 -195.97 Td[(b+b)-222(jbj2 (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z))+ b (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)) )-166(jbj2 (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)))]TJ /F6 11.955 Tf 11.95 0 Td[(1+ (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z))+jbj2!+i)]TJ /F3 11.955 Tf 9.3 0 Td[( (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z))+b (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)))]TJ /F4 11.955 Tf 11.96 0 Td[(b (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z))+ b (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)) )-166(jbj2 (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)))]TJ /F6 11.955 Tf 11.96 0 Td[(1+ (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z))+jbj2=i )-166(j1)]TJ /F4 11.955 Tf 11.95 0 Td[(bj2(1+ (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)))+(b)]TJ ET q .478 w 271.99 -261.33 m 278.45 -261.33 l S Q BT /F4 11.955 Tf 271.99 -271.31 Td[(b)(1)]TJ /F3 11.955 Tf 11.96 0 Td[( (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z))) (1)-222(jbj2)( (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z)))]TJ /F6 11.955 Tf 11.95 0 Td[(1)!=i1+ (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)) 1)]TJ /F3 11.955 Tf 11.96 0 Td[( (J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(z)) j1)]TJ /F4 11.955 Tf 11.96 0 Td[(bj2 1)-222(jbj2!)]TJ /F4 11.955 Tf 11.96 0 Td[(ib)]TJ ET q .478 w 284.17 -301.18 m 290.63 -301.18 l S Q BT /F4 11.955 Tf 284.17 -311.16 Td[(b 1)]TJ ET q .478 w 281.61 -320.72 m 288.07 -320.72 l S Q BT /F4 11.955 Tf 281.61 -330.7 Td[(b21)]TJ /F3 11.955 Tf 11.95 0 Td[( (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z)) 1)]TJ /F3 11.955 Tf 11.95 0 Td[( (J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z))=(J J)]TJ /F7 7.97 Tf 6.58 0 Td[(1)(z)1 =J(b))]TJ /F4 11.955 Tf 11.95 0 Td[(ib)]TJ ET q .478 w 219.17 -338.89 m 225.63 -338.89 l S Q BT /F4 11.955 Tf 219.17 -348.87 Td[(b 1)-222(jbj2=1 =J(b)'(z))]TJ /F4 11.955 Tf 11.95 0 Td[(ib)]TJ ET q .478 w 184.18 -374.37 m 190.64 -374.37 l S Q BT /F4 11.955 Tf 184.18 -384.35 Td[(b b)-222(jbj2=1 =J(b)'(z))]TJ 13.15 8.09 Td[(
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Proof. Since (1)=1,bythedenitionofthemodiedFrostmantransform, b(1)=1.Calculatingtheangularderivativewehave 0b(1)=\limz!1 b(1))]TJ /F3 11.955 Tf 11.95 0 Td[( b(z) 1)]TJ /F4 11.955 Tf 11.95 0 Td[(z=\limz!11)]TJ /F7 7.97 Tf 13.15 4.71 Td[(1)]TJ ET q .359 w 169.57 -82.25 m 174.14 -82.25 l S Q BT /F12 7.97 Tf 169.57 -89.04 Td[(b b)]TJ /F7 7.97 Tf 6.58 0 Td[(1b)]TJ /F5 7.97 Tf 6.58 0 Td[( (z) 1)]TJ ET q .359 w 187.55 -92.01 m 192.12 -92.01 l S Q BT /F12 7.97 Tf 187.55 -98.8 Td[(b (z) 1)]TJ /F4 11.955 Tf 11.95 0 Td[(z=\limz!1(b)]TJ /F6 11.955 Tf 11.96 0 Td[(1)(1)]TJ ET q .478 w 199.35 -119.14 m 205.81 -119.14 l S Q BT /F4 11.955 Tf 199.35 -129.11 Td[(b (z)))]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ ET q .478 w 275.44 -119.14 m 281.9 -119.14 l S Q BT /F4 11.955 Tf 275.44 -129.11 Td[(b)(b)]TJ /F3 11.955 Tf 11.95 0 Td[( (z)) (b)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(1)]TJ ET q .478 w 234.2 -136.6 m 240.66 -136.6 l S Q BT /F4 11.955 Tf 234.2 -146.58 Td[(b (z))(1)]TJ /F4 11.955 Tf 11.96 0 Td[(z)=\limz!1b)-222(jbj2 (z))]TJ /F6 11.955 Tf 11.96 0 Td[(1+ b (z))]TJ /F4 11.955 Tf 11.95 0 Td[(b+ (z)+jbj2)]TJ ET q .478 w 375.58 -154.28 m 382.05 -154.28 l S Q BT /F4 11.955 Tf 375.58 -164.26 Td[(b (b) (b)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(1)]TJ ET q .478 w 266.36 -171.75 m 272.82 -171.75 l S Q BT /F4 11.955 Tf 266.36 -181.73 Td[(b (z))(1)]TJ /F4 11.955 Tf 11.95 0 Td[(z)=\limz!1 (z)(1)-222(jbj2))]TJ /F6 11.955 Tf 11.95 0 Td[((1)-222(jbj2) (b)]TJ /F6 11.955 Tf 11.96 0 Td[(1)(1)]TJ ET q .478 w 199.61 -206.9 m 206.08 -206.9 l S Q BT /F4 11.955 Tf 199.61 -216.88 Td[(b (z))(1)]TJ /F4 11.955 Tf 11.96 0 Td[(z)=\limz!1)]TJ /F6 11.955 Tf 9.3 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[( (z))(1)-221(jbj2) (b)]TJ /F6 11.955 Tf 11.96 0 Td[(1)(1)]TJ ET q .478 w 199.35 -242.05 m 205.81 -242.05 l S Q BT /F4 11.955 Tf 199.35 -252.03 Td[(b (z))(1)]TJ /F4 11.955 Tf 11.95 0 Td[(z)=1)-222(jbj2 1)]TJ /F4 11.955 Tf 11.95 0 Td[(b\limz!11)]TJ /F3 11.955 Tf 11.96 0 Td[( (z) 1)]TJ /F4 11.955 Tf 11.96 0 Td[(z1 1)]TJ ET q .478 w 246.32 -277.2 m 252.78 -277.2 l S Q BT /F4 11.955 Tf 246.32 -287.17 Td[(b (z)=1)-222(jbj2 1)]TJ /F4 11.955 Tf 11.95 0 Td[(b 0(1)1 1)]TJ ET q .478 w 195.43 -312.35 m 201.89 -312.35 l S Q BT /F4 11.955 Tf 195.43 -322.32 Td[(b=1)-222(jbj2 j1)]TJ /F4 11.955 Tf 11.96 0 Td[(bj2 0(1)==J(b) 0(1)==w 0(1). Finallywecometotheproofoftheintegralformulaforthehalf-planecountingfunction. ProofofTheorem 4.2.4 Theproofofthisintegralformulahingesontransportingalltherequisitequantitiesinthehalf-planetotheunitdisk,applyingthetheoremofAhernandClark 4.2.3 ,andthentransportingbacktothehalf-plane.Tothisend,dene tobethefunctionconjugateto'inthedisk, =J)]TJ /F7 7.97 Tf 6.59 0 Td[(1'J.NotethatundertheCayleytransform,J(1)=1andJ)]TJ /F7 7.97 Tf 6.59 0 Td[(1(1)=1,sothatthehypothesis'(1)=1implies (1)=1.Therefore, :D!Dwith (1)=1.Twootherquantitiesneedtobedened;letb=J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(w)andfor 50

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z2Hdeneu=J)]TJ /F7 7.97 Tf 6.58 0 Td[(1(z).Now,weapplyAhern-Clarkto batthevalue=1:j 0b(1)j=X b(u)=01)-221(juj2 j1)]TJ /F4 11.955 Tf 11.95 0 Td[(uj2+2ZT1 j)]TJ /F6 11.955 Tf 11.96 0 Td[(1j2d(). Fromthesimplecomputationabove,1)-221(juj2 j1)]TJ /F4 11.955 Tf 11.95 0 Td[(uj2==J(u)==z,sothatX b(u)=01)-222(juj2 j1)]TJ /F4 11.955 Tf 11.95 0 Td[(uj2=X (u)=b1)-222(juj2 j1)]TJ /F4 11.955 Tf 11.95 0 Td[(uj2=X'(J(u))=J(b)=J(u)=X'(z)=w=z. Werescalethefunction sothatj 0(1)j=1,thenfromLemma 4.2.7 wehavethatj 0b(1)j==w.Theintegralisthemostdifcultpiecetoconsider.Firstfortheintegrand,lett=J()2RsincetheCayleytransformmapstheboundaryofDtotheboundaryofH,thenj)]TJ /F6 11.955 Tf 11.95 0 Td[(1j1=J)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t))]TJ /F6 11.955 Tf 11.96 0 Td[(12=t)]TJ /F4 11.955 Tf 11.96 0 Td[(i t+i)]TJ /F6 11.955 Tf 11.96 0 Td[(12=t)]TJ /F4 11.955 Tf 11.96 0 Td[(i)]TJ /F6 11.955 Tf 11.95 0 Td[((t+1) t+12=)]TJ /F6 11.955 Tf 9.3 0 Td[(2i t+i2=4 (t)]TJ /F4 11.955 Tf 11.96 0 Td[(i)(t+i)=4 t2+1, so1 j)]TJ /F6 11.955 Tf 11.95 0 Td[(1j2=1 4(t2+1).Next,considerthemeasured=)]TJ /F6 11.955 Tf 11.29 0 Td[(logj bjdm+d.BythegeneralizationofFrostman'stheoremduetoRudin,Theorem 2.3.3 bhasnosingularinnerfactor,sod=)]TJ /F6 11.955 Tf 11.29 0 Td[(logj bjdm.Theequivalentmeasureinthehalf-planeisfoundbyconjugating btothehalf-planeandnotingthatnormalizedLebesguemeasureonthecirclecorrespondstothemeasure1 dt 1+t2ontherealline[ 10 ,Chapter8].Puttingallof 51

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thistogetheryields=w=X'(z)=w=z+2ZR1 4(t2+1)()]TJ /F6 11.955 Tf 9.3 0 Td[(1)logj'w(t)j1 dt 1+t2=N'(w))]TJ /F6 11.955 Tf 16.68 8.08 Td[(1 2ZRlogj'w(t)jdt, andrearrangingwehavethedesiredresult. Corollary4.2.8. Let':H!Hbeholomorphicsuchthat'(1)=1and'0(1)=1,thenN'(w)=Xz2')]TJ /F13 5.978 Tf 5.76 0 Td[(1(w)=z=w forallw2H. Theorem4.2.9. Let':H!Hbeholomorphicsuchthat'(1)=1and'0(1)=1,thenC'isboundedaboveonH2(H)andkC'fkkfk forallf2H2(H). Proof. Letf2H2(H),thenfromtheLittlewood-Paleyidentityforthehalf-plane( 2 ),kC'fk2=2ZHjf0(z)j2N'(z)dA(z). BytheLittlewood-styleinequalityforthehalf-plane( 4.2.8 ),kC'fk2ZHjf0(z)j2=zdA(z)=kfk2. 52

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CHAPTER5BOUNDEDBELOW WewillintroducethreetheoremsforacompositionoperatoronH2(D)tohaveclosedrangeinthischapter.Theoriginalproofsofthesetheoremsarecompletelydisjoint;thoughallofthesetheoremsprovideconditionsforC'tohaveclosedrange,itisnotapparentfromtheirproofsthattheyarerelated.WewillproduceadirectgeometricequivalencebetweenthecriterionofCima,Thomson,andWogen[ 5 ]andthatofZorboska[ 26 ]usingtheReproducingKernelThesis. 5.1ClosedRange OneoftheearliestconditionsforacompositionoperatorC'tohaveclosedrangeonH2(D)isfromCima,Thomson,andWogenin1974[ 5 ].Theygaveaconditionthatdependsonlyonthebehaviorofthefunction'ontheboundaryofthediskT.Tounderstandtheircondition,wemustdenesomenotation.Let':D!Dbeholomorphic.First,weidentify'()where2Tastheboundarylimitofpointsinthedisk,thatis'()=limr!1)]TJ /F3 11.955 Tf 8.24 5.81 Td[('(r);thislimitexistsalmosteverywherewithrespecttoLebesguemeasurebyFatou'sTheorem 2.1.1 .Next,wedeneameasureonBorelsetsETby(E)=m(')]TJ /F7 7.97 Tf 6.59 0 Td[(1(E)).ThismeasureisabsolutelycontinuoustoLebesguemeasureonTanditsRadon-Nikodymderivatived dmisinL1(T). Theorem5.1.1(Cima,Thomson,Wogen,1974). [ 5 ]Suppose':D!Disanalyticandnonconstant.ThenC'hasclosedrangeifandonlyifd dmisessentiallyboundedawayfromzero. Attheendoftheirpaper,Cima,Thomson,andWogensaid[i]twouldbeinterestingtocharacterizethecompositionoperatorswithclosedrangebyconsideringtherangeofthemappingonDratherthanonT.In1994,NinaZorboskadidexactlythat;in[ 26 ]sheprovidedacriterionforC'tohaveclosedrangeonH2(D)baseduponpropertiesof'onCarlesonregionsinsidethedisk.HertheoremutilizestheNevanlinnacountingfunctionforthediskintheguiseofthefunction'(z)=N'(z) log1 jzjforz2Dnf'(0)g. 53

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Theorem5.1.2(Zorboska,1994). [ 26 ]AnoperatorC'onH2(D)hasclosedrangeifandonlyifthereexistsapositiveconstantcsuchthatthesetGc=fz2D:'(z)>cgsatisesthefollowingcondition: (PB)Thereexistsaconstant>0suchthatA(Gc\Sa)>A(Sa) foralla2DwhereAdenotesnormalizedLebesgueareameasureonD. Thecondition(PB)isknownasareverseCarlesoninequality.Wecallthiscondition(PB)inreferencetospreadingpeanutbutteronapieceoftoast.IftheareathepeanutbutterisspreadonisrepresentedbyGcandthetoastbySa,thenthisconditiontellsusthatthepeanutbutter,thepreimagesof'arespreadthickenoughoversomepositivefractionofthetoast. In2010,Lefevre,Li,Queffelec,andRodrguez-PiazzagaveanotherconditionforC'tohaveclosedrangeonH2(D)[ 11 ].TheirconditionrequirestheaveragemassoftheNevanlinnaCountingFunctiontobeboundedbelowonCarlesonregions. Theorem5.1.3(Lefevre,Li,Queffelec,Rodrguez-Piazza,2010). [ 11 ]Let':D!Dbeanon-constantanalyticselfmap.ThenthecompositionoperatorC':H2(D)!H2(D),hasclosedrangeifandonlyifthereisaconstantc>0suchthat1 A(Sa)ZSaN'(z)dA(z)c(1)-221(jaj) foralla2D. ThoughZorboskaandLefevre,Li,Queffelec,andRodrguez-Piazzareferencepreviouscontributionstothestudyofclosedrangepropertiesofcompositionoperatorsintheirpapers,theconditionsremainunrelated.Theproofofeachtheoremaboutclosedrangeiscompletelyindependentoftheproofsofeachoftheothertheorems,anditisunclearhowtheseconditionsallgiverisetothesameproperty. 54

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5.2TheReproducingKernelThesis WehavedevelopedanewtoolforprovingclosedrangepropertiesofcompositionoperatorsonH2(D),theReproducingKernelThesis;ithasallowedustoprovideadirectgeometricequivalencebetweenthecriterionofCima,Thomson,andWogenandZorboska'scriterion.TheadvantageoftheReproducingKernelThesisisthatratherthanhavingtoverifyaconditionforC'tohaveclosedrangeonallfunctioninH2(D),itisonlynecessaryverifytheconditiononaexceptionallywell-behavedsubestofH2(D),thesetofSzegokernels. Theorem5.2.1(ReproducingKernelThesis). Givenananalyticmap':D!D,C'isboundedbelowonH2(D)byc>0ifandonlyifC'isboundedbelowonthesetofnormalizedSzegokernelfunctionsf~ka(z)ga2Dbyc,thatisC'~ka2c~ka2=c. InprovingtheReproducingKernelThesis,westartwiththeCima,Thomson,andWogencriterion 5.1.1 ,thenpassthroughseveralotherequivalentconditionsforC'tobeboundedbelowonH2(D)beforearrivingattheresult.TherstthingweneedtodoisrestateCima,Thomson,andWogen'scriterionusingtoolsthatwillhelpusmovefromtheirpropertyontheboundaryofthedisktoapropertyonthereproducingkernels.ThetoolsweneedaretheAleksandrov-Clarkmeasures.RecallfromSection 2.7 : Denition5.2.2. Forananalyticfunction':D!Dandapoint2T,thefunction<+'(z) )]TJ /F3 11.955 Tf 11.96 0 Td[('(z)=1)-222(j'(z)j2 j)]TJ /F3 11.955 Tf 11.96 0 Td[('(z)j2 ispositiveandharmoniconD.ByHerglotz'stheorem1)-222(j'(z)j2 j)]TJ /F3 11.955 Tf 11.95 0 Td[('(z)j2=ZT1)-221(jzj2 j)]TJ /F4 11.955 Tf 11.95 0 Td[(zj2d() forsomeuniquepositiveBorelmeasureonT.Thesetfg2TisthesetofAleksan-drovmeasures,orClarkmeasuresifthefunction'isinner. 55

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TheLebesguedecompositionofisd=hdm+ds whereh()=1)-221(j'()j2 j)]TJ /F3 11.955 Tf 11.95 0 Td[('()j2form-a.e.2T.RecallfromTheorem 2.7.8 thatsiscarriedbythesetf2T:'()=g. Lemma5.2.3. LetsdenotethesingularpartoftheAleksandrov-ClarkmeasureonT,thend dm=kskm)]TJ /F4 11.955 Tf 11.95 0 Td[(a.e.2T. Proof. FixaBorelsetET.Bythedenitionof, (E)=ZEd()=ZTEd(m')]TJ /F7 7.97 Tf 6.58 0 Td[(1)()=ZT')]TJ /F13 5.978 Tf 5.76 0 Td[(1(E)dm().(5) ApplyingtheAleksandrovDisintegrationTheoremyields(E)=ZTZT')]TJ /F13 5.978 Tf 5.76 0 Td[(1(E)d()dm(). For2T,considertheabsolutelycontinuouspartofthemeasure,hdm=1)-222(j'j2 j)]TJ /F3 11.955 Tf 11.96 0 Td[('j2dm.Thismeasureiscarriedbythesetf2T:j'()j<1g,sothat')]TJ /F13 5.978 Tf 5.75 0 Td[(1(E)()=0forhdm-almosteverywhere.Hence,ZT')]TJ /F13 5.978 Tf 5.76 0 Td[(1(E)()h()dm()=0. Nowconsiderthesingularpartofthemeasure,s.NotethatfromProposition 2.7.8 fors-a.e.2T,'()=.Hence,if2E,thensiscarriedbytheset')]TJ /F7 7.97 Tf 6.59 0 Td[(1(fg),whichiscontainedin')]TJ /F7 7.97 Tf 6.59 0 Td[(1(E).Therefore,')]TJ /F13 5.978 Tf 5.76 0 Td[(1(E)()=1s-almosteverywhereandZT')]TJ /F13 5.978 Tf 5.76 0 Td[(1(E)()ds()=ksk. Ontheotherhand,if=2E,then')]TJ /F7 7.97 Tf 6.59 0 Td[(1(fg)\')]TJ /F7 7.97 Tf 6.59 0 Td[(1(E)=;sothat')]TJ /F13 5.978 Tf 5.75 0 Td[(1(E)()=0s-almosteverywhereandZT')]TJ /F13 5.978 Tf 5.75 0 Td[(1(E)()ds()=0. 56

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Puttingthistogether,wehavethatZT')]TJ /F13 5.978 Tf 5.76 0 Td[(1(E)()d()=ZT')]TJ /F13 5.978 Tf 5.76 0 Td[(1(E)()ds()=kskE(). Pluggingthisresultintoequation( 5 )yields(E)=ZTZT')]TJ /F13 5.978 Tf 5.76 0 Td[(1(E)d()dm()=ZTkskE()dm()=ZEkskdm(). Weconcludethatd dm()=kskforLebesguealmostevery2T. ThislemmaleadstoourrstconditionthatisequivalenttotheCima,Thomson,WogenconditionforC'tobeboundedbelow. Proposition5.2.4. C'isboundedbelowbyc>0onH2(D)ifandonlyifessinf2Tkskc>0. Thisresultcanbefurtherrenedsinceinfactessinf2Tksk=inf2Tksk. Lemma5.2.5. essinf2Tksk=inf2Tksk Proof. Supposethatessinf2Tksk=cwherecissomeconstantgreaterthanzero.ThenfromLemma 5.2.3 ,d dm()cform-a.e.2T.Letf2H2(D)andconsiderZTjf('())j2dm()ZTjf()j2d dmdm()cZTjf()j2dm(), whichistrueifandonlyifkC'fk2ckfk2,inotherwordsifC'isboundedbelow.Let~kr(z)denotethenormalizedSzegokernelfunctionq 1)-222(jrj2 1)]TJ ET q .478 w 325.66 -468.75 m 338.69 -468.75 l S Q BT /F4 11.955 Tf 325.66 -476.07 Td[(rzfor2Tand0
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soinf2Tksk=essinf2Tksk. Proposition5.2.6. C'isboundedbelowbyc>0onH2(D)ifandonlyifinf2Tkskc>0. Proof. ThisfollowsdirectlyfromProposition 5.2.4 andLemma 5.2.5 Thisbringsustotheculminationofthissection,theproofoftheReproducingKernelThesis. ProofofReproducingKernelThesis 5.2.1 Theforwarddirectionistrivial:ifC'isboundedbelowonH2(D),thenC'iscertainlyboundedbelowonf~ka(z)ga2D,whichisasubsetofH2(D). Forthereversedirection,supposethatC'isboundedbelowonthesetofkernelfunctions,f~ka(z)ga2D,thenthereexistsaconstantc>0suchthatC'~kr2cforeveryr>0andevery2T.ApplyingProposition 2.7.10 ,ksk=limr!1)]TJ /F10 11.955 Tf 8.74 13.45 Td[(C'~kr2c.Sincethisholdsforevery2T,inf2Tkskc>0.ByProposition 5.2.6 ,C'isboundedbelow. 5.3ANewProofofZorboska'sCondition Zorboska'soriginalproofhingedonLuecking'sTheoremfrom1981[ 13 ];sheusedittoshowtheequivalenceofthegeometriccondition(PB)andboundednessfrombelow. Theorem5.3.1(Luecking,1981). LetGbeameasurablesubsetofDandp>0.ThenthereisaconstantC>0suchthatZDjfjpdACZGjfjpdA forf2Apifandonlyifthereisaconstant>0suchthatA(G\Sa)>A(Sa)foralla2D. HereApreferstotheunweightedBergmanspace;seeLuecking[ 13 ]orCowen,MacCluer[ 7 ]formoreaboutBergmanspaces.Theorem 5.3.1 iseasilyrestatedinoursettingoftheHardyspaceH2ascanbeseeninZorboska'spaper[ 26 ].Theforward 58

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directionofLuecking'stheoremisrelativelyeasytoproveandverygeometricinnature.Wewillemploymanyofthesamesortsoftechniquesheusedinthisimplicationtoprovethenextproposition.However,thereverseimplicationisdifculttoprove,Lueckingevenstatedsohimselfin[ 13 ].Heemploysthreelemmas,themainpurposeofwhicharetogaincontroloverthesizeoffontheCarlesonregionsSa.Byutilizingreproducingkernels,webypassthisissueentirely.Thebehaviorofthekernelfunction~kaiswell-knownontheregionSa,andinfacttheyworktogetherverynicelyaswillweseeinLemma 5.3.4 Theorem5.3.2. C'isboundedbelowonthenormalizedSzegokernelfunctions,f~ka(z)ga2D,ifandonlyifthereexistsapositiveconstantcsuchthatthesetGc=fz2D:'(z)>cgsatisesthefollowingcondition: (PB)Thereexistsaconstant>0suchthatA(Gc\Sa)>A(Sa) foralla2D. Inorderprovethistheorem,afewlemmasarerequired.Forafunctionf2H2(D),recallfromSection 2.6 theLittlewood-Paleyidentityforthenormkfk2=2ZDjf0(z)j2log1 jzjdA(z)+jf(0)j2 andthechangeofvariablesformula,whichwerewriteintermsof':kC'fk2=kf'k2=2ZDjf0(z)j2N'(z)dA(z)+jf('(0))j2=2ZDjf0(z)j2N'(z) log1 jzjlog1 jzjdA(z)+jf('(0))j2=2ZDjf0(z)j2'(z)log1 jzjdA(z)+jf('(0))j2. TherstlemmarelatestheconditionkC'fkckfkonthenormsofC'fandftoaconditionontheintegralsinthechangeofvariablesandLittlewood-Paleyidentities. 59

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Lemma5.3.3. Let':D!Dbeanalyticwith'(0)=0.TheoperatorC'isboundedbelowonthesetkernelfunctionsf~ka(z)ga2Difandonlyifthereexists00andalljaj>r. Proof. Let'(0)=0.ConsidertheLittlewood-Paleyidentityandthechangeofvariablesformulaonanarbitrarykernelfunction~ka(z),wherea2D.~ka2=2ZD~k0a(z)2log1 jzjdA(z)+~ka(0)2C'~ka=2ZD~k0a(z)2'(z)log1 jzjdA(z)+~ka(0)2. Foreaseofnotation,letIC=ZD~ka0(z)2'(z)log1 jzjdA(z)Ik=ZD~ka0(z)2log1 jzjdA(z) Fortheforwarddirection,supposethatC'isboundedbelowonthesetofkernelfunctionsf~ka(z)g,thatisthereexistsc>0suchthatC'~ka2c~ka2.Inthenewnotation,usingtheLittlewood-Paleyidentityandthechangeofvariablesformula,theboundedbelowconditionbecomes2IC+~ka(0)2c2Ik+~ka(0)2, andrearrangingyieldsICcIk)]TJ /F6 11.955 Tf 13.16 8.08 Td[(1 2(1)]TJ /F4 11.955 Tf 11.96 0 Td[(c)~ka(0)2. Sincejaj2 1)-222(jaj2isanincreasingfunctionofjaj,wecanchooser1>0sothatforjaj>r1,jaj2 1)-221(jaj22(1)]TJ /F4 11.955 Tf 11.96 0 Td[(c) c. 60

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Rearrangingwehavethat jaj2 1)-222(jaj22(1)]TJ /F4 11.955 Tf 11.95 0 Td[(c) cc 2jaj2(1)]TJ /F4 11.955 Tf 11.96 0 Td[(c)(1)-221(jaj2)cjaj2)]TJ /F4 11.955 Tf 13.15 8.08 Td[(c 2jaj2(1)]TJ /F4 11.955 Tf 11.96 0 Td[(c)(1)-221(jaj2)cjaj2)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[(c)(1)-222(jaj2)c 2jaj2c1 2jaj2)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2(1)]TJ /F4 11.955 Tf 11.96 0 Td[(c)(1)-222(jaj2)c 21 2jaj2.(5) Notethat~ka(0)2=q 1)-222(jaj2 1)]TJ ET q .478 w 229.24 -231.25 m 235.33 -231.25 l S Q BT /F4 11.955 Tf 229.24 -238.57 Td[(a02=1)-221(jaj2, andas~ka=1, Ik=1 2~ka2)]TJ /F10 11.955 Tf 11.96 13.74 Td[(~ka(0)=1 2jaj2.(5) SubstitutingIkinfor1 2jaj2inthepreviousinequalityandapplying( 5 )givesICcIk)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2(1)]TJ /F4 11.955 Tf 11.95 0 Td[(c)~ka(0)2c 2Ik Letting=c 2yieldsthedesiredintegralcondition. Forthereversedirection,supposethat ICIk.(5) Theideaoftheproofissimilartotheabove.Chooser2>0sothatforjaj>r2,jaj2 1)-222(jaj2)]TJ /F6 11.955 Tf 11.96 0 Td[(2 61

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Again,rearranging,wehavethatjaj2 1)-222(jaj2)]TJ /F6 11.955 Tf 11.95 0 Td[(2 jaj2()]TJ /F6 11.955 Tf 11.95 0 Td[(2)(1)-222(jaj2)2jaj2+2(1)-222(jaj2)jaj2+(1)-221(jaj2)1 2jaj2+1 2(1)-222(jaj2) 21 2jaj2+1 2(1)-221(jaj2. Usingthesubstitutions1)-222(jaj2=~ka(0)2and1 2jaj2=Ik,Ik+1 2~ka(0)2 2Ik+1 2~ka(0)2. Applyingtheintegralcondition( 5 ),IC+1 2~ka(0)2Ik+1 2~ka(0)2 2Ik+1 2~ka(0)2, henceC'~ka2 2~ka2. Finally,bytakingr=maxfr1,r2gweensurethatforjaj>r,C'isboundedbelowonthesetkernelfunctions,f~ka(z)ga2DifandonlyifZD~ka0(z)2'(z)log1 jzjdA(z)ZD~ka0(z)2log1 jzjdA(z) forsome>0. Bycomposingwithanautomorphismofthedisk,Lemma 5.3.3 canbeeasilygeneralizedtothecasewhere'isanyanalyticmapofDtoitself. ThesecondlemmaprovidesalowerboundforthemassofakernelfunctionoveritsmatchingCarlesonregion.Thekernelfunction~kaandtheCarlesonregionSaplayexceedinglywelltogethersincethemaximumvalueof~kaisfoundata jajontheboundary,T,seeFigure 2-2 62

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Lemma5.3.4. Thereexistconstantsr>0andC>0suchthatforallrjaj<1,ZSa~k0a2log1 jzjdA(z)C. Proof. NotethatforzcloseenoughtotheboundaryofD,wecanmaketheestimateZSa~ka0(z)2log1 jzjdA(z)ZSajaj2(1)-222(jaj2) j1)]TJ ET q .478 w 275.68 -121.95 m 281.77 -121.95 l S Q BT /F4 11.955 Tf 275.68 -129.27 Td[(azj4(1)-221(jzj2)dA(z) Leta(z)=jaj aa)]TJ /F4 11.955 Tf 11.95 0 Td[(z 1)]TJ ET q .478 w 132.66 -163.15 m 138.74 -163.15 l S Q BT /F4 11.955 Tf 132.66 -170.47 Td[(azdenotetheautomorphismofthediskthatmapsSaconformallytothelefthalfoftheunitdiskandmapsato0,thatisa(Sa)=fz2D:j1)-222(jajwj<(1)-252(jajw)>1.Finally,choosejwj<1,thenthe 63

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estimatebecomesZa(Sa)jaj2(1)-222(jwj2) j1)-222(jajwj2dA(w)1 2r2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2). LettingC=1 2r2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)yieldsthedesiredresult. OnelastlemmaisnecessarybeforetheproofofTheorem 5.3.2 .Ingeneral,thefunction'(z)=N'(z) log1 jzjisnotboundedontheunitdisk.However,ifwerestrict'toanannulusthatdoesnotcontain'(0),thentheresultingfunctionisboundedonD.Withthisinmind,chooseR>0sothat'(0)=2fz2D:R<>:'(z)jzj>R0jzjR,(5) thenR'isboundedonD. Lemma5.3.5. Thereexistconstantsc>0andk>0suchthatZGc~ka0(z)2R'(z)log1 jzjdA(z)1 kZD~ka0(z)2'(z)log1 jzjdA(z), whereGc=fz2D:'>cg. Thenegationofthislemmais:foreveryconstantc>0orforeveryconstantk>0,ZD~ka0(z)2'(z)log1 jzjdA(z)>kZGc~ka0(z)2R'(z)log1 jzjdA(z). Proof. Bywayofcontradiction,letfcngbeasequenceconvergingto0suchthatZGcn~ka0(z)2R'(z)log1 jzjdA(z))166(!0. Notethatby( 5 ),ZGcn~ka0(z)2R'(z)log1 jzjdA(z)=ZGcn\fR
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Foreaseofnotation,letAn=Gcn\fR
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forsomeconstant00, leadingtoacontradiction. Finallywecometotheproofofthemainpropositionofthissection. ProofofTheorem 5.3.2 First,supposethatC'isboundedbelowonthesetofkernelfunctionf~ka(z)ga2D,thenbyLemma 5.3.3 thereexists00andforalljajr1.ApplyingLemma 5.3.5 tothisresult,thereexistconstantsk>0andc>0suchthatZGc~ka0(z)2R'(z)log1 jzjdA(z) kZD~ka0(z)2log1 jzjdA(z). 66

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FromLittlewood'sinequality 4.1.4 ,undertheassumptionthat'(0)=0,R'(z)=N'(z) log1 jzjlog1 jzj log1 jzj=1, sothatZGc~ka0(z)2log1 jzjdA(z) kZD~ka0(z)2log1 jzjdA(z). Notethat,ingeneral,ZGc~ka0(z)2log1 jzjdA(z)ZD~ka0(z)2log1 jzjdA(z)sinceGcD,hencetheconstant kmustbestrictlylessthan1.Recallfromequation( 5 )ZD~ka0(z)2log1 jzjdA(z)=1 2jaj2, andsubstitutingthisintothelastinequalityyieldsZGc~ka0(z)2log1 jzjdA(z)jaj2 2k. ConsiderZGc\Sa~ka0(z)2log1 jzjdA(z)=ZGc~ka0(z)2log1 jzjdA(z))]TJ /F10 11.955 Tf 11.95 16.28 Td[(ZGcnSa~ka0(z)2log1 jzjdA(z)ZGc~ka0(z)2log1 jzjdA(z))]TJ /F10 11.955 Tf 11.95 16.27 Td[(ZDnSa~ka0(z)2log1 jzjdA(z). ByLemma 5.3.4 ,thereexists0rr,ZSa~ka0(z)2log1 jzjdA(z)1 2r22(1)]TJ /F3 11.955 Tf 12.31 0 Td[(2).FromtheproofofLemma 5.3.4 ,thereissomefreedominthechoiceof,sochoose>r 1)]TJ /F3 11.955 Tf 13.18 8.09 Td[( k,whichislessthan1since0< k<1.ThenwehaveZGc\Sa~ka0(z)2log1 jzjdA(z)r21 2k)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2r22(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2). Settingr=maxfr1,r2g,forallr
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wherethechoiceofaboveensuresthattheconstantontherighthandsideisstrictlypositive.NowthatwehavetheintegralZGc\Sa~ka0(z)2log1 jzjdA(z)boundedbelowbyanabsoluteconstant,weneedtondanupperboundforitsintegrand.RecallfromtheproofofLemma 5.3.4 that~ka0(z)2log1 jzjdA(z)jaj2(1)-221(jaj2) j1)]TJ ET q .478 w 273.2 -121.95 m 279.29 -121.95 l S Q BT /F4 11.955 Tf 273.2 -129.27 Td[(azj4(1)-222(jzj2) forjzjcloseenoughto1.OntheCarlesonregionSa,jajjzj<1,sousingthereversetriangleinequalityonthedenominatorabovegivesj1)]TJ ET q .478 w 195.92 -204.34 m 202.01 -204.34 l S Q BT /F4 11.955 Tf 195.92 -211.66 Td[(azj4j1)-221(j azjj4=(1)-221(jajjzj)4(1)-221(jaj)4. Therefore ~ka0(z)2log1 jzjjaj2(1)-222(jaj2) j1)]TJ ET q .478 w 256.96 -329 m 263.04 -329 l S Q BT /F4 11.955 Tf 256.96 -336.31 Td[(azj4(1)-222(jzj2)jaj2(1)-222(jaj2)(1)-222(jaj2) (1)-222(jaj)4jaj2 (1)-222(jaj)4(1)-221(jaj2)2=jaj2(1)-222(jaj)2(1+jaj)2 (1)-222(jaj)4=jaj2(1+jaj)2 (1)-222(jaj)2=jaj2(1+jaj)2 (1)-222(jaj)2(1+jaj)2 (1+jaj)2=jaj2(1+jaj)4 (1)-222(jaj2)216 (1)-222(jaj2)2(5) 68

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wherethelastestimateusesthefactthatjaj<1.Puttingtogetherthisupperboundontheintegrand( 5 )withthelowerboundontheintegral( 5 )yields16 (1)-222(jaj2)2A(Gc\Sa)ZGc\Sa~ka0(z)2log1 jzjdA(z)r2 2 k)]TJ /F6 11.955 Tf 11.96 0 Td[(1+2. WealsoneedanestimateonA(Sa)=ZSadA(z),thenormalizedLebesgueareameasureofSa,inordertocompletetheproof.Tothisend,applythechangeofvariablesw=a(z)totheintegralasinLemma 5.3.4 ,A(Sa)=Za(Sa)(1)-222(jaj2)2 j1)-222(jajwj4dA(w)(1)-221(jaj2)2Za(Sa))dA(z)=(1)-221(jaj2)21 2A(D)=1 2(1)-222(jaj2)2. SoA(Gc\Sa)r2 2 k)]TJ /F6 11.955 Tf 11.95 0 Td[(1+2(1)-222(jaj2)2 16r2 2 k)]TJ /F6 11.955 Tf 11.95 0 Td[(1+21 8A(Sa) wherer2 2 k)]TJ /F6 11.955 Tf 11.95 0 Td[(1+21 8issomeconstantstrictlygreaterthanzerothatdoesnotdependona,thusprovingtheforwarddirectionforjaj>r. Fortheotherdirection,givenaparameter00and>0suchthatforalla2D,A(Gc\Sa)>A(Sa).LetGc,a=Gc\Saandchoosetsothat 69

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A(Et(a))>(1)]TJ /F5 7.97 Tf 13.34 4.71 Td[( 2)A(Sa).ConsiderA(Et(a)\Gc,a)=1 2[A(Et(a))+A(Gc,a))]TJ /F4 11.955 Tf 11.95 0 Td[(A(Et(a)4Gc,a)]1 21)]TJ /F3 11.955 Tf 13.48 8.09 Td[( 2A(Sa)+A(Sa))]TJ /F4 11.955 Tf 11.95 0 Td[(A(Sa)= 4A(Sa). Now,usingthelemmaswecanmakeseveralreductionstotheconclusionthatC'isboundedbelowontheSzegokernels.FromLemma 5.3.3 itsufcestoshowthatthereexists>0suchthatZD~k0a(z)2'(z)log1 jzjdA(z)ZD~k0a(z)2log1 jzjdA(z) foralljajlargerthansomevaluer1>0.SinceZD~k0a(z)2log1 jzjdA(z)=jaj2 2r21 2 andGc\SaD,itthensufcestoshowthereexists0>0suchthatZGc\Sa~k0a(z)2(1)-221(jzj2)dA(z)0. Tothisend,considerZGc\Sa~k0a(z)2(1)-221(jzj2)dA(z)ZGc\Et(a)~k0a(z)2(1)-222(jzj2)dA(z)ZGc\Et(a)t~k0a(a)2(1)-222(jaj2)dA(z)=t~k0a(a)2(1)-222(jaj2)A(Gc\Et(a))t~k0a(a)2(1)-222(jaj2) 4A(Sa)tjaj2 (1)-222(jaj2)2 4(1)-221(jaj2)2tr2 4. 70

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Theconstant,tr2 4isstrictlypositiveandindependentofa,soapplyingourreductionsabove,weconcludeC'isboundedbelowonkernelfunctions. TogeneralizetoallvaluesofainD,simplynotethatalltheregionsSaandthekernelfunctions~kaareconformallyequivalentforeverya2D.Theassumptionthat'(0)=0isjustaseasilytakencareofbyconjugatingwiththediskautomorphisma. 5.4FutureWork InSections 5.2 and 5.3 weprovidedadirectgeometricequivalencebetweenCima,Thomson,andWogen'scriterion 5.1.1 andZorboska'scriterion 5.1.2 foracompositionoperatortohaveclosedrangeonH2(D).WehopetoprovideaproofofthecriterionbyLefevre,Li,Queffelec,andRodrguez-Piazza 5.1.3 usingtheReproducingKernelThesisaswell. 71

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[16] EricA.Nordgren.Compositionoperators.Canad.J.Math.,20:442,1968. [17] VernI.Paulsen.Anintroductiontothetheoryofreproducingkernelhilbertspaces,2009. [18] ThomasRansford.Potentialtheoryinthecomplexplane,volume28ofLondonMathematicalSocietyStudentTexts.CambridgeUniversityPress,Cambridge,1995. [19] WalterRudin.AgeneralizationofatheoremofFrostman.Math.Scand,21:136(1968),1967. [20] WalterRudin.Realandcomplexanalysis.McGraw-HillBookCo.,NewYork,thirdedition,1987. [21] DonaldSarason.Sub-HardyHilbertspacesintheunitdisk.UniversityofArkansasLectureNotesintheMathematicalSciences,10.JohnWiley&SonsInc.,NewYork,1994.AWiley-IntersciencePublication. [22] JoelH.Shapiro.Theessentialnormofacompositionoperator.Ann.ofMath.(2),125(2):375,1987. [23] JoelH.Shapiro.Compositionoperatorsandclassicalfunctiontheory.Universitext:TractsinMathematics.Springer-Verlag,NewYork,1993. [24] JoelH.Shapiro.Recognizinganinnerfunctionbyitsdistributionofvalues,April1999. [25] JoelH.Shapiro.Whatdocompositionoperatorsknowaboutinnerfunctions?Monatsh.Math.,130(1):57,2000. [26] NinaZorboska.Compositionoperatorswithclosedrange.Trans.Amer.Math.Soc.,344(2):791,1994. 73

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BIOGRAPHICALSKETCH KristinLuerywasborninFallsChurch,VAin1981,partoftheWashington,D.C.suburbs.Shecomesfromabackgroundwheremathwasalwayspartoftheequation;herfatherisastatistician,hermotherwasacomputerprogrammer,hergrandfatherwasanengineering,andhersisterisnowanaccountant.SoitisnosurprisethatKristinwentontoobtainherdoctorateinmathematics.Sheattendedhighschoolatascienceandtechnologymagnetschool,ThomasJeffersonHighSchoolforScienceandTechnology,focusingherstudiesonchemistry,physics,andmathematics.Decidingshewantedtomajorinphysics,Kristinenteredcollegein1999atVirginiaTech,thentransferredafterherrstyeartotheUniversityofVirginia.In2004,shereceivedaB.A.inphysicsandmathematicsfromtheUniversityofVirginia.Kristinthentookseveralyearsofffromschool,tryingtodecidewhethertopursuefurtherstudyinphysicsormathematics.SheeventuallydecidedonmathematicsandenteredgraduateschoolinAugustof2007attheUniversityofFlorida.ShereceivedherM.S.inmathematicsin2009,thencontinuedontocompleteherPh.D.in2013. 74