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Classes of Densely Defined Multiplication and Toeplitz Operators with Applications to Extensions of RKHS's

Permanent Link: http://ufdc.ufl.edu/UFE0045339/00001

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Title: Classes of Densely Defined Multiplication and Toeplitz Operators with Applications to Extensions of RKHS's
Physical Description: 1 online resource (77 p.)
Language: english
Creator: Rosenfeld, Joel A
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

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Subjects / Keywords: analysis -- fock -- hardy -- multipliers -- operator -- sobolev -- toeplitz -- unbounded
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: While bounded multiplication has been extensively researched, unbounded multiplication has received little attention until more recently.  We develop a framework for densely defined multiplication over reproducing kernel Hilbert spaces, and we find an application toward extending reproducing kernels. We also extend a result of Allen Shields, who showed that the multipliers for the Sobolev space are precisely the elements of that space.  We show that this holds even if multipliers are merely densely defined. In connection with multiplication operators, we explore densely defined Toeplitz operators.  Here we find simpler proofs of theorems from Sarason and Suarez.  We also produce a partial answer to a problem posed by Donald Sarason.
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 Joel A Rosenfeld.
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.
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System ID: UFE0045339:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045339/00001

Material Information

Title: Classes of Densely Defined Multiplication and Toeplitz Operators with Applications to Extensions of RKHS's
Physical Description: 1 online resource (77 p.)
Language: english
Creator: Rosenfeld, Joel A
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: analysis -- fock -- hardy -- multipliers -- operator -- sobolev -- toeplitz -- unbounded
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: While bounded multiplication has been extensively researched, unbounded multiplication has received little attention until more recently.  We develop a framework for densely defined multiplication over reproducing kernel Hilbert spaces, and we find an application toward extending reproducing kernels. We also extend a result of Allen Shields, who showed that the multipliers for the Sobolev space are precisely the elements of that space.  We show that this holds even if multipliers are merely densely defined. In connection with multiplication operators, we explore densely defined Toeplitz operators.  Here we find simpler proofs of theorems from Sarason and Suarez.  We also produce a partial answer to a problem posed by Donald Sarason.
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 Joel A Rosenfeld.
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: UFE0045339:00001


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CLASSESOFDENSELYDEFINEDMULTIPLICATIONANDTOEPLITZOPERATORSWITHAPPLICATIONSTOEXTENSIONSOFRKHS'SByJOELA.ROSENFELDADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013JoelA.Rosenfeld 2

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Idedicatethistomother,KatherineVannandmybrother,SpencerRosenfeld. 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadviser,MichaelT.Jury,forhispatienceandadviceasIslowlylearnedoureld.Withouthim,thisworkcouldnothavehappened. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 2PRELIMINARIES ................................... 11 2.1ReproducingKernelHilbertSpacesandCommonExamples ....... 11 2.2ReproducingKernelsandKernelFunctions ................. 13 2.3BoundedMultiplicationOperators ....................... 16 2.4DenselyDenedOperators .......................... 17 2.5Inner-OuterFactorizationofH2Functions .................. 19 3REPRODUCINGKERNELPRESENTATIONS ................... 22 3.1Denitions .................................... 22 3.2RKPExtensions ................................ 27 4CLASSESOFDENSELYDEFINEDMULTIPLICATIONOPERATORS ..... 31 4.1HardySpace .................................. 31 4.2FockSpace ................................... 33 4.3TheDirichlet-HardySpaceandthePolylogarithm .............. 37 5UNBOUNDEDMULTIPLICATIONONTHESOBOLEVSPACE ......... 45 5.1DenselyDenedMultipliersfortheSobolevSpace ............. 45 5.2LocaltoGlobalNon-VanishingDenominator ................. 47 5.3AlternateBoundaryConditions ........................ 50 5.3.1Sturm-LiouvilleBoundaryConditions ................. 50 5.3.2MixedBoundaryConditions ...................... 52 5.4Remarks .................................... 54 6UNBOUNDEDTOEPLITZOPERATORS ...................... 56 6.1Sarason'sProblem ............................... 56 6.2SarasonSub-Symbol ............................. 61 6.3ExtendingCo-analyticToeplitzOperators .................. 69 REFERENCES ....................................... 75 BIOGRAPHICALSKETCH ................................ 77 5

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LISTOFTABLES Table page 5-1MultipliersassociatedwithSturm-Liouvilleboundaryconditions ......... 51 5-2Multipliersassociatedwithmixedboundaryconditions .............. 52 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCLASSESOFDENSELYDEFINEDMULTIPLICATIONANDTOEPLITZOPERATORSWITHAPPLICATIONSTOEXTENSIONSOFRKHS'SByJoelA.RosenfeldMay2013Chair:MichaelT.JuryMajor:MathematicsWhileboundedmultiplicationhasbeenextensivelyresearched,unboundedmultiplicationhasreceivedlittleattentionuntilmorerecently.WedevelopaframeworkfordenselydenedmultiplicationoverreproducingkernelHilbertspaces,andwendanapplicationtowardextendingreproducingkernels.WealsoextendaresultofAllenShields,whoshowedthatthemultipliersfortheSobolevspacearepreciselytheelementsofthatspace.Weshowthatthisholdsevenifmultipliersaremerelydenselydened.Inconnectionwithmultiplicationoperators,weexploredenselydenedToeplitzoperators.HerewendsimplerproofsoftheoremsfromSarasonandSuarez.WealsoproduceapartialanswertoaproblemposedbyDonaldSarasonbyconstructinganalgorithmforrecoveringthesymbolforaclassofdenselydenedToeplitzoperators. 7

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CHAPTER1INTRODUCTIONWehavetwogoalstoreachinthisDissertation.TherstisthediscussionofreproducingkernelpresentationsandextensionsofreproducingkernelHilbertspaces.ThesecondgoalistoexploredenselydenedToeplitzoperatorsovertheHardyspace.ThelinkbetweenbothoftheseideasistheconceptofdenselydenedmultiplicationoperatorsandtheirinterplaywiththereproducingkernelsofareproducingkernelHilbertspace.Thetopicofboundedmultiplicationonfunctionspacesisasubjectthathasreceivedextensivestudy.ThelandmarkresultofPickinthespaceofboundedanalyticfunctionsisapproachingitscentennialanniversary,andthestudyofboundedmultiplicationisstillolderthanthis.AroundthesametimePickandNevanlinnabothprovedwhatisnowcalledtheNevanlinna-PickInterpolationtheorem[ 1 ]: Theorem1.1. Giventwocollectionsofpointsintheunitdiscfz1,z2,...,zkgandfw1,w2,...wkgthereexistsaboundedanalyticfunctionforwhichf(zi)=wiifandonlyifthematrix((1)]TJ /F7 11.955 Tf 13.64 0 Td[(wiwj)(1)]TJ /F7 11.955 Tf 13.09 0 Td[(zizj))]TJ /F4 7.97 Tf 6.59 0 Td[(1)ki,j=1ispositivedenite.Workonthisinterpolationresulthasbroughtmuchattentiontoboundedmultipliers.Inparticularthespaceofboundedanalyticfunctionsinthedisc,H1,ispreciselythecollectionofboundedmultiplicationoperatorsontheHardySpace.Thisworkhasfoundapplicationsnotonlyinpuremathematics,butalsoinsignalprocessingwheremultipliersappearastransferfunctionsforlinearsystems.Therehasalsobeenagreatdealofdevelopmentoftheaptlynamedbranchofcontroltheory:H1-control.HeretheNevanlinna-Pickpropertyisusedexplicitly.Unboundedmultiplicationhasreceivedmuchlessattention.Untilrecentlytherewereonlyahandfulofexceptions.OneisintheFockspace,wherethemultiplicationoperatorMzisdenselydenedandwhoseadjointisthederivative.IntheBergmanspaceoveraregion,Mziswellrecognizedtobeunboundedwhenthedomainitself 8

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isunbounded.In[ 2 ],Kouchekianhasinvestigatedthequestionofwhenthisoperatorisdenselydened.DenselydenedmultiplicationhasalsobeenexploredindeBrangesRovnyakspaces[ 3 ].RecentlyAleman,Martin,andRossworkedonsymmetricandselfadjointdenselydenedoperators(whichincludedmultiplicationoperators)[ 4 ].TherstexplicitcharacterizationofacollectionofdenselydenedmultiplicationoperatorswasgivenbySarasonin2008whenhecharacterizedtheanalyticToeplitzoperatorsovertheHardyspace.HefoundthatthesewerepreciselytheoperatorswithsymbolsintheSmirnovclass,N+.N+:=ff:D!Cjf(z)=b(z)=a(z)whereb,a2H2andaisouter.gHerewebeginasystematicstudyofdenselydenedmultiplicationoperatorsoverreproducingkernelHilbertspaces.Thesemultipliers,whilenotnecessarilycontinuous,areallclosedoperatorsgiventheirnaturaldomain.InthesecondchapterwevisitseveralpreliminaryresultsaboutreproducingkernelHilbertspaces(RKHS),theHardyspace,theFockspace,andunboundedoperators.Chapter3introducesreproducingkernelpresentationsandanapplicationofdenselydenedmultiplierstowardextendingaRKHS.Chapter4explorestheseextensionsanddiscussesdenselydenedmultipliersovertheHardy,FockandthePolylogarithmicHardyspace.Inchapter5,weturnourattentiontoSobolevspaces.WeimproveuponaresultofAllenShieldsfoundinHalmos'AHilbertSpaceProblemsBook[ 5 ].ThereHalmosaskediftherewasaspaceoffunctionswhoseboundedmultiplierswereexactlythefunctionsinthespace.Forinstance,intheHardyspaceonlyapropersubspaceoffunctionsareitsmultipliers.ShieldsshowedthattheSobolevSpaceissuchaspaceoffunctions.Itturnsoutthatinthissamespace,everydenselydenedmultiplicationoperatorisbounded(andhenceinthespace).ThissharpeningofShields'resultisthecontentofchapter5.In 9

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additionweshowthatthereareunboundedmultipliersforsomesubspacesarisingfromchoicesofboundaryconditions.ThissameSobolevspacewasshowntohavetheNevanlinna-Pickpropertyaswell.ThiswasprovedbyAglerin1990[ 6 ].FinallyinChapter6,wereturntoSarason'sUnboundedToeplitzoperators.Attheendofhisexpositoryarticlein2008,SarasonproposedacollectionofalgebraicconditionstoclassifyToeplitzoperators.Heleftwithaquestion:Arethesealgebraicpropertiesenoughtorecoverasymbol(broadlyinterpreted)ofaToeplitzoperator?WeintroducetheSarasonsub-symbolanduseittondsimplerproofsforsomeknownresultsbySuarezandSarason.WealsoshowthatitcanrecoverthesymbolsforalargeclassofdenselydenedToeplitzoperators. 10

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CHAPTER2PRELIMINARIES 2.1ReproducingKernelHilbertSpacesandCommonExamplesLetXbeasetandletHbeaHilbertspaceoffunctionsf:X!C.WesayHisareproducingkernelHilbertspace(RKHS)ifforeachx2XthelinearfunctionalEx:H!CgivenbyExf=f(x)isbounded.RecallthatforanyboundedlinearfunctionalL:H!CcanberepresentedasLf=hf,yiforauniquey2H.Thismeansforeachx2Xthereisauniquekxforwhichhf,kxi=Exf=f(x).Wecallkxthereproducingkernelforx,andwecallK(x,y)=hkx,kyithereproducingkernelfunctionofH.Thekernelfunctionshaveaniceinterplaywithwhatarecalledfunctiontheoreticoperators.Weareprincipallyconcernedwithmultiplicationoperators,whicharedenedasfollows. Denition1. Let:X!Cforwhichf2Hforeveryf2H.WesaythattheoperatorM:H!HgivenbyMf=fisa(bounded)multiplicationoperator.WhenMcanbeappliedtoallofthefunctionsinH,theclosedgraphtheoremtellsusthatMisaboundedoperator.BoundedmultiplicationisanextensivelystudiedareaofoperatortheorythatgoesatleastasfarbackasPickin1916[ 1 ].AcomprehensivereferenceforthisstudycanbefoundinAglerandMcCarthy'sbookpublishedin2002[ 7 ].Thisalsoincludesmuchofwhatwewilldiscussaboutreproducingkernels.However,thestandardreferenceonreproducingkernelswaspublishedbyAronszajnin1950[ 8 ].WewillbediscussingseveralRKHSsinthisdissertation,butwewillfocusonthreeinparticular.TherstistheHardyspaceH2,whichcanbeviewedasthosefunctionsfinL2(T)=f:T!CjZ)]TJ /F12 7.97 Tf 6.58 0 Td[(jf(ei)j2d<1forwhich1 2Z)]TJ /F12 7.97 Tf 6.59 0 Td[(f(ei)e)]TJ /F8 7.97 Tf 6.58 0 Td[(ind=hf,zni=0 11

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foralln<0.AnotherconvenientwaytodenetheHardyspaceisasthosefunctionsfwhichareanalyticinthedisc(writingf(z)=P1n=0anzn)andforwhichthequantity(2))]TJ /F4 7.97 Tf 6.59 0 Td[(1Z)]TJ /F12 7.97 Tf 6.59 0 Td[(jf(rei)j2d=1Xn=0r2njanj2staysboundedasr!1)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(.Fatou'stheoremtellsusthatforsuchf,theauxiliaryfunction~f(ei):=limr!1)]TJ /F3 11.955 Tf 8.74 -.3 Td[(f(rei)isdenedalmosteverywhereand~f2L2(T).MoreoverD~f,znE=anforn0.Thisenablesustoexpresstheinnerproductintwoforms.Takef=P1n=0anznandg=P1n=0bnznwewritetheirinnerproductas:hf,gi=1 2Z)]TJ /F12 7.97 Tf 6.59 0 Td[(f(ei) g(ei)d=1Xn=0an bn.Notethatweknowfzng1n=0isanorthonormalbasisforH2.ThisfollowsfromthefactthatitisaclosedsubspaceofL2(T).ThepropertiesofH2thatwewillbeexploringinthesepreliminariescanbefoundinDouglas'sbookBanachAlgebraTechniquesinOperatorTheory[ 9 ],Hoffman'sBanachSpacesofAnalyticFunctions[ 10 ],andalsoDuren'sTheoryofHpspaces[ 11 ].WecanalsondtheminRudin'sRealandComplexAnalysis[ 12 ].TheHardyspacewasrstintroducedbyReiszin1923[ 13 ].AnotherspaceofanalyticfunctionsthatisimportanttoourdiscussionsistheFockspace,denotedbyF2.ThisisalsoknownastheBargmann-Segalspace.TheFockspaceisaHilbertspaceofentirefunctionsf(z)forwhichthefollowinginequalityholds:kfk2F2=(2))]TJ /F4 7.97 Tf 6.58 0 Td[(1ZCjf(z)j2ejzj2dA(z)<1.HerethequantitydA(z)indicatesthatwearetakingthestandardareaintegraloverthecomplexplane.ThisiscontrastedwiththeHardyspace,wherewewereintegratingwithrespecttoarclengthonthecircle.AswithanyHilbertspace,theinnerproductforthe 12

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Fockspacecanbedeterminedbyitsnormandisgivenby:hf,giF2=(2))]TJ /F4 7.97 Tf 6.58 0 Td[(1ZCf(z) g(z)ejzj2dA(z).ItiseasytoseethatthemonomialsareintheFockspaceandaremutuallyorthogonal(thisfollowsafteracomputationinpolarcoordinates).ThenormofzncanbecomputedrecursivelybytheuseofintegrationbypartsandyieldskznkF2=p n!.Thesetfzn=p n!gisanorthonormalbasisfortheFockspace.MoreovertheTaylorpolynomialsoffconvergetofintheF2norm.WecanusethistodetermineifanentirefunctionisinF2: Lemma1. Letfbeanentirefunction.Thenf2F2ifff(z)=P1n=0anznanditsTaylorcoefcientssatisfyP1n=0n!janj2<1.ThesefactscanbefoundinKeheZhu'sbookAnalysisontheFockSpaces[ 14 ],andinJ.Tung'sdissertationFockSpaces[ 15 ].AlotoftheinitialworkonthisspacewasdonebyBargmannin1961[ 16 ].FinallythethirdspacewewillbeconsideringistheSobolevspaceW1,2[0,1].ThiswillbethefocusofChapter5.TheSobolevspaceisthecollectionoffunctionsf:[0,1]!CthatareabsolutelycontinuousandwhosederivativesareinL2[0,1].Absolutecontinuityguaranteestheexistenceofthederivativealmosteverywhere(withrespecttoLebesguemeasure),anditalsoisthetypeofcontinuitythatguaranteesf(x)=Rx0f0(t)dt+f(0).Theinnerproductisgivenbyhf,giW1,2[0,1]=Z10f(t) g(t)+f0(t) g0(t)dt. 2.2ReproducingKernelsandKernelFunctionsHavinglaidoutthedenitionsofthespacesintheprevioussection,wenowsetouttodeterminethekernelfunctionsofeachspace.ThereisausefultheoreminthisdirectionfordeterminingK(x,y)forageneralRKHSH. 13

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Theorem2.1(Bergman). LetK(z,w)bethereproducingkernelforaRKHS,H,overasetX.IffengisanorthonormalbasisforHthenwecanwriteK(z,w)=P1n=0 en(w)en(z). Proof. Foreachw2XwehaveK(z,w)=kw(z)2H.Wecanthenwritekw(z)=1Xn=0hkw,enien(z)=1Xn=0 hen,kwien(z)=1Xn=0 en(w)en(z). Thisexpeditestheprocessofndingthekernelfunctiononceaspaceisknowntohaveboundedpointevaluations.NoticethatfortheHardyspacethemonomialsformanorthonormalbasis.Takeen(z)=znandweseethatforw2D,KH2(z,w)=1Xn=0 en(w)en(z)=1Xn=0wnzn=1 1)]TJ /F7 11.955 Tf 13.64 0 Td[(wz.WecanprovethattheHardyspacehasboundedpointevaluationsretroactively.Letw2Dandsetkw=(1)]TJ /F7 11.955 Tf 13.64 0 Td[(wz))]TJ /F4 7.97 Tf 6.59 0 Td[(12H2.Iff2H2,thenjEwfj=jf(w)j=1Xn=0anwn=1Xn=0an wn=jhf,kwijkfkkkwk.ThusbyCauchy-Schwarzweseeforarbitraryw2DwehavethatEwisaboundedfunctional.SimilarlywecanproducekernelsfortheFockspacebythesamecalculationsndingthatKF2(z,w)=1Xn=0wnzn n!=ewz=kw(z).Itcanbeveriedthatkw(z)2F2forallw2CbyLemma 1 .IntheFockspace,thereproducingkernelshaveaphysicalsignicance:ThereproducingkernelsoftheFockspacearethecoherentstatestothequantumharmonicoscillator. 14

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FortheSobolevspacethereproducingkernelfors2[0,1]isthefunctionKW1,2[0,1](t,s)=ks(t)=8><>:aset+bse)]TJ /F8 7.97 Tf 6.59 0 Td[(t:tscset+dse)]TJ /F8 7.97 Tf 6.59 0 Td[(t:tswhereas=bs=es+e2e)]TJ /F8 7.97 Tf 6.59 0 Td[(s 2(e2)]TJ /F7 11.955 Tf 11.96 0 Td[(1),cs=es+e)]TJ /F8 7.97 Tf 6.58 0 Td[(s e2)]TJ /F7 11.955 Tf 11.96 0 Td[(1,andds=e2cs.Thiscanbefoundin[ 6 ]amongotherplaces.ThereproducingkernelissometimescalledtheGreen'sfunctioninthiscontext.NowthatwehaveestablishedsomeexamplesofRKHSs,let'stakealookatamoreabstractdenition.RecallthatannmatrixAispositiveifforeveryvector=(1,...n)T2CnwehavehA,i=nXi,j=1jiAi,j0.Letusxx1,x2,...,xn2CandconsiderthematrixA=(K(xi,xj))ni,j=1.Thefollowingcomputationtellsusthatthismatrixispositive:hA,i=nXi,j=1jiK(xi,xj)=nXi,j=1jikxi,kxj=*nXi=1ikxi,nXj=1jkxj+=nXi=1ikxi20..Thisleadsustothedenitionofakernelfunction. Denition2. LetXbeasetandK:XX!Cbeafunctionoftwovariables.ThenKiscalledakernelfunctionifforeverynandeverychoiceofdistinctpointsfx1,...,xng2Cnthematrix(K(xi,xj))ni,j=10.TheMoore-AronszajnTheoremtellsusthateverykernelfunctionisareproducingkernelforsomeRKHS.TheproofofthisinvolvesaGNSstyleconstructiontomaketheHilbertSpace,andcanbefoundinPaulsen'snotes[ 17 ].Ineveryinstanceweuseakernelfunctionthespacewillbeexplicitlydened. 15

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2.3BoundedMultiplicationOperatorsWewillbeexploringthemultiplicationoperatorsoverthesespaces.Thefollowinglemmahelpsusseeanimportantdifferencebetweenthemultipliersofthesespaces. Lemma2. IfHisareproducingkernelHilbertspaceoverasetXandisthesymbolofaboundedmultiplicationoperatorM,thenMkx= (x)kxforallx2X.Moreover,isaboundedfunction. Proof. Letf2Handx2X.Nowletusconsiderf,Mkx=hf,kxi=(x)f(x)=(x)hf,kxi=Df, (x)kxE.Thisholdsforallf2H,soMkx= (x)kxasdesired.Toseethatisbounded,recallthatthespectralradiusofaboundedoperatorVisr(V)=supfjj2CjV)]TJ /F11 11.955 Tf 11.96 0 Td[(Iisnotinvertibleg.Inparticularr(V)isatleastaslargeasanyeigenvalue.Alsothenormofanoperatorisingenerallargerthanthespectralradius.Thusforallx2Xwehave:j(x)jr(M)kMk. Wealsoneedthefollowinglemma: Lemma3. IfDisadensesubspaceofaRKHSHoverasetX,thenforallx2Xthereisafunctionf2Dforwhichf(x)6=0. Proof. Supposethiswerenottrue,thenthereisanx2Xsuchthatforallf2Dwehavehf,kxi=f(x)=0.ThusDfkxg?andisnotdense. FromthiswecanconcludethatifMisaboundedmultiplicationoperatorforaRKHSHofanalyticfunctionsonasetX,thenisanalyticonXaswell.Indeed,notethatforeachx2Xthereisafunctionf2Hsuchthatf(x)6=0.SinceMf=f2H, 16

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thefunctionh:=fisanalyticonX.Finally(x)=h(x)=f(x)isaratioofanalyticfunctionwherethedenominatorisnonzero.ThusisanalyticonX.NowletuslookattheHardyspace.IfisthesymbolforaboundedmultiplicaitonoperatorM:H2!H2,then=M12H2.Moreover,isaboundedanalyticfunctiononthedisc.Ontheotherhand,ifisaboundedanalyticfunctiononthedisc,thenthereisanM>0sothatj(z)j0suchthatforally2NwehavekLykCkyk.Inparticular, 17

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LisLipshitzcontinuouswithrespecttothenormmetriconN.Astrongerconnectionisgivenbytheclassicresult: Theorem2.3. [ 18 ]LetNbeavectorspaceandL:N!Nalinearoperatoroverthatspace.Thefollowingareequivalent: 1. Lisbounded. 2. Liscontinuous. 3. Liscontinuousat0.Inthissensewhenwesayanoperatorisunboundeditissynonymouswithsayingitisdiscontinuous.Theunboundedoperatorswewillbediscussingherearealldenselydened. Denition3. LetHbeaHilbertspaceandletD(T)beadensesubspaceofH.WesayanoperatorwhosedomainisD(T),T:D(T)!H,isdenselydened.Whenwedeneadenselydenedoperator,wemustalsodenethedomainitisover.IfTandT0areoperatorsonHwithdomainsD(T)D(T0)andTf=T0fforallf2D(T),wesaythatT0extendsT.WedenotethiswithTT0. Example1. LetPbethesetofallpolynomials,andletd dz:P!H2betheoperationofdifferentiationappliedtothepolynomials.SincePisdenseinH2,weseethatd dzisdenselydened.However,itisclearthatthenaturaldomainofthedifferentiationoperatorshouldbelargerthanjustthespaceofpolynomials.Indeed,considerthespaceoffunctionsf(z)=P1n=0anzninH2forwhichPn2janj2<1.ThederivativeclearlytakesthiscollectionoffunctionsintoH2.WewillcallthiscollectionbyD(d dz).Theoperatord dz:D(d dz)!H2extendsd dzandsowewrited dzd dz.Whileadenselydenedoperatormaynotbecontinous,wemaystillholdouthopeforsomesortoflimitproperties. 18

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Denition4. LetT:D(T)!HbeadenselydenedoperatoronaHilbertspaceH.WesaythatTisaclosedoperatorifwheneverffngD(T)withfn!f2HandTfn!h2Hwehavef2D(T)andTf=h.Equivalently,wedeneanoperatorasclosedifitsgraphG(T)=f(f,Tf)jf2D(T)gisaclosedsubsetofHHwithrespecttothenormtopology.TheclosedgraphtheoremfromFunctionalAnalysistellsusthatifTiseverywheredenedandclosed,thenTisbounded.Itcanbeshownthatd dzisclosed,whereasd dzisnot.Thedenitionofadjointsfordenselydenedoperatorsisalittlebitmoredelicate.Wedenethedomainoftheadjointasthosefunctionsh2HforwhichthefunctionalL:D(T)!CgivenbyL(f)=hTf,hiiscontinuous.WewritethisdomainasD(T).TheRieszRepresentationTheoremtellsusthereisauniquevector,callitThforwhichL(f)=hf,Thi.InmanycasesitmightturnoutthatD(T)containsonlythezerovector.WewouldliketohaveaconditionthatwouldguaranteethatD(T)isnotonlynontrivialbutisalsodenseinH.Thefollowingisthiscondition.NotebyclosablewemeanthatThasaclosedextension. Theorem2.4. LetT:D(T)!HbeadenselydenedoperatoronH.IfTisclosable,thenTisclosedandenselydened.Moreover,TisthesmallestclosedextensionofT.AmoreextensivetreatmentofdenselydenedoperatorscanbefoundinPedersen'sAnalysisNOW[ 19 ]andConway'sACourseinFunctionalAnalysis[ 20 ]. 2.5Inner-OuterFactorizationofH2FunctionsAwellknownresultfromComplexAnalysisistheWeierstrassfactorizationtheorem.Thistheoremtellsusthatanyentirefunctionfcanbefactoredintotwootherentirefunctionsoneofwhichisnon-vanishing,andtheotherisaparticularkindofproductcontainingallofthezerosoff.Specically: 19

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Theorem2.5(WeierstrassFactorizationTheorem). Iffisanentirefunctionwitharootoforderkatzeroandf!ng1n=1isthesequenceofnon-zerorootsoff,thenf(z)= zk1Yn=1En(z=!n)!eh(z)whereh(z)isanentirefunctionandEn(z)=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(z)expz+z2 2+z3 3++zn n.ThereisananalogoustheoremfortheHardyspacewithoneadditionaladvantage.TheexponentialpieceoftheWeierstrassfactorizationcanaffecttheaveragegrowthoffinanobviousway.Itislessobviousthattheaveragedensityofthezerosalsocontributetothegrowthofourentirefunctionf.IntheHardyspace,theinner-outerfactorizationallowsustoseparatethezerosfromthegrowthofafunctioninthedisc.Let2D.WecallafunctionoftheformB(z)=jj )]TJ /F3 11.955 Tf 11.95 0 Td[(z 1)]TJ /F7 11.955 Tf 12.9 0 Td[(zaBlaschkefactor.ABlaschkefactorhasazeroatandasimplepoleat1=outsidethedisc.Noticethatforall:jB(ei)j=)]TJ /F3 11.955 Tf 11.96 0 Td[(ei 1)]TJ /F7 11.955 Tf 12.9 0 Td[(ei=je)]TJ /F8 7.97 Tf 6.58 0 Td[(ij)]TJ /F3 11.955 Tf 11.96 0 Td[(ei e)]TJ /F8 7.97 Tf 6.58 0 Td[(i)]TJ /F7 11.955 Tf 12.9 0 Td[(=j)]TJ /F3 11.955 Tf 11.96 0 Td[(eij j ei)]TJ /F11 11.955 Tf 11.95 0 Td[(j=1.Iff2H2andfng1n=1isthezerosequenceoff,thenwecallB(z)=QBn(z)theBlaschkefactoroff. Proposition2.1. IfBistheBlaschkefactoroff,thenB(z)hasthefollowingproperties: 1. limr!1)]TJ /F2 11.955 Tf 8.74 -.3 Td[(jB(rei)j=1foralmosteveryandjB(z)j<1forallz2D. 2. B2H2andkBk=1. 3. f=B2H2,f=Bisnonvanishing,andkf=Bk=kfk. 20

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Wesaythatafunctionsatisfying(1)intheabovepropositionisaninnerfunction.Forafunctionf2H2,wecallF(z)=exp1 2Z)]TJ /F12 7.97 Tf 6.59 0 Td[(ei+z ei)]TJ /F3 11.955 Tf 11.95 0 Td[(zlogjf(ei)jdtheouterfactoroff.WewillalsocallF(z)anouterfunction. Proposition2.2. F(z)doesnotvanishinsidethedisc.Moreover,F(z)2H2isanouterfunctioniffthesetfF(z)p(z):p(z)isapolynomial.gisdenseinH2.Finallyifwewishtofactorf,weneedoneothertypeoffactorcalledthesingularinnerfactoroff.Thisisaninnerfunctionwithoutzerosthatispositiveattheorigin. Theorem2.6(Beurling'sInner-OuterFactorization[ 10 ]). Everyfunctionf2H2canbeexpressedasf=BSFwhereBistheBlaschefactoroff,Fistheouterfactor,andSisasingularinnerfactor.WecallBStheinnerfactoroff. 21

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CHAPTER3REPRODUCINGKERNELPRESENTATIONSLetfbeananalyticfunctionofthedisc,andsupposethatwehaveasequenceofpointsfngthatconvergeattheorigin.Atheoremfromcomplexanalysistellsusthatiff(n)=0foralln,thenfisidenticallyzero.Thisalsotellsusthatiftherearetwofunctionsf,ganalyticinthediscandf(n)=g(n)foralln,thenf(z)=g(z)forallz2D.Wecallsuchasetauniquenesssetforanalyticfunctionsinthedisc.Inparticularwhatthistellsusisthatfiscompletelydeterminedbyasmallsubsetofitsdomain.Giventhatweknowfisananalyticfunctionofthedisc,wecouldstartwiththerestrictionofftofngandrecoverfbyextension.InthischapterwesetouttodeneanotionofextensionofaRKHSandworkoutareasonableextensiontheory.OneofthekeyfeaturesofaRKHSHoverasetXisthatthereproducingkernelshavedensespaninsideofH.Forinstancesupposethathf,kxi=0forallx2X.Thismeansthatf(x)=0foreachxbythereproducingpropertyandhencef0.Sometimesthesetofkernelsisreferredtoasanover-completebasis.AjusticationofthisterminologycanbeseenbyconsideringtheHardyspaceH2takingasubsetfkng(fngasabove)ofit'skernels.Bytheargumentintheaboveparagraphs,weseethatthiscollectionalsohasadensespaninH2.Themainquestionofthissectionisgivenasmallsubsetofthekernelfunctions,isthereawayofrecoveringthewholesetofkernels?IfwearegivenaRKHSHoverasetX,canwedetermineamorenaturaldomainYthatcontainsXforthefunctionsinH?Inthischapter,wewillanswerthisquestionafrmatively. 3.1Denitions Denition5. LetHbeaHilbertSpaceandXHsuchthatspanXisdenseinH.Thereprodcingkernelpresentation(RKP)isdenedtobethepair(H,X).ItisviewedasareproducingkernelHilbertspacewhereevaluationisdenedas:f(a)=hf,aiforallf2Handa2X. 22

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GivenanyreproducingkernelHilbertspaceHwhosefunctionstakevaluesinCanditscollectionofkernelsX,thentheRKPdenedby(H,X)isthesamereproducingkernelHilbertspace(wherewesayf(kw)f(w)).InthiswayweareabstractingthenotionofaRKHS. Denition6. Let(H,X)beaRKPandsupposeg:X!C.IfthesetD(Mg)=ff2Hj9h2Hst.h(x)=g(x)f(x)8x2XgisdenseinH,thenwesaythatMg:D(Mg)!H,Mgf=gf,isadenselydenedmultiplier.Thecollectionofallmultipliersfor(H,X)isdenotedbyMX.Whenthereisnoconfusion,wemaysimplysaysuchafunctiongisamultiplier.Alsoforsimplicitywewilloftensayg2MXforMg2MX.Foradenselydenedoperatortherstquestiontoaskisifitisclosed.Itturnsoutthatthisfollowsimmediatelyfromthereproducingproperty. Proposition3.1. Let(H,X)beanRKPandletMgbeamultiplicationoperatoronHwithdomainD(Mg)givenabove.TheoperatorMgwiththisdomainisclosed. Proof. LetffngbeasequenceoffunctionsinD(Mg)thatconvergeinnormtoafunctionf2H,andsupposethatMgfn!h2H.Wewishtoshowthatf2D(Mg)andgf=h.Westartwithh(x):h(x)=hh,kxi=limn!1hgfn,kxi=limn!1g(x)fn(x)=limn!1g(x)hfn,kxi=g(x)hf,kxi=g(x)f(x).Thelimitsabovearejustiedsinceweakconvergence(convergenceinsidetheinnerproduct)iscontrolledbynormconvergence.Hencewehaveshownthath(x)=g(x)f(x)forallx2X,whichmeansf2D(Mg)andMgf=h. Onceweknowthatanoperatorisclosedanddenselydened,wealsoknowthatMgisclosedanddenselydened.Liketheirboundedcounterparts,thereproducingkernelsofaRKHSareeigenvectorsofMg. 23

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Proposition3.2. Let(H,X)beanRKP,andletg2MX.Everyx2XisaneigenvectorforMgwitheigenvalue g(x). Proof. Inordertoplacethekernelx2XinsideofD(Mg)wemustshowthatthelinearoperatorL:D(T)!CgivenbyL(f)=hMgf,xiiscontinuous.NotethathMgf,xi=g(x)f(x)forallf.Sincefn!finnormimpliesthatfn!fpointwise:L(fn)=g(x)fn(x)!g(x)f(x).ThusLiscontinuous.Moreover,f,Mgx=hgf,xi=g(x)f(x)=g(x)hf,xi=Df, g(x)xEforallf2D(Mg).SinceD(Mg)isdenseinH,thismeansMgx= g(x)x. Wenowseethatthereisaconnectionbetweendenselydenedmultiplicationoperatorsandreproducingkernels.WedeneanextensionofaRKPasfollows: Denition7. Let(H,X)beaRKPandsupposeXYH,thentheRKP(H,Y)iscalledanextensionof(H,X).GivenanRKPitiseasytondanextension.If(H,X)isanRKPthenforanyYforwhichXYHthepair(H,Y)immediatelysatisesthedenitionofaRKP.Forthisreasonalone,extensionsasdenedabovearenotinteresting.However,thereisamorecompellingreasontolookforanotherdenitionofextension. Example2. ConsiderthetrivlalRKP(H,H),andsupposethatg2MH.MgisaclosedoperatoronHandhasacloseddenselydenedadjointforwhicheveryvectorf2Hisaneigenvector.Immediatelyitfollowsthatgmustbeaconstantfunction.Wecanquicklyjustifythisbytakingtwolinearlyindependentvectorsinf,h2H.ThismeansMgf= g(f)fandMgh= g(h)h,butalsoMg(f+h)= g(f+h)(f+h).Expandingthelefthandsidewend: g(f)f+ g(h)h= g(f+h)(f+h) 24

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movingeverythingtotheleft,( g(f))]TJ ET q .478 w 158.39 -37.18 m 202.48 -37.18 l S Q BT /F3 11.955 Tf 158.39 -47.82 Td[(g(f+h))f+( g(h))]TJ ET q .478 w 270.95 -37.18 m 315.03 -37.18 l S Q BT /F3 11.955 Tf 270.95 -47.82 Td[(g(f+h))h=0.Sincefandharelinearlyindependent,g(f)=g(f+h)=g(h).Thevectorsfandhwerearbitrarylinearlyindependentvectors,thusgisconstant.Fromthisexampleweseethatifwearetryingtoextend(H,X)to(H,H)thenMHcanbemuchsmallerthanMX.IfwewishtohaveaninterestingconnectionbetweenanRKPanditsextension,weshouldstartbyexaminingthedenselydenedmultipliers.Beforewedenethisstrongerextension,weneedalemma. Lemma4. Suppose(H,Y)isanRKPextending(H,X).Ifg1,g22MYandg1jX=g2jX,theng1=g2. Proof. Letg1andg2beasinthehypothesis.Letg=g1jX=g2jX.ThengisadenselydenedmultiplicationoperatorfortheRKP(H,X).LetD(Mg)bethedomainofMgasamultiplierinMX,andtakeD(Mg1)tobethedomainofMg1asamultiplierinMY.NotethatD(Mg1)D(Mg).Takeapointy2Yandletf2D(Mg1)beafunctionforwhichf(y)=hf,yi6=0.Ifwexh=Mg1f=Mgf,thismeanshh,yi=g1(y)hf,yiandg1(y)=hh,yi hf,yi.However,thevaluesofhh,yiandhf,yidonotdependonhowgwasextended. Lemma 4 tellsusthatXisasetofuniquenessforthemultipliersinMY.Comparethistotheexampleatthebeginningofthechapterwherewesawthatasequenceofpointsaccumulatingattheoriginwasasetofuniquenessforfunctionsanalyticinthedisc. Denition8. Let(H,Y)beanextensionof(H,X).IfeverymultiplierinMXcanbeextendedtobeamultiplierinMY,thenwesay(H,Y)isarespectfulextensionof(H,X). 25

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FromExample 2 weseethatnoteveryextensionisarespectfulextension,andthereshouldbesomeupperboundtohowfarwemayextendagivenpresentation.Inthenextsectionwewillexploretheseextensions,andndthemaximalrespectfulextensionofapresentation.Thefollowingtwolemmashelpussimplifyoursearchforanextension: Lemma5. Let(H,X)beanRKPandx1,x22X.Ifx1+x22Xandg2MX,theng(x1)=g(x2)=g(x1+x2). Proof. Letf2D(Mg)andseth=gf2HFirstwehave:h(x1+x2)=g(x1+x2)f(x1+x2)=g(x1+x2)hf,x1+x2i=g(x1+x2)(f(x1)+f(x2)).Wealsocanseethath(x1+x2)=hh,x1+x2i=h(x1)+h(x2)=g(x1)f(x1)+g(x2)f(x2).Rearrangingterms,wendthefollowingrelation:(g(x1+x2))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x1))f(x1)=(g(x2))]TJ /F3 11.955 Tf 11.96 0 Td[(g(x1+x2))f(x2).SincefisanarbitraryelementofD(Mg)andD(Mg)isadensesubspaceofH,thistellsusthatg(x1+x2))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x1)=g(x1+x2))]TJ /F3 11.955 Tf 11.95 0 Td[(g(x2)=0andg(x1)=g(x1+x2)=g(x2). Lemma6. If(H,X)isaRKPand:X!Cisafunctionsuchthat(x)6=0forallx2X,thenMX=MXwhereX=f(x)xjx2Xg. 26

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Proof. Letg2MX,thenidentifyg(x)g((x)x).Foreachelementf2D(Mg)andforgf=h2Hthefollowingholdsforeveryx02X:hh,(x0)x0i=g(x0)hf,(x0)x0ibutthenmultiplyingbothsidesby (x0))]TJ /F4 7.97 Tf 6.58 0 Td[(1:hh,x0i=g(x0)hf,x0iThusweseeg2MX,withthesamedensedomain.Theotherinclusionisshownidentically. Lemma 6 allowsustoonlyconsiderx2Xforwhichkxk=1.Thisalsomeansforelementinspan(x)\Xweonlyneedtoconsiderasinglerepresentative,sinceeachoftheseelementsareequivalentunderthedenselydenedmultipliers.Weendthissectionwithanotherstraightforwardexample.ThisiscomplimentarytoExample 2 Example3. LetX=fengn2NwherefengisanorthonormalbasisforaseparableHilbertspaceH.ConsidertheRKP(H,X)anditssetofmultipliersMX.Inthiscasewecanvieweachmultipliergasasequenceindexedbythenaturalnumbers.Inthiscase,everysequencefg(en)gginthecomplexplanewillbeamultiplier.WecanseethisbyshowingthatthesubsetH0:=ff2Hjf=PNn=0nengisinthedomainofg.SincethissubsetisdenseinH,themultiplierwillbedenselydened.Thisfollowstrivially,sinceh=PNn=0g(en)nen2Hisafunctionforwhichh(en)=g(en)f(en)wheref=PNn=0nen. 3.2RKPExtensionsNowthatwehavesomeconceptofanextensionofaRKPournextgoalistodiscoverhowwemightndarespectfulextension.Thefollowingisaminimaltypeofextensionthattakesadvantageofcontinuity.Inasense,ittellsusthatifwearenotalreadyworkingwithaweaklyclosedsetofkernels,thenourcollectionistoosmall. 27

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Theorem3.1(TheTrivialExtension). Let(H,K)beaRKP,andsupposeg2MK.Inthiscase,gisweaklycontinuousonKnf0Hg.Moreovergextendstobeamultiplieron Kwknf0Hg. Proof. LetfkngKHandsupposekn!wkk2K.Thereforewehavehf,kni!hf,kiforallf2H.ThusfisweaklycontinuousonKwhenwedenef(z):=hf,zi.Letg2MK,andletz02Knf0Hg.Thereexistssomef2D(Mg)suchthatf(z0)6=0,sinceD(Mg)isadensesubsetofH.Leth2Hwhereh(z)=g(z)f(z).Fromthepreviousparagraph,handfareweaklycontinuousatz0,soh(z)=f(z)=g(z)isweaklycontinuousatz0.Sincez0wasanarbitraryelementofK,gisweaklycontinuousonK.Letz02 Kwknf0HgandsupposefkngisasequenceinKconvergingtoz0weakly.Letf2D(Mg)suchthatf(z0)6=0andleth(z)=g(z)f(z)2Hforallz2K.Sincef(kn)!f(z0)6=0andh(kn)!h(z0)thismeansh(kn)=f(kn)convergesinC.Inadditionthevaluesofh(kn)=f(kn)=g(kn)areindependentofthechoiceoff.Thuswecandeneg(z0)uniquelyasthislimit,andgextendstobeamultiplieron Kwknf0Hg. Theorem 3.1 tellsusthatwecanextendourmultipliersfromKtoitsweakclosure.ItalsoshowsusthateverymultipliermustbecontinuousonKattheveryleast.Thisnotionhasalotofvaluewhenwesetouttoclassifycollectionsofdenselydenedmultipliers.Forthegeneralcase,wecanonlyextractthecontinuityofourmultipliers.However,ifwearedealingwithaspaceofanalyticfunctions,thenthesameproofyieldsthatmultipliersmustalsobeanalytic.Inparticularifg2MCfor(F2,C),thengmustbeentire.Sowecanconcludethat,looselyspeaking,ifg2MXforsomeRKP(H,X),thengisassmoothasthemembersofH.WewillseethatthismethodofextensionisweakerthantheonewewillndfortheHardySpaceinExample 4 .TherewewilltakeadvantageoftheanalyticityofthefunctionsinH2toextendthemultipliers.Thisbringsupthequestion:howfarcanwe 28

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expecttoextendthespaceinthegeneralcase?Weknowthereissomelimitation,sincenotallextensionsarerespectful.JuryandMcCulloughfoundthatthereisamaximalextensionthatcanbefoundthroughthedenselydenedmultiplicationoperatorsthemselves. Theorem3.2(JM-ExtensionTheorem). Let(H,X)beanRKPwithmultipliersMX.ThenthereisauniquelargestsetYwithXYHsuchthat(H,Y)isarespectfulextensionof(H,X);thisYisequaltothesetofallcommoneigenvectorsfortheoperatorsMg,g2MX. Proof. Supposethatg2MX.IfgcanbeextendedtobeinMY,theneachy2YisaneigenvectorforMgbyProposition 3.2 .OntheotherhandsupposeYisthecollectionofallvectorsyforwhichy2D(Mg)forallg2MXandyisaneigenvectorforeachg2MX.Foreachg2MXwehaveMgy=y.Takeg(y)=.WemustshowthateachgextendedinsuchafashionisinMY.LetDdenotethedomainofMgfortheRKP(H,Y).WewillshowthatD=D(Mg),whichmeansthatg2MY.ItfollowsimmediatelybytakingrestrictionstoXthatDDX(Mg)(thedomainasamultiplierinMX).FortheotherinclusionwewillusethefactthatMgwiththedomainDX(Mg)isacloseddenselydenedoperator,henceMg=Mg.Thusf2D(Mg),anditfollowsfromthedenitionofthedomainofanadjointthatthefollowinglinearfunctionalisboundedonspanY:Lf:Xcjyj!Df,MgXcjyjE=Xcjg(yj)hf,yji.Thereforethereisauniqueh2HforwhichLfy=hh,yiforally.Inparticular,h(y):=hh,yi=f,Mgy=g(y)hf,yi=g(y)f(y).Thusf2DandDX(Mg)D.HenceD=DX(Mg)andg2MY. 29

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FinallynoticethatsinceXY,h(x)=g(x)f(x)forallx.WhichmeansMgf=hwhethergistakentobeinMXorMY.ThesamehthatworkedforXworksforY. Thistheoremprovidesatargetforourextensiontheory.WewishnotonlytondarespectfulextensionofagivenRKP,butwewanttondamaximalone.Usingthistheoremcanbeoverwhelming.Oftenthespaceofmultipliersisuncountable,sobarringspecialcircumstances,wewouldnotexpecttobeabletondoutifagivenRKPisindeedmaximal.Inspiteofthis,laterwewillshowthat(H2,D)isamaximalRKP. 30

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CHAPTER4CLASSESOFDENSELYDEFINEDMULTIPLICATIONOPERATORS 4.1HardySpaceRecallthatafunction2N+istheratiooftwoH2functions,wherethedenominatorcanbechosentobeouter.Thechoiceoffunctionscanbemademuchmorespecicandleadstoacanonicalrepresentationfor. Theorem4.1. [ 21 ]If2N+,thenthereexistsauniquepairoffunctionsb,a2H1forwhichjbj2+jaj2=1,aisouterand=b=a.InSarason'spaperUnboundedToeplitzOperators[ 21 ]hecharacterizedthedenselydenedmultiplicationoperatorsofH2asfollows: Theorem4.2(Sarason). Thefunction:D!CisthesymbolforadenselydenedmultiplicationoperatoronH2iffisintheSmirnovclassN+.Moreover,if=b=aisthecanonicalrepresentationof,thenD(M)=aH2. Proof. LetD(M)bethedomainofM.SinceMisdenselydened,thismeansthatD(M)isnontrivial.Letf2D(M)andsupposef60.Then=f=fistheratiooftwoH2functionsandisintheNevanlinnaclassN.Nowsupposethat= =where andhaverelativelyprimeinnerfactors.Takef2D(M),andsetg=f.Theng= f.Nowlet 0and1betheouterfactorfor andtheinnerfactorforrespectively.Since andhaverelativelyprimeinnerfactorsthismeansthat 0f21H2,andhencef21H2.ThusweconcludethatD(M)1H2.SinceD(M)isdense,and1H2isdenseonlyif1isconstant.Thuswehave1isconstant.Weconcludethatisanouterfunction.Fortheotherdirection,suppose2N+.Let=b=abeitscanonicalrepresentationwhereaisouter,thenaH2isdenseinH2.Moreover,aH2D(M)trivially,andisthesymbolforadenselydenedmultiplicationoperator. 31

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Fortheothercontainment,takef2D(M).Thenjfj2=jbj2jfj2 jaj2=f a2)-222(jfj2.SincefandfareinL2,soisf=a.Moreover,f=aisanalyticinthedisc,sinceaisouter.Thusf=a2H2.FromthisweconcludethatD(M)aH2.Thiscompletestheproof. WenowwilluseSarason'stheoremasalaunchingpointforextensionsofRKPsinvolvingtheHardyspace. Example4. LetH2betheusualhardyspaceofthediscandletK=fkwn(z)gforsomeuniquenesssetfwng2D.AuniquenesssetisacollectionofpointsinthediscforwhichP(1)-278(jwnj)=1.Thesearepointswhodonotapproachtheboundaryquickly.Forexample,ifasequenceconvergesintheinteriorofthedisc,thenitisauniquenesssetforH2.FortheRKP(H2,K),itcanbeshownthatMKconsistsoffunctionsintheSmirnovclassthathavebeenrestrictedtofwng.ThisfollowsfromtheproofofTheorem 4.2 .FormallyelementsofMKarefunctionsdenedonK,notnecessarilyonD.However,sinceN+isaquotientofH2functions,ithasthesameuniquenesssetsasH2.ThereforeeverymultiplierinMKextendstobeamultiplierinMDandthisextensionisunique.Thus(H2,D)isanextensionthatrespects(H2,K).ThenotionsusedinthisexamplestemfromH2asaspaceofanalyticfunctions.Setsofuniquenessexistforeveryspaceofanalyticfunctions,inthecaseofH2thisisanysetfwngDwhereP(1)-242(jwnj)=1.WealsousedthefactorizationtheoryofH2functionsimplicitlythroughininvocationofTheorem 4.2 .BothoftheseideasshouldcarryovertoalargeextentforanyRKHSofanalyticfunctions.However,forspacesthatdonotinvolveanalyticfunctions,othertechniquesmustcomeforward,aswewillseewhenwecharacterizethedenselydenedmultiplication 32

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operatorsfortheSobolevspace.NowweshowthatH2ismaximalinregardstorespectfulextensions. Example5. Let(H2,Y)bearespectfulextensionof(H2,D).ThismeansMX=MYandMz2MY.ByLemma 3.2 ,everyvectorinYH2isaneigenvectorofMz.IfweconsidertheeigenvectorequationforMzwend:Mzf(z)=f(z))]TJ /F3 11.955 Tf 11.96 0 Td[(f(0) z=f(z).Thisyields:f(z)=f(0)(1)]TJ /F7 11.955 Tf 12.98 2.66 Td[(z))]TJ /F4 7.97 Tf 6.59 0 Td[(1,whichissimplyascaledcopyofoneoftheoriginalreproducingkernels.Thus(H2,Y)=(H2,D)anditismaximal.ThismethodofextensionisstrictlystrongerthantheTrivialExtensioninTheorem 3.1 .Forexampleiffwngwasasequenceofdistinctpointsconverginginthedisc,theTrivialExtensionwouldonlyaddthelimitpoint. 4.2FockSpaceAswasshowninthepreliminaries,thesymbolsforboundedmultipliersovertheFockspaceweretheconstantfunctions.ThismeansboundedmultiplicationontheFockspaceisnotaninterestingconcept.However,itiseasytondsomeexamplesofdenselydenedmultiplicationoperatorsovertheFockspace.ForinstanceawellstudiedoperatorisMz.Wecanseethisoperatorisdenselydened,sincethepolynomialsareadensesubspaceofF2.MoreinterestinglythedenselydenedoperatorMzisd=dz.ItfollowsthattheeigenvectorsforMzarepreciselykw=ewz,andtheFockspaceRKP(F2,C)ismaximal.ThisalsodemonstratestheneedofdenselydenedmultipliersintheextensionsofRKPs,sinceifweonlycollectedthecommoneigenvectorsoftheboundedmultiplicationoperatorswewouldextendto(F2,F2).Thetrivialityoftheboundedmultipliersfolloweddirectlyfromthefactthatanymultipliermustbeentire(boundedordenselydened).Itturnsouttheproblemswiththe 33

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Fockspacearedeeperthanthis.ThenormontheFockspaceinducesagrowthlimitfortheentirefunctionsinthisspace.Inarecentpreprint[ 22 ],KeheZhuexploredwhatarecalledmaximalzerose-quences.WesayasequenceisazerosequenceforF2ifthereisafunctioninF2thatvanishesatexactlythosepointswithappropriatemultiplicity.ForazerosequenceZwedeneIZ:=ff2F2jf(w)=0forallw2Zg.ThefollowingisCorollary8in[ 22 ]. Theorem4.3(Zhu). LetZbeazerosequenceforF2andk2N.Thefollowingareequivalent: 1. dim(IZ)=k 2. Foranyfa1,...,akgthesequenceZ[fa1,...,akgisauniquenesssetforF2,butthesequenceZ[fa1,...,ak)]TJ /F4 7.97 Tf 6.59 0 Td[(1gisnot. 3. Forsomefa1,...,ak)]TJ /F4 7.97 Tf 6.59 0 Td[(1gthesequenceZ[fa1,...,ak)]TJ /F4 7.97 Tf 6.59 0 Td[(1gisnotauniquenesssetforF2,butZ[fb1,...,bkgisauniquenesssetforsomefb1,...,bkg.KeheZhushowedtheexistenceofthesemaximalzerosequencesviatheWeierstrasssigmafunction.ThefollowingarestandardinthetheoryofFockspacesandcanbefoundinZhu[ 14 22 ]amongothers. Denition9. Forw,w0,w12Cwecallthesubsetofthecomplexplane(w,w0,w1)=fw+nw0+mw1jn,m2Zgalatticeofpointscenteredatw. Denition10. Fixthelattice(0,p ,ip )==fwmn=p (m+in)j(m,n)2Z2g.WedenetheWeierstrasssigmafunctionby(z)=zY(m,n)6=(0,0)1)]TJ /F3 11.955 Tf 19.95 8.09 Td[(z wmnexpz wmn+z2 2w2mn. Lemma7. j(z)jejzj2isdoublyperiodicwithperiodsp andip Proposition4.1. [ 22 ]TheWeierstrasssigmafunctionisnotinF2.Moregenerallyiff2F2andfvanishesonthenf0.Notethatthedoubleperiodicityofj(z)jejzj2impliesthat(z)62F2.However,(z)=z(z)]TJ /F3 11.955 Tf 12.2 0 Td[(w1,1)isinF2sincetheintegrandink(z)=z(z)]TJ /F3 11.955 Tf 12.21 0 Td[(w1,1)kF2willbeO(1=z2)for 34

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largez.Thus0:=nf0,w1,1gisazerosetforF2,butbyProposition 4.1 isnotazerosetofF2.Hence,0isanitedimensionalzeroset.ThisyieldsthecounterintuitiveresultthatmultiplicationbyapolynomialcanaffectthegrowthrateofaFockspacefunctionenoughsothattheproductisnotintheFockspace.WecandrawacorollarytoTheorem 4.3 thatcanapplytodenselydenedmultiplicationoperators. Corollary1. IfZisamaximalzerosequenceforF2andg2IZ,thengisnotadenselydenedmultiplicationoperator. Proof. IfD(Mg)=ff2F2jgf2F2gweredense,thenD(Mg)isinnitedimensional.MoreovergD(Mg)IZ.Thuswehaveacontradiction. ThisdemonstratesthatunliketheHardyspace,noteveryFockspacefunctionisasymbolforadenselydenedmultiplier.Wecandrawanotherconclusion.NotethatwearetreatingZasasequenceandnotasetofpoints.ThusifZisazerosequencefortheFockspacewithdim(IZ)=1,thendim(IZ0)=1forZ0=Z[f0,0,...,0| {z }ntimesg.Thisleadsustothefollowingconjecture: Conjecture1. IfZisazerosequenceforF2forwhichdim(IZ)=1,thenIZcontainsthesymbolforsomedenselydenedmultiplicationoperator.Wenowtakeasmalldiversiontodiscusssomealgebraicpropertiesofmultipliers.WewillndthatthecollectionofdenselydenedmultipliersovertheFockspaceisnotanalgebra.Fromourexperiencewithboundedmultipliers,wearetrainedtoviewcollectionsofmultipliersasalgebras.ThisintuitionwouldnotbetrayusifwerestrictedourviewtothemultipliersovertheHardyspaceoreventheSobolevspace.InfactwewillseefortheSobolevspaceallofourmultipliersarebounded.WesetouttoshowthatthemultipliersovertheFockspacelackthesealgebraicproperties.Webeginwithanoptimisticstatement,thatproductsofdenselydenedmultiplierswithboundedmultipliersalwaysyieldnewmultipliers. 35

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Proposition4.2. Let(H,X)beanRKP.Ifg2MX,M2B(H),thenMMgandMgMaredenselydenedoperatorswithdomainD(Mg). Proof. Miseverywheredened,soMMgisdenselydened.Nowsupposethatf2D(Mg).Forsuchafunctiongf2HandsinceM:H!H,(gf)2H.Noticethat(gf)=g(f)2H,sobydenitionf2D(Mg).WenowseethatleavesD(Mg)invariant,andMgM:D(Mg)!Hisadenselydenedoperator. ForspacesliketheFockspace,thisshedslittlelight.TheFockspacehasonlyconstantboundedmultipliers,byLiouville'stheorem.Thistellsusforanydenselydenedmultiplierwecanmultiplybyaconstantandhaveanewdenselydenedmultiplier.Thisisnotaninterestingresult.InthiscaseweseeProposition 4.2 doesnothelpus.ItisinfactmuchworsefortheFockspace.WeuseLemma 7 andProposition 4.1 toprovethefollowing. Theorem4.4. ThedenselydenedmultiplicationoperatorsintheFockspacearenotanalgebra. Proof. Weprovethisbyconstructingacounterexample.Thefunction(z)isentireandhaszerosexactlyatthepointsin.Moreoverthefunctionj(z)jejzj2isdoublyperiodicwithperiodsp andip .Considerthefunction(z=2).Thisfunctionhaszerosatthepointsw(2m)(2n)in.Fromthestatementinthepreviousparagraph,j(z=2)jejzj2=4isalsodoublyperiodic.ThefollowingcalculationdemonstratesthatM(z=2)isdenselydenedmultiplierfortheFockspacebyestablishingthatthepolynomialsareinitsdomain:ZCjzk(z=2)j2ejzj2dA(z)=ZCjzj2kj(z=2)ejzj2=4j2ejzj2=2dA(z)CZCjzj2k ejzj2=2dA(z)<1.Theinequalityfollowsfromdoubleperiodicityandcontinuity.ThespaceofpolynomialsaredenseintheFockspace,soD(z=2)isadensesubspaceofF2. 36

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SimilarcalculationsshowM((z)]TJ /F4 7.97 Tf 6.58 0 Td[(1)=2),M((z)]TJ /F8 7.97 Tf 6.59 0 Td[(i)=2),andM((z)]TJ /F4 7.97 Tf 6.59 0 Td[((i+1))=2)aredenselydenedmultipliers.However(z=2)((z)]TJ /F7 11.955 Tf 12.02 0 Td[(1)=2)((z)]TJ /F3 11.955 Tf 12.03 0 Td[(i)=2)((z)]TJ /F7 11.955 Tf 12.03 0 Td[((i+1))=2)f2F2forf2F2onlyiff0.ThisisbecausetherearenononzeroFockspacefunctionswiththezeroset. 4.3TheDirichlet-HardySpaceandthePolylogarithmADirichletseriesisaseriesoftheformf(s)=P1n=1ann)]TJ /F8 7.97 Tf 6.58 0 Td[(swherean2Cforalln.TheseseriesarecentraltothestudyofAnalyticNumberTheory,andinOperatorTheoryhavebeenstudiedasastartingpointforRKHS's.In[ 23 ]Hedenmalm,LindqvistandSeipshowedthatifthecoefcientsanaresquaresummable,thenf(s)convergesinthehalfplane>1=2.(Weadoptthestandardconventionthat=Re(s).)WesaythatsuchfunctionsareintheDirichlet-HardyspacewhichwewilldenotebyH2d.ThereproducingkernelforH2disgivenby:K(s,t)=1Xn=11 ns+t=(s+t).In[ 23 ]theyalsoshowedthattheboundedmultiplicationoperatorsoverH2darepreciselythosefunctionsthatarerepresentableasaDirichletseriesintherighthalfplane>0andareboundedinthathalfplane.McCarthyalsodidsomeworkinthisdirectionin[ 24 ],butgeneralizedtoweightedDirichlet-Hardyspaces.In[ 24 ]McCarthyalsofoundaweightthatwouldyieldaspacewiththeNevanlinna-Pickproperty.NowweintroduceanewRKHS.WebeginconstructingitbytakingthetensorproductofH2withH2d.ThisisaRKHSwithorthonormalbasisznm)]TJ /F8 7.97 Tf 6.59 0 Td[(sforalln0andm1.ThetensorproductspacehasreproducingkernelK(z,w,s,t)=(s+t) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(zw.Weareconcernedwiththeclosedsubspacemadefromthosebasisvectorsforwhichn=m.Inparticularwedenethefollowing: 37

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Denition11. ThePolylogarithmicHardySpaceistheHilbertspaceoffunctionsintwovariablesgivenbyPL2=(f(z,s)=1Xk=1akzk ks:1Xk=1jakj2<1)ThespacePL2isaRKHS,sinceitisaclosedsubspaceofaRKHS.Moreover,eachf(z,s)inheritsconvergenceonjzj<1and>1=2.ThereproducingkernelforthisspaceisK(z,w,s,t)=1Xk=1zkwk ks+t=Ls+t(zw).HereLs(z)denotesisthepolylogarithm.ThespacePL2hassomeadditionalproperties.Firstnotethatiff(z,s)=P1k=1akzkk)]TJ /F8 7.97 Tf 6.58 0 Td[(s2PL2,thenf(0,s)=0.Alsosinceak!0andjzj<1,theDirichletseriesconvergesabsolutelyforallsinC.Inotherwordsforxedz0,f(z0,s)hasanabscissaofabsoluteconvergenceofandf(z0,s)isentire.Thisbuysusthefollowingproposition: Proposition4.3. If(z,s)isthesymbolforaboundedmultiplicationoperatoronPL2then(z,s)mustbeconstantinbothvariables. Proof. Let(z,s)beathesymbolforaboundedmultiplicationoperatoronPL2.Firstxz02Dnf0gands02C.Takethefunctionf(z,s)=P1k=1akzkk)]TJ /F8 7.97 Tf 6.58 0 Td[(s2PL2forwhichf(z0,s0)6=0.SinceMisamultiplieronPL2,takeh(z,s)=P1k1bkzkk)]TJ /F8 7.97 Tf 6.59 0 Td[(s=Mf.Wemaywrite(z0,s)=h(z0,s)=f(z0,s)and(z0,s)isanalyticats0.Sinces0wasarbitrary,(z0,s)mustbeentirewithrespecttos.Thefunction(z0,s)isbounded,sinceitisthesymbolforaboundedmultiplier.ByLiouville'stheorem,(z0,s)isconstant.Therefore(z,s)=(z)2H1.Now(z)f(z,s)=1Xk=1ak((z)zk)k)]TJ /F8 7.97 Tf 6.58 0 Td[(s=1Xk=1bkzkk)]TJ /F8 7.97 Tf 6.59 0 Td[(s. 38

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BytheuniquenessofDirichletserieswehave(z)ak=bkforallk.Sincef60wehavethat(z)isconstant.Hencetheproposition. ThispropositionissubsumedunderTheorem 4.6 .Therewewillestablishthattherearenonon-constantdenselydenedmultiplicationoperatorsoverthisspace.Sinceboundedmultipliersarealldenselydened,itfollowsthatanyboundedmultiplierisconstantinbothvariables.TheproofofTheorem 4.6 ismorealgebraicinavorthansomeofthepreviousresults.WewillalsoneedsomefactsaboutDirichletseries.TobeginwithwewillbetakingadvantageoftheDirichletconvolutionfortheproductofDirichletspaces.Recallthatiff(s)=P1k=1akk)]TJ /F8 7.97 Tf 6.58 0 Td[(sandg(s)=P1k=1bkk)]TJ /F8 7.97 Tf 6.59 0 Td[(saretwoDirichletseriesthatconvergeinsomecommonhalfplane,theninthathalfplanewehave(fg)(s)=P1k=1ckk)]TJ /F8 7.97 Tf 6.59 0 Td[(swhereck=Pnm=kanbm.WewillalsobeconcernedwiththeDirichletseriesforthemultiplicativeinverseoff(s).Itisastandardexercisetoshowthatiff(s)=P1k=1akk)]TJ /F8 7.97 Tf 6.58 0 Td[(sanda16=0,thenwecandeneaformalinverseseriesg(s)=P1k=1bkk)]TJ /F8 7.97 Tf 6.58 0 Td[(srecursively.However,thereisnoguaranteethatthisserieswillconvergeforanys.TondaconvegenthalfplaneweutilizethefollowingtheoremduetoHewittandWilliamson[ 25 ]: Theorem4.5(HewittandWilliamson). SupposethatP1k=1jakj<1.IftheDirichletseriesf(s)=P1k=1akk)]TJ /F8 7.97 Tf 6.59 0 Td[(sisboundedawayfromzeroinabsolutevalueon0then(f(s)))]TJ /F4 7.97 Tf 6.59 0 Td[(1=P1k=1bkk)]TJ /F8 7.97 Tf 6.59 0 Td[(sfork0whereP1k=0jbkj<1andthisseriesconvergesin0.Thisisaspecializationoftheirresult,butitiswhatweneedinordertoproceed.Recallthatiff(s)=P1k=1akk)]TJ /F8 7.97 Tf 6.59 0 Td[(sconvergesforsomesandisnotidenticallyzero,thenthereissomehalfplanewheref(s)isnonvanishing.Moreoverf(s)!a1as=Re(s)!1.Thismeanswhena16=0wecangraduatetheresultfromthetheoremtothishalfplaneandnotjust0.OtherfactsaboutDirichletseriescanbefoundinApostol'sbookIntroductiontoAnalyticNumberTheory[ 26 ].AsawarmuptoTheorem 4.6 werstestablishthefollowingproposition: 39

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Proposition4.4. NofunctioninPL2isthesymbolforadenselydenedoperatoroverPL2. Proof. Letf(z,s)=P1k=1akzkk)]TJ /F8 7.97 Tf 6.58 0 Td[(sand(z,s)=P1k=1bkzkk)]TJ /F8 7.97 Tf 6.59 0 Td[(sbeinPL2.Supposethattheirproducth=fisinPL2,whichmeansh=P1k=1ckzkk)]TJ /F8 7.97 Tf 6.59 0 Td[(swhereck=Pnm=kanbmzm+n.ThisfollowsfromDirichletconvolution.Foreachckthelargesttermis(akb1+bka1)zk+1.Thiscoefcientmustbezerosinceckisascalar.Thuswendthatbk=b1=)]TJ /F3 11.955 Tf 9.3 0 Td[(ak=a1foreachk.However,thismeansthatthefunctionsinthedomainofMmusthavethesameratiosofitscoefcientsas.Inparticular,ifg(z,s)=P1k=1dkzkk)]TJ /F8 7.97 Tf 6.59 0 Td[(swemusthaved2=d1=)]TJ /F3 11.955 Tf 9.3 0 Td[(b2=b1.FromthisweseethatD(M)(f2PL2:f(z,s)=a1z)]TJ /F3 11.955 Tf 13.15 8.09 Td[(bn b1znn)]TJ /F8 7.97 Tf 6.58 0 Td[(s+1Xk>1:k6=nakzkk)]TJ /F8 7.97 Tf 6.58 0 Td[(s).Thisimpliesthat(bn=b1)z+znn)]TJ /F8 7.97 Tf 6.58 0 Td[(s2D(M)?.FinallyweconcludethedomainisnotdenseinPL2,sinceotherwiseD(M)?=f0g. Aconvenienceofthelastpropositioncamefromknowingtheformofbeforehand.Inordertoproceedtothefulltheoremwemustshowthatif(z,s)isamultiplieroverPL2,then(z,s)isoftheformP1k=1k(z)zkk)]TJ /F8 7.97 Tf 6.59 0 Td[(swherek(z)issomefunctionofz2D.Tothisendwebeginwiththefollowinglemmata: Lemma8. Letz02D,>0andf(z,s)2PL2.Forevery>0thereisa>0sothatforallz2B(z0)andallsforwhich=Re(s)3=2+wehavejf(z,s))]TJ /F3 11.955 Tf 11.96 0 Td[(f(z0,s)j<. 40

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Proof. Fixz02Dand>0.Andconsiderjf(z,s))]TJ /F3 11.955 Tf 11.96 0 Td[(f(z0,s)j=1Xk=1akzk)]TJ /F3 11.955 Tf 11.96 0 Td[(zk0 ks=jz)]TJ /F3 11.955 Tf 11.96 0 Td[(z0jXk=1akzk)]TJ /F4 7.97 Tf 6.59 0 Td[(1z00+zk)]TJ /F4 7.97 Tf 6.59 0 Td[(2z10++z0zk)]TJ /F4 7.97 Tf 6.59 0 Td[(10 ksjz)]TJ /F3 11.955 Tf 11.96 0 Td[(z0j1Xk=1jakj1 k)]TJ /F4 7.97 Tf 6.58 0 Td[(1jz)]TJ /F3 11.955 Tf 11.96 0 Td[(z0j 1Xk=1jakj2!1=2 1Xk=11 k2)]TJ /F4 7.97 Tf 6.58 0 Td[(2!1=2=jz)]TJ /F3 11.955 Tf 11.96 0 Td[(z0jkfk(2)]TJ /F7 11.955 Tf 11.95 0 Td[(2)1=2.Thezetafunctionisdenedfor>3=2,and(2)]TJ /F7 11.955 Tf 11.95 0 Td[(2)isadecreasingfunctionof.For=Re(s)3=2+wehave:jf(z,s))]TJ /F3 11.955 Tf 11.95 0 Td[(f(z0,s)jK. Proof. Fixarealnumber0<>0.ByvirtueofLemma 8 ,wecanchoosezsothatitvariescontinuouslywithrespecttoz(possiblyrequiringz3=2+with>0xed).InparticularztakesamaximumvalueonK,wecallthisK.Ineveryhalfplane>Kthefunctionjf(z,s)jisboundedawayfromzero.Foreveryz2D,P1k=1jakzkj<1.WenowapplyTheorem 4.5 toobtaintherepresentationfor(f(z,s)))]TJ /F4 7.97 Tf 6.58 0 Td[(1. 41

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Theorem4.6. TheonlydenselydenedmultipliersoverPL2aretheconstantfunctions. Proof. Supposethat(z,s)isadenselydenedmultiplieroverPL2.LetKbeacompactsubsetofthepunctureddisk.Letf(z,s)=Pakzkk)]TJ /F8 7.97 Tf 6.59 0 Td[(s2D(M)andseth(z,s)=(f)(z,s)=Pbkzkk)]TJ /F8 7.97 Tf 6.58 0 Td[(s.Further,assumethata16=0.SuchafunctionmustbeinD(M)sinceitisdenseinPL2.LetKbethehalfplanedescribedinLemma 9 .Then(f(z,s)))]TJ /F4 7.97 Tf 6.59 0 Td[(1=P1k=1k(z)k)]TJ /F8 7.97 Tf 6.59 0 Td[(sinthishalfplaneand(z,s)=h(z,s)=f(z,s).Inparticular(z,s)=P1k=1k(z)k)]TJ /F8 7.97 Tf 6.58 0 Td[(sforallz2K,wherek(z)issomefunctiononK.WewillshowthatmustbeaconstantfunctionbytakingadvantageoftheincompatibilityofDirichletconvolutionandtheconvolutionofcoefcientsofpowerfunctions.Theproofproceedsbyinductiononthenumberofprimefactorsofk.Herewewillshowthateachcoefcientk(z)withk>1iszero.Alongthewaywewillalsoestablish2k(z)=c2kz2k)]TJ /F4 7.97 Tf 6.59 0 Td[(1,butc2kwillturnouttobezerointhenextstep.Forallknotprime,inproofbelowitisessentialtonoticethatifk=p1p2pnistheprimefactorizationfork,thenk+1>2k pi+pi)]TJ /F7 11.955 Tf 12.57 0 Td[(1.Thiswillisolateaterminthepolynomials,andtheresultwillfollow.Westartwithk=1,usingDirichletconvolutionwendthecoefcientb1z1=1(z)a1z1.Thismeans1(z)=b1=a1:=c1.Nowifk=2wendthatb2z2=2(z)a1z1+1(z)a2z2.Since1(z)isaconstantfunction,thismeansthat2(z)=c2z1forsomec2.Ifwetakek=4wendthatb4z4=4(z)a1z1+2(z)a2z2+1(z)a4z4=4(z)a1z1+c2a2z2+c1a4z4.Thus4(z)=b4)]TJ /F3 11.955 Tf 11.96 0 Td[(c1a4 a1z3)]TJ /F3 11.955 Tf 13.15 8.09 Td[(c2a2 a1z.However,sinceMisdenselydened,thereisanotherfunctiong=Pdkzkk)]TJ /F8 7.97 Tf 6.59 0 Td[(s2D(M)forwhichd16=0andd2=d16=a2=a1.Usingthesamealgorithmwewouldndthatthezcoefcientisc2d2d)]TJ /F4 7.97 Tf 6.59 0 Td[(11.Since4(z)isxed,wemusthavec2d2d)]TJ /F4 7.97 Tf 6.59 0 Td[(11=c2a2a)]TJ /F4 7.97 Tf 6.59 0 Td[(11. 42

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ThusweseethatsinceMisdenselydenedc2=0and4(z)=c4z3.Inthesamemannerwecanshowthatp(z)=0foreveryprimepand2p(z)=c2pz2p)]TJ /F4 7.97 Tf 6.58 0 Td[(1.Supposeforeachm
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Corollary2. Nomultiplicationoperator(withnonconstantsymbol)onthetensorproductofH2withH2dhasPL2asaninvariantsubspace.Letustakeamomenttosummarizetheseresults.FortheFockspacewedidnothaveanynontrivialboundedmultipliersbecauseofLiouville'stheorem,butwefoundthatthereisalargecollectionofnontrivialdenselydenedmultipliers.TheHardyspacehastheNevanlinnaclassasitscollectionofdenselydenedoperators,andthuseveryelementoftheHardyspaceandmanymorefunctionsaresymbolsofdenselydenedmultiplicationoperators.InthecaseofthePolylogarthmicHardyspace,evenwhenweallowforamultipliertobedenselydened,westillhaveonlytrivialmultipliers.ThoughthisfactfollowsfromthealgebraicpropertiesofDirichletseriesandnotfromtheanalyticpropertiesofthespace. 44

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CHAPTER5UNBOUNDEDMULTIPLICATIONONTHESOBOLEVSPACEHerewewillinvestigateunboundedmultiplicationovertheSobolevSpace.ThatisthespaceW1,2[0,1]ofabsolutelycontinuousfunctionson[0,1]whosealmosteverywheredenedderivativeiscontainedinL2[0,1].TheSobolevspacehasaNevanlinna-Pickkernel,asfoundbyAglerin[ 6 ].ThismakesitagoodnextstepaftertheclassicationoftheHardyspace.ItwasshownbyShields[ 5 ]thatthecollectionofboundedmultipliersovertheSobolevspaceistheSobolevspaceitself.Wesharpenthisbyshowingthatbyrelaxingboundedtodenselydened,thecollectionofmultipliersremainstheSobolevspace.Inparticular,nodenselydenedmultiplierovertheSobolevspaceisanunboundedmultiplier. 5.1DenselyDenedMultipliersfortheSobolevSpaceFirstwewillclassifythedenselydenedmultiplicationoperatorsforthesubspaceW=ff2W1,2[0,1]jf(0)=f(1)=0g.Theinvestigationofthisspacewillexposethecomplicationsthatcanariseinthegeneralcase,butultimatelyturnoutnottobepresent. Example6. Aswewillseeinthefollowingtheorem,thedenselydenedmultipliersofWarethosefunctionsthatarewellbehavedeverywherebuttheendpointsof[0,1].Takeforinstancethetopologist'ssinecurveg(x)=sin(1=x).Onanyintervalboundedawayfromzero,sin(1=x)iswellbehaved.TodeterminethatD(Mg)isdense,itisenoughtorecognizethatthesetoffunctionsthatvanishinaneighborhoodofzeroareinD(Mg)andthiscollectionoffunctionsisdenseinW. Example7. Twootherexamplesofexoticfunctionsthataresymbolsfordenselydenedmultiplicationoperatorsareg(x)=1=xandexp(1=x). 45

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Theorem5.1. Afunctiong:(0,1)!CisthesymbolforadenselydenedmultiplieronWiffg2W1,2[a,b]forall[a,b](0,1). Proof. FirstsupposethatgisadenselydenedmultiplieronW.Foreachx02(0,1)thereisafunctionf2D(Mg)suchthatf(x0)6=0,thisfollowsfromthedensityofthedomain.Leth=Mgf,sothatg(x)=h(x)=f(x).Thefunctionshandfaredifferentiablealmosteverywhereinaneighborhoodofx0,sothenisg.Sincex0isarbitrary,gisdifferentiablealmosteverywhereon(0,1).Fix[a,b](0,1).Bywayofcompactness,thereexistsanitecollectionff1,...,fkgD(Mg)togetherwithsubsets[a,t1),(s2,t2),(s3,t3),...,(sk)]TJ /F4 7.97 Tf 6.59 0 Td[(1,tk)]TJ /F4 7.97 Tf 6.59 0 Td[(1),(sk,b]sothatthesubsetscover[a,b]andfidoesnotvanishon[si,ti],herewetakes1=aandtk=b.Sincefidoesnotvanishon[si,ti],gisabsolutelycontinuousoneach[si,ti]andhenceon[a,b].Wewishtoshowthatg2W1,2[a,b]sowesetouttoshowgandg0areinL2[a,b].Sethi=gfiandbytheproductrulewendh0i=g0fi+f0igalmosteverywhere.Sincegiscontinuouson[si,ti],g2L2[si,ti].Thefunctionf0isalsoinL2[si,ti]whichimpliesgf02L2[si,ti],becausegisbounded.Thereforeh)]TJ /F3 11.955 Tf 12.48 0 Td[(gf0=g0f2L2[si,ti].Byconstruction,fidoesnotvanishon[si,ti],soinf[si,ti]jfi(x)j2Z10jg0j2dxZ10jg0fij2dx<1.Thusg02L2[si,ti]andg2W1,2[a,b].Fortheotherdirection,supposethatg2W1,2[a,b]forall[a,b](0,1).Letf2Wsuchthatfhascompactsupportin(0,1).Let[a,b]beacompactsubsetof(0,1)containingthesupportoff.Outsideof[a,b],fisidenticallyzeroandsof00aswell.ThefunctiongfisinL2[a,b]sinceitiscontinuous.Alsothefunctiongf02L2[a,b]sincegiscontinuousandf02L2[a,b],andg0f2L2[a,b]fortheoppositereason.Thus 46

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h:=gf2W1,2[a,b],andsinceitvanishesoutsidetheinterval,h2W.Thereforef2D(Mg),andcompactlysupportedfunctionsaredenseinW.Thusgisadenselydenedmultiplicationoperator. Itwasessentialtotheproofabovetondforeachpointx0(6=0or1)afunctionf2D(Mg)thatdidnotvanishatthatpoint.Wecanthenconcludethatforsomeclosedneighborhood[a,b]ofx0thesymbolgisinW1,2[a,b].InthecasewhereweconsiderthewholeSobolevspace,wecanndf,~f2D(Mg)sothattheydonotvanishat1and0repsectively.Thisproducesthefollowingtheorem: Theorem5.2. FortheSobolevspace,W1,2[0,1],thecollectionofsymbolsofdenselydenedmultipliersisW1,2[0,1].Inparticular,allthedenselydenedmultipliersarebounded.Thusfarthisistheonlynontrivialcollectionofdenselydenedmultipliersforwhicheveryoneisbounded.Thesamemethodscanbeusedtoshowthefollowingcorollaries: Corollary3. Letfx1,...,xng[0,1]andVW1,2[0,1]suchthatV=ff2W1,2[0,1]jf(xi)=0foreachig.Afunctiong:[0,1]nfx1,...,xng!Cisthesymbolofadenselydenedmultiplicationoperatoriffg2W1,2(E)forallcompactsubsetsof[0,1]nfx1,...,xng. Corollary4. GiventheSobolevspaceW1,2(R),afunctiongisamultiplierfor(W1,2(R),R)iffg2W1,2(E)forallEacompactsubsetofR. 5.2LocaltoGlobalNon-VanishingDenominatorIdeallygivenanydenselydenedmultiplicationoperatoroveraHilbertfunctionspaceH,wewouldliketoexpressitssymbolasaratiooftwofunctionsfromHsuchthatthedenominatorisnon-vanishing.FortheHardyspacethiswasachievedthroughanapplicationoftheinner-outerfactorization,butthereisnosuchfactorizationtheoremforfunctionsintheSobolevspace.Thusweneedtotrysomethingalittledifferent. 47

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WesawinTheorem 5.1 thatforanypointx2(0,1)wecanndafunctioninthedomainthatdoesnotvanishinaneighborhoodofthatpoint.Inotherwords,weusedalocalnon-vanishingproperty.NowthatwehaveanexplicitdescriptionofthedenselydenedmultipliersofW,wecansharpenthistondingaglobalnonvanishingdenominator.LookingatourthreeexoticfunctionswecanrewritethemasquotientsoffunctionsinWasfollows:sin(1=x)=x2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)sin(1=x) x2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)1=x=x(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x) x2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)exp(1=x)=x(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x) x(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)exp()]TJ /F7 11.955 Tf 9.29 0 Td[(1=x)Theproofofthefollowingtheoremfeelslikeanapplicationofl'Hopital'sruleappliedbyhand: Theorem5.3. IfgisadenselydenedmultiplierforW,thenthereexistsf2D(Mg)suchthatf(x)6=0on(0,1). Proof. Firstwesupposethatg62W1,2[0,1].Inourconstructionwewillonlyconsiderthelefthalfoftheinterval.Theotherhalfcanbehandledidentically.WenowknowbyTheorem 5.1 thatg2W1,2[a,1 2]foreacha>0,butg62W1,2[0,1].Thismeansg62L2[0,1 2]and/org062L2[0,1 2].SupposeR1=20jgj2+jg0j2dx=1.Sincebothgandg0areinL2[a,1 2]foralla>0:R.5ajgj2+jg0j2dx<1.Usingthisinformation,wewillconstructasequence:an=Z1 2n1 2n+1jgj2+jg0j2dx.Byourconstruction,Panisadivergentseries.Denebn=minf(an))]TJ /F4 7.97 Tf 6.59 0 Td[(1,(an)]TJ /F4 7.97 Tf 6.58 0 Td[(1))]TJ /F4 7.97 Tf 6.58 0 Td[(1,1g.Noticethatanbn+1,anbnandbn1foralln. 48

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Nowwecanbeginconstructingournon-vanishingfunctionf.DeneLn(x)by:Ln(x)=8><>:4(bn))]TJ /F4 7.97 Tf 6.58 0 Td[((bn+1) 2n+1(x)]TJ /F7 11.955 Tf 11.95 0 Td[(2)]TJ /F8 7.97 Tf 6.59 0 Td[(n)+(2)]TJ /F4 7.97 Tf 6.59 0 Td[(2n(bn)):x2(2)]TJ /F4 7.97 Tf 6.59 0 Td[((n+1),2)]TJ /F8 7.97 Tf 6.59 0 Td[(n]0:otherwiseNowsetf=P1n=1Ln(x).Inotherwordsfinterpolatesthepoints(1 2n,bn 22n)1n=1linearly.Thefunctionfiscontinuouson[0,1 2]anddifferentiablealmosteverywhere.Furtherf,f02L2[0,1 2].Thusf2W1,2[0,1 2]andf(0)=0.Thefunctiongfiscontinuouson(0,1=2)anddifferentiablealmosteverywhere.Wewishtoshowthatbothgfand(gf)0=g0f+f0gareinL2[0,1 2].Z0.50jgfj2dx=1Xn=1Z1=2n1=2n+1jgLnj2dx1Xn=1anmax(bn 22n2,bn+1 22(n+1)2)<1Z0.50jg0fj2dx=1Xn=1Z1=2n1=2n+1jg0Lnj2dx1Xn=1anmax(bn 22n2,bn+1 22(n+1)2)<1Z0.50jgf0j2dx=1Xn=1Z1=2n1=2n+1jgL0nj2dx1Xn=1an4(bn))]TJ /F7 11.955 Tf 11.96 0 Td[((bn+1) 2n+12<1Hereweseeeachintegralisdominatedbyageometricseries,andsogf,(gf)02L2[0,1=2].Thusfarwehaven'tyetestablishedthatfisinDg.Indeedgfmaynotvanishat0.Torectifythisnoticethatgfand(gf)0isinL2[0,1=2]andgfisabsolutelycontinuouson[a,1=2]forany0
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Onenalnotetothisproof.Itmighthappenthatg2W1,2[a,b]forall[a,b](0,1)andg,g02L2[0,1].Inthiscasetheaboveargumentyieldsx2(1)]TJ /F3 11.955 Tf 11.17 0 Td[(x)2g2W,x2(1)]TJ /F3 11.955 Tf 11.17 0 Td[(x)22D(Mg)andisnon-vanishingon(0,1). 5.3AlternateBoundaryConditionsWecanalsoconsiderothersubspacesofW1,2[0,1]wherewedonothavezeroboundaryconditions.Theseideascanbeextendedtoshowthatcertainsubspacessatisfyingotherboundaryconditionsleadtoeverydenselydenedmultiplierbeingbounded. 5.3.1Sturm-LiouvilleBoundaryConditionsConsidertheSturm-Liouvilleboundaryconditions:8><>:py(0)+qy0(0)=0sy(1)+ty0(1)=0LetW(p,q,s,t)bethesubspaceofW1,2[0,1]offunctionssatisfyingtheaboveconditions.Underthisdenition,thespaceWwehavebeenstudyingcorrespondstoW(1,0,1,0).Wecanexploreallofthesespaceswithonlyafewcases.Forexampleifa6=0andc6=0thenW(p,q,s,t)=W(1,q=p,1,t=s).ThereforeifwewishtoexamineallofthesespaceswemustcheckW(1,q,1,t),W(1,q,0,1),W(0,1,1,t),andW(0,1,0,1).Moreover,becausetheboundaryconditionsareseparated,itissufcienttodeterminewhathappenstothemultipliersat0forW(1,q,,)andW(0,1,,)bysymmetry.InthefollowingtheoremwewillndthatsomeofthesubspaceswithSturm-Liouvilleboundaryconditionshaveonlyboundedmultipliers.Thisgivesusmorenontrivialcollectionsofdenselydenedmultiplierswherenoneareunbounded.Fortheothersubspaces,wendthattheyhaveunboundeddenselydenedmultipliersinthestyleofW. 50

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Theorem5.4. Wehavethefollowingconnectionsbetweenthesesubspacesandtheirdenselydenedmultiplicationoperators. Table5-1. MultipliersassociatedwithSturm-Liouvilleboundaryconditions SubspaceDenselyDenedMultipliers W(1,0,1,0)\00(disregardingtherightboundarycondition.)Alltheotherscanbecomparedthroughsymmetry. 51

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5.3.2MixedBoundaryConditionsInadditiontotheseparatedSturm-Liouvilleboundaryconditions,wecanexaminethemixedboundaryconditions:8><>:py(0)+qy(1)=0sy0(0)+ty0(1)=0WewilldesignatethesubspacesofW1,2[0,1]correspondingtotheseconditionsasW0(p,q,s,t). Theorem5.5. Wehavethefollowingconnectionsbetweenthesesubspacesandtheirdenselydenedmultiplicationoperators. Table5-2. Multipliersassociatedwithmixedboundaryconditions SubspaceDenselyDenedMultipliers W0(1,0,1,0)\00.Herewecanseethatthefunctionsf2W0(1,0,1,0)forwhichf(x)=0inaneighborhoodofzeroareinthedomainofthesemultipliers.ThespaceW0(1,0,0,1)W0(1,0,0,1)W0(1,0,0,1):AgainwendthemultipliersforthisspacearethosefunctionsinW1,2[a,1]foralla>0.Inaddition,ifgisthesymbolforadenselydened 52

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multiplieronthisspaceandf2D(Mg)withh=gf,then0=h0(1)=g0(1)f(1)+g(1)f0(1)=g0(1)f(1).Thisistrueforallf2D(Mg)whichmeansg0(1)=0.ThespaceW0(1,0,1,t),t6=0W0(1,0,1,t),t6=0W0(1,0,1,t),t6=0:JustasabovethemultipliersforthisspacemustbeinW1,2[a,1]foralla>0.Ifgisthesymbolforadenselydenedmultiplieronthisspaceandf2D(Mg)sothath=fg,thenwehavetwosubcases.Ifghasalimitatzero(andredeninggtobecontinuousifnecessary)then)]TJ /F3 11.955 Tf 9.3 0 Td[(tg(0)f0(1)=g(0)f0(0)=h0(0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(th0(1)=g(1)f0(1)+g0(1)f(1)andsof0(1)(tg(0)+g(1))=)]TJ /F3 11.955 Tf 9.3 0 Td[(g0(1)f(1).Ifg0(1)6=0thenf0(1)tg(0)+g(1) g0(1)=f(1).Thefractioninthisequationisaxedquantityforagiveng,andtheequalitymustholdforeveryf2D(Mg).ThisimpliesthatMgisnotdenselydened.Ifg0(1)=0then)]TJ /F3 11.955 Tf 9.3 0 Td[(tg(0)f0(1)=g(1)f0(1)andg(1)+tg(0)=0.Whichgivesusboundaryconditionsong.Ontheotherhand,usingthecommondensedomainoffunctionsfthatvanishinaneighborhoodofzerowecanndunboundedmultipliers.LetfbesuchafunctioninW0(1,0,1,t)andleth=gf.Inthiscaseh0(0)=f0(0)=0.Thismeansthath0(1)=f0(1)=0aswell.Usingthiswecanndtheboundaryconditionat1ong:0=h0(1)=f0(1)g(1)+g0(1)f(1)=g0(1)f(1).Thisholdsforallfunctionsinthedomainofgwhichimpliesg0(1)=0.ThespaceW0(1,0,0,0)W0(1,0,0,0)W0(1,0,0,0):ThisisthespaceofSobolevspacefunctionswheref(0)=0.ThiscanbehandledbyCorollary 3 53

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ThespaceW0(1,q,0,0)W0(1,q,0,0)W0(1,q,0,0):NowletusconsiderthedenselydenedmultiplicationoperatorsoverW0(1,q,0,0),whereq6=0.Iff2D(Mg)foradenselydenedmultiplierMgandlettingh=gf,then)]TJ /F3 11.955 Tf 9.3 0 Td[(qg(1)f(1)=)]TJ /F3 11.955 Tf 9.3 0 Td[(qh(1)=h(0)=g(0)f(0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(qg(0)f(1),andg(0)=g(1).AlsonotethatsinceafunctioninW0(1,q,0,0)mayhavenonzeroboundaryvalues,g2W1,2[0,1].Thusg2W0(1,)]TJ /F7 11.955 Tf 9.3 0 Td[(1,0,0).ThespaceW0(1,q,1,t)W0(1,q,1,t)W0(1,q,1,t):NextletusconsiderthespaceW0(1,q,1,t)forq,t6=0.Againwendthatg(0)=g(1)foreverydenselydenedmultiplicationoperator.Supposealsothatf2D(Mg)andh=fg,theng0(0)f(0)+f0(0)g(0)=h0(0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(th0(1)=)]TJ /F3 11.955 Tf 9.29 0 Td[(t(g0(1)f(1)+f0(1)g(1)),andso)]TJ /F3 11.955 Tf 9.3 0 Td[(qg0(0)f(1))]TJ /F3 11.955 Tf 11.96 0 Td[(tf0(1)g(0))=)]TJ /F3 11.955 Tf 9.3 0 Td[(tg0(1)f(1))]TJ /F3 11.955 Tf 11.95 0 Td[(tf0(1)g(0).Thisyieldsqg0(0)=tg0(1).Thusg2W0(1,)]TJ /F7 11.955 Tf 9.3 0 Td[(1,q,)]TJ /F3 11.955 Tf 9.29 0 Td[(t).ThespaceW0(0,0,1,t)W0(0,0,1,t)W0(0,0,1,t):FinallyexaminingthespaceW0(0,0,1,t)fort6=0.Letg,fandhbeasbefore.Thisyieldsg0(0)f(0)+f0(0)g(0)=h0(0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(th0(1)=)]TJ /F3 11.955 Tf 9.3 0 Td[(t(g0(1)f(1)+f0(1)g(1)),andsog0(0)f(0))]TJ /F3 11.955 Tf 11.95 0 Td[(tf0(1)g(0)=g0(1)f(0))]TJ /F3 11.955 Tf 11.96 0 Td[(tf0(1)g(1).Rearrangingtermswendf(0)(g0(0))]TJ /F3 11.955 Tf 12.31 0 Td[(g0(1))=)]TJ /F3 11.955 Tf 9.3 0 Td[(tf0(1)(g(0))]TJ /F3 11.955 Tf 12.31 0 Td[(g(1)).Thisholdsforallf2D(Mg)=W0(0,0,1,t),whichmeansg0(0)=g0(1)andg(0)=g(1).Thusg2W0(1,)]TJ /F7 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F7 11.955 Tf 9.3 0 Td[(1). 5.4RemarksWeleavewithonelastnoteconcerningdenselydenedmultipliersontheSobolevspace.WeknowfromLemma 2 thatifamultiplierisbounded,thenit'ssymbolisboundedbythenormoftheoperator.Thequestionarrises,ifgisknowntobeadensely 54

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denedmultiplicationoperatoroverWandsupx2(0,1)jg(x)j<1isMgaboundedmultiplier?Theansweris:notnecessarily.Wecanproduceacounterexamplebyexaminingg(x)=p 1=4)]TJ /F7 11.955 Tf 11.95 0 Td[((x)]TJ /F7 11.955 Tf 11.95 0 Td[(1=2)2whichisboundedon[0,1]by1/2.Bytheworkabove,weknowthatMgisadenselydenedmultiplier,sinceg2W1,2[a,b]forevery[a,b](0,1).However,sinceg0isnotboundedon[0,1],Mg1=g162W.Thereforeeventhoughgisaboundedfunction,themultiplierMgisnotbounded. 55

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CHAPTER6UNBOUNDEDTOEPLITZOPERATORS 6.1Sarason'sProblemToeplitzoperatorscomeupnaturallyinmanyareasofmathematics.Theyalsoappearsinphysics,signalprocessingandmanyothercontexts.InthenitedimensionalcasebyToeplitzoperatorwemeananoperatorwhosennmatrixisconstantdownthediagonals.Extendingonthisidea,wesaythataboundedoperatorTonH2isToeplitzifthematrixrepresentationforT(withrespecttothestandardorthonormalbasis)isconstantdownthediagonals.Moreoverwehavethefollowingequivalentdenitions: Theorem6.1. LetTbeaboundedoperatoronH2.Thefollowingareequivalent: 1. TisaToeplitzoperator. 2. Forsome2L1wehaveT=T:=PH2M 3. STS=TMostoftheworkinvolvingToeplitzoperatorsisconcernedwiththoseoperatorswhohaveboundedsymbol.Ina1960paper[ 27 ]Rosenblumdemonstratedtheabsolutecontinuityofsemi-boundedToeplitzoperators.In1997Suarez[ 28 ]characterizedthecloseddenselydenedoperatorsthatcommutewithS,andSarason[ 21 ]investigatedtheanalyticunboundedToeplitzoperators.Helson[ 29 ]alsoinvestigatedthepropertiesofsomesymmetricToeplitzoperators,whichwewilluseasexamplesshortly.FortheunboundedToeplitzoperators,thepropertiesinTheorem 6.1 arenolongerequivalent.Nottomentionthesepropertiesneedsomerewording.WestartbylookingforToeplitzoperatorsintheformof(2)intheabovetheorem.Let'sstartwithoneplausibledenitionofanunboundedToeplitzoperator. Denition12. WesaythatTisaToeplitzoperatorofmultiplicationtypeifthereisathatisthesymbolforadenselydenedmultiplicationoperatorforwhichT=T:=PH2MwithdomainD(M). 56

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Notethatif2H1thenPH2M=M.WecallsuchanoperatorananalyticToeplitzoperator,andweextendthisterminologytoTwith2N+.WecallTaco-analyticToeplitzoperatorifT=Mforsome2N+.ThenextexamplewilldemonstratethatMwithdomainD(M)isnotthesameastheoperatorM withdomainD(M ). Example8. Ifweexaminethefunction0=i(1+z)(1)]TJ /F3 11.955 Tf 11.45 0 Td[(z))]TJ /F4 7.97 Tf 6.59 0 Td[(1asHelsondid,weseethat02N+.Thisisbecause1+z2H2and1)]TJ /F3 11.955 Tf 11.96 0 Td[(zisanouterfunctioninH2.Inaddition 0(ei)= i1+ei 1)]TJ /F3 11.955 Tf 11.96 0 Td[(ei=)]TJ /F3 11.955 Tf 9.3 0 Td[(i1+e)]TJ /F8 7.97 Tf 6.58 0 Td[(i 1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F8 7.97 Tf 6.58 0 Td[(i=)]TJ /F3 11.955 Tf 9.3 0 Td[(iei+1 ei)]TJ /F7 11.955 Tf 11.95 0 Td[(1=i1+ei 1)]TJ /F3 11.955 Tf 11.96 0 Td[(ei=0(ei).NowsupposethatfandgareinD(M0).WecanshowdirectlythatT0issymmetric:hT0f,gi=1 2Z)]TJ /F12 7.97 Tf 6.59 0 Td[(0(ei)f(ei) g(ei)d=1 2Z)]TJ /F12 7.97 Tf 6.59 0 Td[(f(ei) 0(ei)g(ei)d=hf,T0gi.IfT0werenotonlysymmetricbutalsoselfadjoint,thenwewouldhaveT0=T0andD(T0)=D(T0).Onewayofdeterminingwhetheraclosedsymmetricoperatorisselfadjointistoshowitsdefectindicesare0.Wherethedefectindicesaregivenbythequantities:+=dimfker(T0+iI)g)]TJ /F7 11.955 Tf 10.4 1.8 Td[(=dimfker(T0)]TJ /F3 11.955 Tf 11.95 0 Td[(iI)g.However,thedefectindices+and)]TJ /F6 11.955 Tf 10.41 1.79 Td[(arebothnonzero.Theindex+=0,buttheindex)]TJ /F7 11.955 Tf 10.88 1.8 Td[(=1.ThismeansthatT06=T0=T 0.TheoperatorT0containsthedomainofT0,whichtellsusthattherearef2D(T0)thatarenotinD(M0).Amoreexhaustivetreatmentofsymmetricnon-selfadjointoperatorscanbefoundinAleman,MartinandRoss's2013paper[ 4 ].Suarez[ 28 ]showedthatthedomainforTisthedeBranges-RovnyakspaceH(b),wherebistheH1functioninthenumeratorofthestandardformb=a=2N+.NotethatthedeBranges-RovnyakspacecontainsaH2,thedomainforM.ThisindicatesthatweneedanotionmoregeneralthanmultiplicationtypeToeplitzoperatorsifwewishtoincludeallofthenaturalcases.Let'sgiveanametoeachofthesenotionsofToeplitzness. 57

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Denition13. 1. IfanoperatorTcontainsthepolynomialsinitsdomain,anditsmatrixrepresenta-tionwithrespecttothestandardorthonormalbasisisconstantdowndiagonalswecallTamatrix-typeToeplitzoperator. 2. IfanoperatoristheadjointofamultiplicationtypeToeplitzoperatorwithanalyticsymbolwesaythatTisacoanalyticToeplitzoperator.InhisexpositoryarticleonUnboundedToeplitzOperators[ 21 ],DonaldSarasoninvestigatedseveralclassesofunboundedToeplitzOperators.Ineachcase,hefoundthattheseToeplitzOperatorssharedsomeofthealgebraicpropertiesoftheirboundedcounterparts.Attheendofthepaperheposedthefollowingasaproblemforcharacterization: Problem1(TheSarasonProblem). CharacterizethecloseddenselydenedoperatorsTonH2withtheproperties: 1. D(T)isS-invariant, 2. STS=TjD(T), 3. IffisinD(T)andf(0)=0thenSf2D(T).IseverycloseddenselydenedoperatoronH2thatsatisestheaboveconditionsdeterminedinsomesensebyasymbol?Ifwetakeb=atobethecanonicalrepresentationfor2N+,thenweseethatthedomainofT=MisaH2.Thisisshiftinvariant.ThedomainofTisH(b),whichisalsoshiftinvariant.Bothofthesespacesalsosharepropertytwo.Sinceco-analyticandanalyticToeplitzoperatorscommutewithSandSrespectively,propertythreealsofollows.ThustheseSarasonconditionsencompassbothanalyticandco-analyticToeplitzoperators.WewillcallanoperatorthatsatisesSarason'sconditionsaSarason-Toeplitzoperator.RecallthatforaboundedToeplitzoperatorSTS=Tcompletelycharacterizestheseoperators.Bythiswemean,ifanoperatorsatisesthisalgebraiccondition,thenthematrixrepresentationofsuchanoperatorisconstantdownthediagonals(with 58

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respecttothestandardbasis).ItcanalsoberepresentedbyT=T=PH2Mforsome2L1.Howeverintheunboundedcase,havingsuchamatrixrepresentationdoesnotguaranteethatTwillagreewithPMevenonadensedomainforany.Ifmultiplicationbythismatrixonpolynomialsisclosable,theclosedoperatorwillhavepropertythree.In[ 28 ]theauthorcharacterizedallsuchmatricesasthosewhosecoefcientsarefromananalyticfunctioninN+.Wewillprovideadifferentproofinthischapter.Firstlet'sestablishthefollowinglemma: Lemma10. IfTsatisestheSarasonconditions,thensodoesT. Proof. SinceTsatisestheSarasonconditions,itisclosedanddenselydened.Thusit'sadjointisdenselydenedandclosedaswell.Nowletusestablishtheshiftinvarianceofthedomain.Takeg2D(T).WewanttoshowthattheoperatorL:D(T)!CgivenbyL(f)=hTf,zgiiscontinuous.Weknowthat~L(f):=hTf,giiscontinuoussince,g2D(T).Firstsupposethatf2SD(T).Thatisf=Shforsomeh2D(T).InthiscasewecanseethatL(f)=hTf,zgi=hSTSh,gi=hTh,gi=~L(h)iscontinuous,sinceitcoincideswith~L.ThespaceSD(T)isco-dimension1asasubspaceofD(T).Thismeansthereisavectorf0inD(T)suchthatforeachf2D(T)thereexists2Candh2SD(T)sothat:f(z)=f0(z)+h(z).WeknowthatLiscontinuousonthespanoff0sincetherestrictiontothisspaceisonedimensional.LisalsocontinuousonSD(T)aswehaveestablished.ThereforeLiscontinuousonD(T)andzg(z)2D(T).Nowweneedtoshowthatifg2D(T)andg(0)=0thenSg2D(T).Takesuchag2D(T).WenowsetouttoshowtheoperatorF:D(T)!CgivenbyF(f)=hTf,Sgiiscontinuous.Whatwehavetoworkwithisthattheoperator~F(f)=hTf,giiscontinuous,sinceg2D(T). 59

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Thistimetakef2D(T).Thisyields:F(f)=hTf,Sgi=hSTSf,Sgi=hTSf,SSgi=hTSf,gi=~F(Sf).ThelastequalitybetweentheinnerproductsfollowssinceSSg=gforeveryfunctionsatisfyingg(0)=0.~FiscontinuousbyconstructionandSisaboundedoperator.ThusweseethatFiscontinuousonD(T).Finallythesecondconditionfollowsbytakingadjoints. AsforrecoveringasymbolforsuchaToeplitzoperator,agoodrstattemptwouldbetousetheBerezintransformofT.TheBerezintransformisafunctionobtainedfromanoperatorwhosepropertiesreectthatoftheoperatoritcamefrom.TheBerezintransformwasrstusedinthecontextoftheFockspace.Moreinformationcanbefoundin[ 14 ]. Denition14. LetTbeanoperatorwithk2D(T)foreach2D.WedenetheBerezintransformofTtobe~T()=(1)-222(jj2)hTk,ki.InmanycasesthiswillreproducethesymbolofaToeplitzoperator.Forinstanceif2H1wehavethefollowing:~T()=(1)-222(jj2)hTk,ki=(1)-222(jj2)Dk, ()kE=()1)-222(jj2 1)-222(jj2=().TheBerezintransformwillreproducethesymbolinmanyothercases.However,forageneraldenselydenedToeplitzoperatorwecannotapplytheBerezintransformasdenedabove.TheproblemisthatweknowkwillnotbeinthedomainofTforgeneralunboundedToeplitzOperators.InfacttheyarenotinthedomainofamultiplicationtypeToeplitzoperatorswithanalyticsymbol. 60

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Inthiscasethereisstillhope.Intheequationsabove,wesawthatweusedthefactthatkwasaneigenvalueforMwhichisstilltruefordenselydenedmultiplicationoperators.InthiscasewecanseethatforadenselydenedanalyticToeplitzoperatorwehaveadualBerezintransform:~T(z)y:=(1)-221(jj2)k,Tk=()whichwouldreproducethesymbolofananalyticToeplitzoperator.InthenextsectionwewilluseanothermethodofrecoveringthesymbolofaToeplitzoperatorthatusesonlythealgebraicpropertiesstatedintheSarasonProblem. 6.2SarasonSub-SymbolLetusbeginwiththeboundedcase.ThisisthetraditionalToeplitzoperatorwithL1symbol:T.Wesetouttorecoverbyplayingwithitsdomain.Firstwebeginwithanexample. Example9. Let=P1n=nzn2L1andconsidertheboundedToeplitzoperatorT.Thematrixrepresentationofthisoperatorisgivenby0BBBBBBB@0)]TJ /F4 7.97 Tf 6.59 0 Td[(1)]TJ /F4 7.97 Tf 6.58 0 Td[(210)]TJ /F4 7.97 Tf 6.58 0 Td[(1210......1CCCCCCCAGivenonlythismatrix,ifwewishedtondthenthFouriercoefcientofforn0thenwewouldapplythismatrixto(1,0,0,0,...)T12H2andtaketheinnerproductwith(0,0,...0,0,1,0,0,...)Tzn2H2.Thisintuitioniseasilyveried:hT1,zniH2=h,zniL2=n.Similarlyifwewishedtondthe)]TJ /F3 11.955 Tf 9.3 0 Td[(nthcoefcientforweneedtocomputehTzn,1iH2=,z)]TJ /F8 7.97 Tf 6.59 0 Td[(nL2=)]TJ /F8 7.97 Tf 6.59 0 Td[(n. 61

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ThusforaboundedToeplitzoperatorwecanreproducethesymbolofthisoperatorbywriting(z)=1Xn=1hTzn,1izn+1Xn=0hT1,znizn.Usingthismethodweseethatwerequirezntobeinthedomainoftheoperatorforalln0.Thisisnotgenerallytrueintheunboundedcase,butwithashiftinvariantdomainasinSarason'sProblemwecanproducesomesymbol.WedenetheSarasonSub-Symbolasfollows: Denition15(SarasonSub-Symbol). LetV:D(V)!H2beanoperatoronH2forwhichzD(T)D(T).TheSarasonSub-Symboldependingonf2D(T)isdenedasRf=h=fwherehisgivenformallyby:h1Xk=1Vfzk,1zk+1Xk=0Vf,zkzkItshouldbeemphasizedthatthissymbolisdenedonlyformallyintermsofFourierseriesonthecircle.Inmostcasesthenegativefrequencycomponentsofharenotsquaresummable,andthehypotheseswewillchooseforthesucceedingtheoremsallowustondhinL2.OfcoursewhenhisinL2,thesub-symboliswelldened.EveryexampleofToeplitzoperatorsexploredsofaradmitsadensecollectionoffunctionsfforwhichhinthesub-symbolisinL2.WecallthiscollectionD2(T)forgeneraloperatorsTwithshiftinvariantdomain.IthasnotyetbeenproventhatforanoperatorwithshiftinvariantdomainhasnontrivialD2(T).Wehopethattheclosabilityofanoperatorisenoughtoguaranteethis.Intheboundedcase,itisstraightforwardtondelementsinD2(T),andweusethistoprovethefollowingtheorem.ThisaddsyetanotherequivalentcharacterizationofToeplitzoperatorstothelistinTheorem 6.1 Theorem6.2. LetVbeaboundedoperatoronH2.Inthiscase,theSarasonSub-SymbolisuniqueiffVisaToeplitzoperator. 62

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Proof. Fortheforwardimplication,supposethatVisnotaToeplitzoperator.Thisyieldstwocases.Intherstcaseweassumethereisapairn,m2Nwithn
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recovertheToeplitzsymbolbysimplyusingthesubsymbol1.WhichwouldmaketheSarasonSub-symbolredundant.ThisisnotthecaseforunboundedToeplitzOperators,sincewecannotguaranteethat1isinthedomainoftheoperator. Theorem6.3. GivenanSarason-ToeplitzoperatorT,thereexistsasymbol2N+forwhichT=TiffhTfzm,1i=0forallm>0andallf2D(T).Moreoverthissymbolisunique. Remark1. ThesufcientconditionhereisequivalenttoTbeingS-analytic.ThatisTS=ST. Proof. TheforwarddirectionfollowssinceT=Mfor2N+,andMfisananalyticL2functionforf2D(T).Fortheotherdirectionletf1=Panznandf2=PbnznbenonzerofunctionsinthedomainofT,anddenetwocorrespondingfunctionsh1andh2asthenumeratorsoftheSarasonsub-symbolscorrespondingtof1andf2respectively:h1,2=1Xn=1hTf1,2zn,1izn+1Xn=0hTf1,21,znizn.Howeverbyourhypothesis:h1,2=1Xn=0hTf1,21,znizn.Thefunctionsh1andh2areinH2,sincetheyaresimplytheimagesoff1andf2respectively.Ourgoalistoshowthath1=f1=h2=f2andthusthesymbolwendforTisindependentofthechoiceoff2D(T).Equivalentlywewilldemonstratethath1f2)]TJ /F3 11.955 Tf 12.74 0 Td[(h2f1=0bywayofitsFourierseries.TheFourierseriesforh1f2andh2f1can 64

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becomputedbyusingtheconvolutionproduct:h1f2=1Xn=0 nXk=0Tf11,zn)]TJ /F8 7.97 Tf 6.58 0 Td[(kbk!zn=1Xn=0 nXk=0Tf1zk,znbk!znh2f1=1Xn=0 nXk=0Tf2zk,znak!znWenowhavetheFourierseriesforh1f2)]TJ /F3 11.955 Tf 11.95 0 Td[(h2f1(forbrevitycallthisfunctionH):H=h1f2)]TJ /F3 11.955 Tf 11.96 0 Td[(h2f1=1Xn=0 nXk=0Tf1zk,znbk)]TJ /F8 7.97 Tf 18.3 14.95 Td[(nXk=0Tf2zk,znak!zn.Wenowsetouttoshowthateachofthecoefcientsarezero.Letusexaminethenthcoefcient:^H(n)=nXk=0Tf1zk,znbk)]TJ /F8 7.97 Tf 18.3 14.95 Td[(nXk=0Tf2zk,znak=nXk=0\012Tf1zk,znbk)]TJ /F13 11.955 Tf 11.95 9.68 Td[(Tf2zk,znak=*T f1nXk=0bkzk)]TJ /F3 11.955 Tf 11.96 0 Td[(f2nXk=0akzk!,zn+=*T f1"nXk=0bkzk)]TJ /F3 11.955 Tf 11.96 0 Td[(f2#)]TJ /F3 11.955 Tf 11.95 0 Td[(f2"nXk=0akzk)]TJ /F3 11.955 Tf 11.96 0 Td[(f1#!,zn+TheH2functioninsideofTisinfactinthedomainofTbyourassumptions.Moreoveritisafunctionwithazeroatzeroofordergreaterthann.Writeitaszn+1Fnandbythehypothesiswearriveat:^H(n)=Tzn+1FN,zn=hTzFN,1i=0. 65

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Thush1=f1=h2=f2forallchoicesof(nonzero)f1andf2.Callh1=f1=.isanalytic,sinceforanyf2D(T),Tf2H2andforanyz2Dthereissomef2D(T)suchthatf(z)6=0,thus=(Tf)=fisanalyticateverypointz2D.Furthernotethatforanyf2D(T),=(Tf)=fandPH2Mf=PH2Tf=Tf.ThusTisadenselydenedToeplitzoperatorwithananalyticsymbol,whichmeans2N+. Thestrategyusedabovecanbesummarizedasfollows.FromtheSarasonconditionswebelievethatTisinsomesensea(denselydened)Toeplitzoperator.Assuch,thematrixrepresentationofTshouldbeconstantdownthediagonals,andwecanndtheFouriercoefcientsofitssymbolbysimplyreadingtheentriesintherstrowandcolumn.However,sinceTdoesnotnecessarilycontainthemonomialsinitsdomain(itdoesintheco-analyticcase,butnottheanalyticcase)wersttakeanelementfromthedomainofTandinsteadexaminetheoperatorTMf.ThelatteroperatorcontainsthepolynomialsandsatisesSarason'sconditions.Withthenewoperator,wemakeanunboundedToeplitzoperatorthatagreeswithTonff(z)p(z)jp(z)isapolynomialg.Inthecasethatfisouter,TextendsthedenselydenedToeplitzoperatorwehavecreated. Lemma11. LetbeafunctionontheunitcirclethatcanbewrittenastheratioofanL2functionandanH2outerfunction.LetD(M)=ff2H2jf2L2g.TheoperatorM:D(M)!L2isacloseddenselydenedoperator. Proof. Write=h=gwhereh2L2andg2H2withgouter.Sincehp2L2foreverypolynomialp(z),weseethatgp2D(M),sothedomainofMisdenseinH2. 66

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NowsupposethatffngD(M)andfn!f2H2.SupposefurtherthatMfn!F2L2.Wewishtoshowthatf2D(M)andf=F.Sincefn!finL2norm,theremustbeasubsequencefnj!falmosteverywhere.Sincegisanouterfunction,g(ei)6=0foralmostevery.Thusfnj!falmosteverywhere.Thesubsequencefnj!FinnormandhasitselfasubsequenceconvergingtoFalmosteverywhere.Callthissubsequencefnjk.Howeverthissubsequencealsoconvergestof.Thusf=Falmosteverywhere,andtheconclusionfollows. Theorem6.4. LetTsatisfySarason'sconditions.Supposefurtherthatthereisanouterfunctionf2D(T)suchthatP1n=1hTfzn,1izn2L2.Inthiscase,TextendsancloseddenselydenedToeplitzoperatorwithsymbolthatisaratioofanL2functionandf. Remark2. TheseconditionsonthedomainofTarenotunreasonable.ForaboundedToeplitzoperatorthisholdstrivially,andfortheunboundedToeplitzoperatorsexaminedbySarasontheconditionsalsohold.Ineachofthesecases,thesymbolproducedisexactlytheonewestartwith. Proof. Letfbeasinthestatementabove.Deneh=1Xn=1hTfzn,1izn+1Xn=0hTf,znizn.Forbrevitywewillwriteh=P1n=bnznandbytheToeplitznessofTwehavebn)]TJ /F8 7.97 Tf 6.59 0 Td[(m=hTzm,zni.Finallydeclare=Rf=h=f,andconsidertheunboundedToeplitzoperatorT=PH2MwiththedomainF.BytheSarasonconditions,f(z)p(z)2D(T)foreverypolynomialp(z).ThismeansthatD(T)containsthedensesubsetofH2:F=ff(z)p(z)jp(z)isapolynomialg.ThefollowingcalculationsshowthatTagreeswithTonthisdomain. 67

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Letp(z)=akzk++a1z+a0.h(z)p(z)=1Xn= kXn=0bn)]TJ /F8 7.97 Tf 6.58 0 Td[(mam!zn=1Xn= kXn=0hTfamzm,zni!zn=1Xn=hTfp(z),znizn=w(z)+T(fp(z)).Herew(z)2 H20byourhypothesis.InparticularthismeansT(fp(z))=PH2(hp(z))=T(fp(z)).HenceTagreeswithTonadensedomain,andTextendsTjF.Finallynotethatbyvirtueofthepreviouslemma,TjFisclosable.FurtherTjFTwhichmeansTjFT=T. Theaboveargumentdoesn'tdependonThavingtheToeplitzpropertiesasmuchasTMfhavingtheseproperties.Wecouldadjustthehypothesisaccordingly,andwecanalsoarriveatthefollowingcorollary. Corollary5. SupposeVisaboundedoperatoronH2.IffisanouterfunctionsuchthatVMfisaToeplitzoperatorandV12D(Mf),thenV=TRf. Proof. Giventhehypothesis,weseethatP1n=1jhVfzn,1ij2=P1n=1jhzn,(VMf)1ij2<1since(VMf)12H2.ThusbythepreviouspropositionVagreeswithTRfonadensedomain.HenceV=TRfbycontinuity. ThistheoremgivesananswertowhetherTcanberelatedtoasymbol.However,wehadtoassumethattherewasnotonlyanouterfunctionf2D(T)butalsooneforwhichP1n=1hTfzn,1iznwasinL2.Withouttheaboveconditions,wecanstillndthatTextendsapointwiselimitofunboundedToeplitzoperators. 68

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Theorem6.5. SupposeTsatisesSarason'sconditions,letf2D(T)andconsiderthesetF=ff(z)p(z):p(z)polynomialg.ThereexistsasequenceofdenselydenedmultiplicationtypeToeplitzoperatorsthatconvergestronglytoTonF.MoreovertheseToeplitzoperatorsaredenselydenedwithcommondomainffo(z)p(z):p(z)polynomialgwherefoistheH2outerfactoroff. Proof. Letf2D(T)anddenehN=PNn=1hTfzn,1izn+P1n=0hTf,znizn=P1n=)]TJ /F8 7.97 Tf 6.59 0 Td[(Nbnzn.AlsodeneN=hN=f.Letp(z)=akzk++a1z+a0,andconsidertheproducthNp(z):hN(z)p(z)=1Xn=)]TJ /F8 7.97 Tf 6.58 0 Td[(N min(k,n+N)Xm=0bn)]TJ /F8 7.97 Tf 6.59 0 Td[(mam!zn=k)]TJ /F8 7.97 Tf 6.59 0 Td[(N)]TJ /F4 7.97 Tf 6.59 0 Td[(1Xn=)]TJ /F8 7.97 Tf 6.58 0 Td[(NTf)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(a0++an+Nzn+N,znzn+1Xn=k)]TJ /F8 7.97 Tf 6.59 0 Td[(NhTfp(z),znizn.Writecn=Tf)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(a0++an+Nzn+N,zn.ThusMNfp(z)=hNp(z)2L2andTN(fp(z))=PH2(hNp(z))=0@)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(c0+c1z++ck)]TJ /F8 7.97 Tf 6.58 0 Td[(N)]TJ /F4 7.97 Tf 6.58 0 Td[(1zk)]TJ /F8 7.97 Tf 6.59 0 Td[(N)]TJ /F4 7.97 Tf 6.59 0 Td[(1+1Xn=max(k)]TJ /F8 7.97 Tf 6.59 0 Td[(N,0)hTfp(z),znizn1A!T(fp(z))wherethelimithappensasN!1.ThelimitispossiblesimplybecausethesequenceisconstantforlargeN.Forthecommondomain,letf=fifobetheinner-outerfactorizationoffinH2.Let~hN=hN=fi.Thefunction~hN2L2(T)sincefihasmodulusoneonthecircle.ThusNfop(z)=~hNp(z)2L2andTNfop(z)=PH2(~hNp(z)). 6.3ExtendingCo-analyticToeplitzOperatorsFollowingSarason,webeginastudyofCo-analyticToeplitzoperatorsbyusingamatrixrepresentationoftheoperator.Thoughitisnotstrictlytrue,weoftenthinkofthismatrixasbeingtheconjugatetransposeofthematrixofadenselydenedmultiplicationoperatorwhosecoefcients(withrespecttothestandardorthonormalbasisforH2) 69

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iscomposedofitsFouriercoefcients.Aswehaveseenbefore,thesymbolforthismultiplicationmustbeanalyticinthedisc,andthisputsaconstraintonthecoefcients.Herewewillviewthematrixasanobjectabstractedfromamultiplicationoperator.Insteadthiswillbeanuppertriangularmatrixthatisconstantdownthediagonals:0BBBBBBB@012001000......1CCCCCCCA.IfthesewerethecoefcientsforanL1function,thenthiswouldbethematrixrepresentationfortheadjointofaboundedmultiplicationoperatorM.However,ifthisisnotsuchasequencethentheoperatorcanonlybedenselydened.Usingtheanalogyofmatrixmultiplication,wecanseethatmultiplyingthismatrixtoavectorcorrespondingtoapolynomialyieldsanotherpolynomial.Hencethisoperatorisdenselydenedwiththepolynomialsinitsdomain.WemayextendthisoperatorbytheoperatorTdenedas:Tf=1Xm=0 1Xn=0n^f(n+m)!zmwherewesaythedomainofTisthecollectionoffunctionsinH2forwhichTfisinH2.Thisoperatorextendstheoperatorwedenedonthepolynomialsabove,andforalargenumberofcaseswecandescribethedomainsimply.InSarason'spaper,hewasconcernedonlywithfunctionsthatweretheadjointsfordenselydenedmultiplicationoperators.Theseplaceconditionsonthesequencefng.First,thesemustbetheTaylorcoefcientsforafunctionthatisanalyticinthedisc.Second,thedomainofthisoperatormustcontainthekernelfunctions(ifitisclosed).Thefollowingcanbefoundin[ 21 ]: 70

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Theorem6.6. Iff2H(D),theneachseriesP1n=0n^f(m+n)convergesabso-lutelyform=0,1,2,...,andthefunctionrepresentedbythepowerseriesTf=P1m=0P1n=0n^f(m+n)zmisinH(D).NotethatifafunctionisinH(D),thesetoffunctionsanalyticinaneighborhoodoftheclosedunitdisc,thenitisaboundedanalyticfunctionandhenceinH2.ThissetcontainsnotonlythepolynomialsbutalsoalloftheH2kernelfunctions.Ifwenolongerrequirethecoefcientstobethatofananalyticfunctionofthedisc,thedomainoftheoperatorTbecomesmucheasiertodescribe.Firstlet'sexaminethecasewithn=n!.ThisisclearlynotcoveredbyTheorem 6.6 ,sincethepowerseriesP1n=0n!znhasaradiusofconvergenceof0. Theorem6.7. LetTbetheextensionofmultiplicationbyanuppertriangularmatrixgivenbyTf=P1m=0P1n=0n!^f(n+m)zm.WedenethedomainofTtobeD(T)=ff2H2jTf2H2g.Everyfunctionf2D(T)isanentirefunctionandcanbewrittenasf(z)=P1n=0anzn n!whereP1n=0anconverges. Lemma12. Thesequencefcm=P1n=1(n+1))]TJ /F8 7.97 Tf 6.59 0 Td[(mg1m=2isanl2sequence. Proof. ByvirtueoftheintegraltestwecanboundcmbyZ101 (1+x)mdx=limb!1Zb01 (1+x)mdx=limb!11 m)]TJ /F7 11.955 Tf 11.96 0 Td[(11)]TJ /F7 11.955 Tf 37.84 8.09 Td[(1 (1+b)m)]TJ /F4 7.97 Tf 6.59 0 Td[(1=1 m)]TJ /F7 11.955 Tf 11.96 0 Td[(1.Thusweboundcmbyaharmonicsequence,whichbyagiftfromEulerisinl2. ProofofTheorem 6.7 Firstwesupposethatf2D(T).BydenitionthezerothcoefcientofTfisgivenbyP1n=0n!^f(n),whichmustbeaconvergentseries.Declaringan=n!^f(n)weseethatPanisconvergent.Moreover,sincef(z)=P1n=0anzn n!thefunctionfmustbeentirebytheroottest. 71

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Fortheotherdirection,supposethatf(n)=P1n=0anzn n!wherePanconverges.Calld0=P1n=0n!^f(n)=Pan.NotealsothatsincePanconverges,sodoesd1=1Xn=0n!^f(n+1)=1Xn=0an+1 n+1,anddm=1Xn=0n!^f(n+m)=1Xn=0an+m (n+1)(n+2)(n+m).Thisfollowssince1 (n+1)(n+2)(n+m)isdecreasingtozeromonotonically.ThisenablesustodeneTfformallyasP1m=0dmzm.FinallywecanshowthatTfisinH2,bythefollowingobservation:dm=am m!+1Xn=1an+m (n+1)(n+2)(n+m):=sm+tm.Thesequencefsmgisinl2sincethesearetheFouriercoefcientsoff.Forthetmsequenceweusethecomparison:1Xn=11 (n+1)(n+2)(n+m)1Xn=11 (n+1)m=cm.Thustmisanl2sequencebycomparisonwithLemma 12 .Thiscompletesthetheorem. Providedthen'saregrowingfastenough,wecanapplythesameprooftondthedomainofthisoperator.Wearriveatthefollowing: Theorem6.8. Letnbeasequenceofpositiverealnumberssuchthatn+1(n+1)nforalln,andletTbetheoperatorgivenbyTf=P1m=0P1n=0n^f(n+m)zm.WedenethedomainofTtobeD(T)=ff2H2jTf2H2g.Everyfunctionf2D(T)isanentirefunctionandcanbewrittenasf(z)=P1n=0anzn nwhereP1n=0anconverges. 72

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Usingthesametechniqueswecanndaslightlyweakerresultinthecasethatthesequencefngiscomplexwiththegrowthconditionjn+1j>(n+1)jnj. Theorem6.9. LetfngbeasequenceofcomplexnumbersasdescribedaboveanddenetheoperatorTf=P1m=0(P1n=0n^f(n+m))zmwiththedomainD(T)=ff2H2jTf2H2g.Inthiscaseiff(z)=P1n=0anzn nandPjanj<1thenf2D(T). Proof. Letfbeasstatedinthehypothesis.Denedm=P1n=0n^f(n+m).Thismakesd0=Pan,andiswelldenedsincetheseriesisabsolutelyconvergent.WeseethatdmiswelldenedbycomparisonwithPan,jdmj=1Xn=0nan+m n+m1Xn=0jan+mj (n+1)(n+2)(n+m)1Xn=0jan+mj.Toverifythatfdmgisanl2sequence,wesplititasinTheorem 6.7 NowthatwehavetheentiredomainfortheoperatorinTheorem 6.7 ,wecanarriveatanalarmingconclusion.Thedomainisnotshiftinvariant.Toseethistakef2D(T)andconsiderzf(x)=z1Xn=0an n!zn=1Xn=0(n+1)an (n+1)!zn+1.IfzfwasinD(T),theseriesP1n=0(n+1)anmustbeconvergent.Howeversimplysettingan=()]TJ /F7 11.955 Tf 9.3 0 Td[(1)n+1(n+1))]TJ /F4 7.97 Tf 6.58 0 Td[(1,weseethatf2D(T)butzfisnot.Wehaveyettoaddresstheclosureoftheseco-analyticToeplitzoperators.Ifthengrowtoofast,thenitisclearthatclosabilityshouldbemuchharder.WeconcludebyprovingthatifmultiplicationbyaToeplitzmatrixisclosable,thenthesymbolthematrixcoefcientscomefrommustbeinN+.ThiswasrstrecognizedbySuarez,buttheproofhereisdifferent. Theorem6.10. LetTbetheoperatorcorrespondingtotheuppertriangularmatrixwiththepolynomialsasitsdomain.IfTisclosable,thenthecoecientsinthematrixaretheTaylorcoefcientsofaSmirnovclassfunction.Inparticular,theyarethecoefcientsofafunctionanalyticinthedisc. 73

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Proof. IfTwasclosable,thenTisclosed,denselydened,andsatisesSarason'sConditionsbytheproofofLemma 10 .SinceTcommuteswithS,thismeansTcommuteswithS.ByTheorem 6.3 weseethatTisadenselydenedmultiplierontheHardyspacewithsymbol2N+.Thiscompletestheproof. 74

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REFERENCES [1] G.Pick,UberdieBeschrankungenanalytischerFunktionen,welchedurchvorgegebeneFunktionswertebewirktwerden,Math.Ann.77(1)(1915)7. [2] S.Kouchekian,ThedensityproblemforunboundedBergmanoperators,IntegralEquationsOperatorTheory45(3)(2003)319. [3] D.Sarason,Sub-HardyHilbertspacesintheunitdisk,UniversityofArkansasLectureNotesintheMathematicalSciences,10,JohnWiley&SonsInc.,NewYork,1994,aWiley-IntersciencePublication. [4] A.Aleman,R.Martin,W.T.Ross,OnatheoremofLivsic,J.Funct.Anal.264(4)(2013)999. [5] P.R.Halmos,AHilbertspaceproblembook,2ndEdition,Vol.19ofGraduateTextsinMathematics,Springer-Verlag,NewYork,1982,encyclopediaofMathematicsanditsApplications,17. [6] J.Agler,Nevanlinna-PickinterpolationonSobolevspace,Proc.Amer.Math.Soc.108(2)(1990)341. [7] J.Agler,J.E.McCarthy,PickinterpolationandHilbertfunctionspaces,Vol.44ofGraduateStudiesinMathematics,AmericanMathematicalSociety,Providence,RI,2002. [8] N.Aronszajn,Theoryofreproducingkernels,Trans.Amer.Math.Soc.68(1950)337. [9] R.G.Douglas,Banachalgebratechniquesinoperatortheory,2ndEdition,Vol.179ofGraduateTextsinMathematics,Springer-Verlag,NewYork,1998. [10] K.Hoffman,Banachspacesofanalyticfunctions,DoverPublicationsInc.,NewYork,1988,reprintofthe1962original. [11] P.L.Duren,TheoryofHpspaces,PureandAppliedMathematics,Vol.38,AcademicPress,NewYork,1970. [12] W.Rudin,Realandcomplexanalysis,3rdEdition,McGraw-HillBookCo.,NewYork,1987. [13] F.Riesz,UberdieRandwerteeineranalytischenFunktion,Math.Z.18(1)(1923)87. [14] K.Zhu,AnalysisonFockspaces,Vol.263ofGraduateTextsinMathematics,Springer,NewYork,2012. [15] Y.-c.J.Tung,Fockspaces,ProQuestLLC,AnnArbor,MI,2005,thesis(Ph.D.)UniversityofMichigan. 75

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[16] V.Bargmann,OnaHilbertspaceofanalyticfunctionsandanassociatedintegraltransform,Comm.PureAppl.Math.14(1961)187. [17] V.I.Paulsen, Anintroductiontothetheoryofreproducingkernelhilbertspaces ,online.(September2009).URL http://www.math.uh.edu/~vern/rkhs.pdf [18] G.B.Folland,Realanalysis,2ndEdition,PureandAppliedMathematics(NewYork),JohnWiley&SonsInc.,NewYork,1999,moderntechniquesandtheirapplications,AWiley-IntersciencePublication. [19] G.K.Pedersen,Analysisnow,Vol.118ofGraduateTextsinMathematics,Springer-Verlag,NewYork,1989. [20] J.B.Conway,Acourseinoperatortheory,Vol.21ofGraduateStudiesinMathematics,AmericanMathematicalSociety,Providence,RI,2000. [21] D.Sarason,UnboundedToeplitzoperators,IntegralEquationsOperatorTheory61(2)(2008)281. [22] K.Zhu,MaximalzerosequencesforFockspaces,preprint.arXiv:1110.2247(October2011). [23] H.Hedenmalm,P.Lindqvist,K.Seip,AHilbertspaceofDirichletseriesandsystemsofdilatedfunctionsinL2(0,1),DukeMath.J.86(1)(1997)1. [24] J.E.McCarthy,HilbertspacesofDirichletseriesandtheirmultipliers,Trans.Amer.Math.Soc.356(3)(2004)881(electronic). [25] E.Hewitt,J.H.Williamson,NoteonabsolutelyconvergentDirichletseries,Proc.Amer.Math.Soc.8(1957)863. [26] T.M.Apostol,Introductiontoanalyticnumbertheory,Springer-Verlag,NewYork,1976,undergraduateTextsinMathematics. [27] M.Rosenblum,TheabsolutecontinuityofToeplitz'smatrices,PacicJ.Math.10(1960)987. [28] D.Suarez,Closedcommutantsofthebackwardshiftoperator,PacicJ.Math.179(2)(1997)371. [29] H.Helson,Largeanalyticfunctions,in:Linearoperatorsinfunctionspaces(Timisoara,1988),Vol.43ofOper.TheoryAdv.Appl.,Birkhauser,Basel,1990,pp.209. 76

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BIOGRAPHICALSKETCH JoelRosenfeldwasborninGainesville,Florida.BorntoparentsKatherineVannandPerryRosenfeldwithonesibling,SpencerRosenfeld.Hehasspentthepastsevenyearsstudyingmathematicsandthelastveingraduateschool.JoelhasenjoyedhistimeattheUniversityofFloridaimmensely.Beforehebecameamathematician,heworkedasagraphicdesignerforseveralyearsataneducationaltechnologycompany.Healsohasworkedasaprogrammerinseveralcontractjobs.Joelispassionateabouteducation,mathematics,andthehistoryofmathematics.HereceivedhisPh.D.degreefromtheUniversityofFloridainthespringof2013. 77