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Functions of Positive Real Part of the Unit Ball of a Normed Space

Permanent Link: http://ufdc.ufl.edu/UFE0044528/00001

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Title: Functions of Positive Real Part of the Unit Ball of a Normed Space
Physical Description: 1 online resource (65 p.)
Language: english
Creator: Castillo-Gil, Miriam S
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: ball -- cauchy -- classes -- commutative -- contractions -- duality -- fantappie -- functional -- herglotz -- inequality -- kernel -- lie -- neumann -- operators -- pairing -- polydisk -- positive -- riesz -- schur -- transform -- unit
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: We study some classes of holomorphic functions of positive real part on domains Omega that are the unit ball for some norm over C^d and characterize these classes through operator-valued Herglotz formulas and through von Neumann-type inequalities. Inspired on the work of J.E. McCarthy and M. Putinar we extend results valid for the unit ball, due to  M.T. Jury, to the polydisk and other unit balls by defining a family of Fantappie pairings on Omega in order to establish duality relations between certain pairs of classes.
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 Miriam S Castillo-Gil.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Jury, Michael.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044528:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044528/00001

Material Information

Title: Functions of Positive Real Part of the Unit Ball of a Normed Space
Physical Description: 1 online resource (65 p.)
Language: english
Creator: Castillo-Gil, Miriam S
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: ball -- cauchy -- classes -- commutative -- contractions -- duality -- fantappie -- functional -- herglotz -- inequality -- kernel -- lie -- neumann -- operators -- pairing -- polydisk -- positive -- riesz -- schur -- transform -- unit
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: We study some classes of holomorphic functions of positive real part on domains Omega that are the unit ball for some norm over C^d and characterize these classes through operator-valued Herglotz formulas and through von Neumann-type inequalities. Inspired on the work of J.E. McCarthy and M. Putinar we extend results valid for the unit ball, due to  M.T. Jury, to the polydisk and other unit balls by defining a family of Fantappie pairings on Omega in order to establish duality relations between certain pairs of classes.
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 Miriam S Castillo-Gil.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Jury, Michael.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044528:00001


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FUNCTIONSOFPOSITIVEREALPARTONTHEUNITBALLOFANORMEDSPACEByMIRIAMSALOMECASTILLOGILADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012MiriamSalomeCastilloGil 2

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Tomymom,whoseunconditionalloveandsupporthasbroughtmethisfarinlife 3

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ACKNOWLEDGMENTS IwouldliketothankmyadviserDr.MichaelJuryforhisinnitepatienceandexcellentmentoring.Thankyoutoallthemembersofmygraduatecommittee,Dr.LouisBlock,Dr.ScottMcCullough,Dr.PaulRobinson,Dr.Li-ChienShenandDr.MeeraSithatram,fortheirsuggestions,commentsandavailability.SpecialthankstothemembersofthestaffintheDepartmentofMathematics,fortheirhelpandformakingalltheadministrativestepsgosmoothly.Thankstoallmyfriendsinthisdepartmentandoutsideofthedepartmentfortheirsupportandconstantencouragingwords.Lastbutnotleastspecialthankstomywonderfulmother,Profr.MiriamGilVazquez,whowassounderstandingandpatientthroughthisprocess,whosupportedmemorallyandnanciallywhenneeded,whoencouragedmeandinculcatedinmeapassionforlearning. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 2BACKGROUND ................................... 11 2.1CauchyTransform ............................... 13 2.1.1TheSpaceofCauchyTransforms ................... 14 2.1.2TheDualofO(D) ............................ 14 2.2Riesz-HerglotzTheorem ............................ 16 2.3FantappieTransform .............................. 20 2.4OperatorSpaces ................................ 24 2.4.1TheOperatorSpacesMIN(V)andMAX(V) ............ 26 2.4.2ExtensionTheorems .......................... 27 3MAINRESULTS ................................... 29 3.1Duality ...................................... 29 3.2Operator-ValuedHerglotzKernels ...................... 36 3.2.1RealizationofS+E() .......................... 37 3.2.2DualoftheClassS+E() ........................ 39 3.2.3VonNeuman-typeInequalities ..................... 44 4EXAMPLES ...................................... 48 4.1ThePolydisk .................................. 48 4.2Thel1-UnitBall. ............................... 50 4.2.1E=MIN(l1d) ............................... 50 4.2.2E=MAX(l1d) .............................. 53 4.3Anon-ReinhardtDomainExample ...................... 54 5FUTURERESEARCH ................................ 58 5.1OperatorSpaces ................................ 58 5.2KernelsontheLieBall ............................. 58 5.3TheSpaceofFantappieTransformsF .................... 59 REFERENCES ....................................... 63 BIOGRAPHICALSKETCH ................................ 65 5

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LISTOFFIGURES Figure page 3-1DistinctcharacterizationsofthedualclassesS+E()andR+E() ........ 47 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyFUNCTIONSOFPOSITIVEREALPARTONTHEUNITBALLOFANORMEDSPACEByMiriamSalomeCastilloGilAugust2012Chair:MichaelT.JuryMajor:MathematicsWestudysomeclassesofholomorphicfunctionsofpositiverealpartondomainsthataretheunitballforsomenormoverCdandcharacterizetheseclassesthroughoperator-valuedHerglotzformulasandthroughvonNeumann-typeinequalities.InspiredontheworkofJ.E.McCarthyandM.Putinarweextendresultsvalidfortheunitball,duetoM.T.Jury,tothepolydiskandotherunitballsbydeningafamilyofFantappiepairingsoninordertoestablishdualityrelationsbetweencertainpairsofclasses. 7

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CHAPTER1INTRODUCTIONTheresearchpresentedinthisdissertationhasitsoriginsinthestudyoftheclassofholomorphicfunctionswithpositiverealpart.Thesefunctionsplayanimportantroleinmanyapplicationsofcomplexanalysisandhavebeenwidelystudied,especiallyintheonevariablecaseovertheunitdiskinwhichacompleteclassicationwasgivenbytheRiesz-Herglotztheoremin1911.Thistheoremsayseveryfunctionholomorphicinthediskandhavingpositiverealpartcanbewrittenasfollows: f(z)=Z@D(1+z w)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(z w))]TJ /F6 7.97 Tf 6.59 0 Td[(1d(w)+it,(1)forsomepositivemeasureontheunitcircle@Dandsomerealnumbert(whichcorrespondstothevalueoftheimaginarypartoff(0)).Thestudyofthesefunctionsdoesnotendwiththisclassication;forinstance,thereisasurprisingconnectionwithvonNeumann'sinequality.vonNeumann'sinequalitystatesthat,givenaholomorphicfunctionfonthediskwithpositiverealpart,andamatrixTwithnormlessthanone,wehavethattherealpartoff(T)ispositive.Itfollows,fromthevonNeumanninequality,thatallholomorphicfunctionsontheunitdiskwithpositiverealpartarealsocontainedinthepositiveSchurclassS+,thatconsistsofthefunctionsholomorphiconthediscforwhichthekernelf(z)+ f(w) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(zwispositivedenite(seedenitioninSection 2.2 ).Summingup,inonedimension,itfollowsthatallholomorphicfunctionswithpositiverealpartcanalsobedescribedasafunctionwithrepresentationasinEquation 1 (Riesz-Herglotz)andasmembersoftheSchurclass,wewillseethisthoroughlyinSection 2.2 .Asnotedabove,threeverydifferentlookingconditionsdescribethesameclassinthecaseoftheunitdisk;howeverthisisnotthecaseinhigherdimensions.InspiredbytheconnectionobservedintheonevariablecasebetweenthevonNeumanninequalityandthepositiveSchurclass,anewpointofviewhasemergedoverthepastfew 8

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decades,thatitisworthwhilestudyingholomorphicfunctionsintermsoftheiractionsonmatricesoroperatorsandnotjustonscalars.Ingeneral,thesetofallfunctionswithpositiverealpartonagivendomaininCdcanbehardtoclassify,butsmallerconesdenedbyvonNeumann-typeinequalitiesaremoretractable.OneexampleofthisapproachistheworkofJ.E.McCarthyandM.Putinarontheunitball[ 14 ].TheyconsideredthepositiveSchurclassinseveralvariables,whichconsistsofalltheholomorphicfunctionsontheballsuchthatthekernelk(w,z)=f(z)+ f(w) 1)-221(hz,wiispositivesemidenite.Thisclasscanberepresentedintwoways;ononehandDruryandArveson[ 3 9 ]showedthatthepositiveSchurclassisequaltotheclassoffunctionsf,suchthattherealpartoff(T)ispositive,whereT=(T1,T2,...Td)isad-tupleofcommutingboundedoperatorsoversomeauxiliaryHilbertspaceHsatisfyingI)]TJ /F10 11.955 Tf 12.95 8.97 Td[(Pdi=1TiTi0;d)]TJ /F1 11.955 Tf 9.3 0 Td[(tuplesofoperatorssatisfyingthisinequalityarecalledrowcontractions.Ontheotherhand,itturnsoutthatthefunctionsinthisclasscanalsoberepresentedthroughthefollowingformula,whichwewillrefertoasanon-commutativeHerglotzrepresentation:f(z)=h(1+dXi=1ziSi)(1)]TJ /F7 7.97 Tf 18.02 14.94 Td[(dXi=1ziSi))]TJ /F6 7.97 Tf 6.59 0 Td[(1,i+it,wheretheinnerproductistakenovertheauxiliaryHilbertspaceH,tisarealnumberandSanyrowcontraction(notnecessarilycommutative).Similarresults,duetoJ.Agler[ 1 ],areknownforthepolydisk,wheretheoperatorsinvolvedwiththevonNeumann-typeinequalityarealld-tuplesofcommutativecontractionsT;thatis,TconsistsofcommutingoperatorssatisfyingjjTijj<1foralli=1,...,d.Theinvestigationsonthedualityrelationshipsbetweentheseclassesstudiedin[ 12 14 ]alsoinvolveanoperatortheoryapproach.Inparticularwendin[ 12 ]thatthesamefamilyofoperatorscandescribeaclassoffunctionsthroughanon-commutativeHerglotzrepresentation,whilecharacterizingthecorrespondingdualclassthroughavonNeuman-typeinequality.Forinstance,thedualofthepositiveSchurclasscan 9

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becharacterizedbyallrowcontractionsthroughavonNeumann-typeinequalityandasymmetricfunctionalcalculusnecessarytodenetheevaluationofaholomorphicfunctiononanon-commutatived-tupleofoperators.Themaingoalinthisdissertationisextendtheresultsfoundin[ 14 ]and[ 12 ]thatholdfortheuniteuclideanball,tootherdomains,inparticulartodomainsthatcorrespondtotheunitballofanygivennormedspace.Oneofthedifcultiesinextendingtheexistingresultsovertheballisthatmoregeneraldomainscouldlackcertainsymmetrypropertiesneededtocomputethedualitypairing;wesaysuchdomainsarenon-Reinhardt.Losingtheself-dualityoftheeuclideannormwhenpassingtootherdomains(unitballsofarbitrarynorms)isanothercrucialdifferencethatneedstobeconsidered.Thekeytoachievingthisgeneralizationwasdeninganappropriatedualitypairingthataccountsforthesedifferencesandextendsthedualityrelationusedintheeuclideancase.WedenedthedualitythroughafamilyofFantappiepairingsbasedonresultsfromJ.E.McCarthyandM.Putinar[ 14 ].Thedualitypairingpresentediscomputedinanoticeablydifferentmanneronlywhenthedomainisnon-Reinhardt;thusweillustratethesignicanceofthisgeneralizationthroughsomeexamples.SincethedomainsweconsiderareunitballsofsomenormedspaceandsincemostofthecanonicalexamplesofunitballsturnouttobeReinhardtweconcludewithanexamplewhereweconsidertheunitballoftheLienormwhichisanon-Reinhardtdomain.Inordertoobtainoperator-valuedrepresentations,analogoustothoseforthecasesoftheballorthepolydisk,foranalyticfunctionswithpositiverealpartovertheunitballofsomevectorspaceV,itwasnecessarytoviewtheproblemfromamoregeneralperspective.ThemainresultoverthepolydiskmentionedabovefollowsfromafactorizationtheoremduetoAgler[ 1 ].SimilarlyTheorem6.1in[ 13 ],duetoM.Jury,whichextendsthatofAgler'stothemoregeneralsettingofoperatorspaces,enabledustoobtaintherequiredrepresentationbyconsideringanoperatorspacestructureoverthenormedspaceV. 10

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CHAPTER2BACKGROUNDWewillusethefollowingnotation:Cdenotesthesetofcomplexnumbers,byz2Cdwemeanz=(z1,...,zd)witheachzi2C,wedenotetheusualhermitianinnerproducthz,wi=Pdj=1zj wjwithcorrespondingnormjzj=p hz,zi.WewillwriteBfortheeuclideanball,thatis,B=fz2Cd:jzj<1gandDforthepolydisk,D=fz2Cd:kzk1<1g.Wewillalsoneedthefollowingmulti-indexnotation.Let=(1,...,d),whereeachi(fori=1,,d)isapositiveinteger,byzwemeanz=z11zdd.Forexample,ifd=3,=(2,1,4),thenz=z21z2z43.Wealsodenejj=1++dand!=1!2!d!,forinstance,withanddasabovejj=7and!=2!1!4!=48.ThroughthisdissertationwewilldenotethespaceofholomorphicfunctionsonsomedomainGbyO(G)withthetopologyofuniformconvergenceoncompactsubsetsofG,sothatO(G)isalocallyconvextopologicalvectorspace.Wedenotetherealpartofafunctionfby<(f)andthesubsetofholomorphicfunctionsonthedomainGthathavepositiverealpartbyO+(G).WhenweomitthedomainGwemeanGistheballB.Ourstudyismainlyconcernedwithpositiveclassesoffunctionswhichwedenebelow. Denition1. [ 12 ,Denition2.1]ApositiveclassonBisasetoffunctionsPO+whichisaclosed,convexconeinOandisclosedunderdilations,thatis,iff2P,thenfr2Pforall0r1,wherefrisdenedbyfr(z):=f(rz).Asourmaingoalistoextendcertainresultsfoundin[ 12 ]thatapplyforholomorphicfunctionsdenedontheeuclideanballinddimensionsBCdtootherdomains,wediscusswhichkindsofdomainswewillbeinterestedin.ThedomainsthatwewillbeworkingwithcorrespondtounitballsofothernormsoverCd,suchasthel1andl1 11

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norms.TheextensionoftheresultsisnotstraightforwardwhenweconsiderdomainswhicharenotReinhardt. Denition2. AnopensubsetGofCdiscalledaReinhardtdomainif(z1,...,zd)2Gimpliesthat(ei1z1,...,eidzd)2G,forallrealnumbers1,...,d.ForinstancetheeuclideanballBandthepolydiskDareReinhardtdomains,butwewillseeinSection 4.3 thattheunitballcorrespondingtotheLienorm(seeDenition 37 )turnsouttobeanon-Reinhardtdomain.Theextensionsoftheresultsmentionedabove,inparticulartheproofofTheorem 3.5 ,becomesamuchmoreinterestingprobleminthecasethatthedomaininquestionisnon-Reinhardt.ThemainreasonwillbethefactthatthemonomialszconstituteanorthonormalbasisforfunctionspacesovertheReinhardtdomains,whichallowstheoftenusefulexpansionofthefunctionsinsuchspacesaspowerseries,howeverthesemonomialslackorthogonalitywhenworkingwithfunctionspacesovernon-Reinhardtdomains.Aswementionedabove,weworkwithspacesofholomorphicfunctionsdenedonsomedomain.InparticularwerecurrentlymentionandusetwowellknownfunctionHilbertspaces,theBergmanspaceandtheDrury-Arvesonspace.Webrieydescribethesetwospaces,see[ 2 ]forafurtherdiscussion.TheBergmanspaceoversomedomain,whichwewilldenotebyA2(),isdenedasA2():=L2(,dV)\O(),wheredVistheLebesgue(volume)measure.Giventwofunctionsf,g2A2(),theinnerproductoffandginA2()isdenedbyhf,giA2()=Rf gdV.TheDrury-Arvesonspace,denotedbyH2d,istheHilbertfunctionspacewhichconsistsoffunctionsthatareholomorphicontheballBandwithinnerproductdenedasfollows.Sayfandg2H2d,sincetheyareholomorphicontheballtheyhaveTaylorseriesexpansionsf=Pczandg=Pdz(wherethesumsareoverallpossiblemulti-indices=(1,...,d)),thendenetheinnerproductoffandginH2dbyhf,gid=Xc d! jj!. 12

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Denition3. LetHbeaHilbertspace,ifthereisavectork2Hwiththepropertythathf,ki=f()forallf2H,thenwesaythatHisareproducingkernelHilbertspacewithreproducingkernelk.Sincek2H,wehavethatk()=hk,kiwhichwedenotebyk(,).IfHisaholomorphicspace,thenkisholomorphicintherstvariableandconjugateholomorphicinthesecond[ 2 ].WecanseethattheDrury-ArvesonspaceH2disareproducingkernelHilbertspacewithreproducingkernelk(z,w)=1 1)-222(hz,wi.Forthisweuse[ 2 ,Proposition2.18],whichstatesthatareproducingkernelk(z,w)=Pi2I ei(z)ei(w),wherethesetfeigi2IisanorthonormalbasisforthecorrespondingreproducingkernelHilbertspaceH.Thus,since1 1)-222(hz,wi=1Xn=1(hz,wi)n=Xjj! !zw,andthemonomialszareanorthogonalbasisforH2d,wehavethatkisitsreproducingkernel. 2.1CauchyTransformInthissectionweworkovertheeldofthecomplexnumbersinonedimension,sobyDwejustmeantheunitdiskonC.LetMbethesetofnite,complex,BorelmeasuresontheunitcircleT:=@D,thentheCauchytransformofthemeasure2MisthefunctiongivenbyK(z)=ZT1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(zd(),withz2DNotethatforallz2Dandall2Twehavethatjzj=jzjjj=jzj<1sothat1)]TJ /F3 11.955 Tf 12.25 0 Td[(znevervanishes,thustheCauchytransformofisananalyticfunctiononthediskD.WedenotebyK:=fK:2MgthespaceofCauchytransformsandrecallthatH1(T)isthesetofholomorphicfunctionsL1integrable,thatis,H1(T)=fanalyticfjsup0
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2.1.1TheSpaceofCauchyTransforms Denition4. Foraxedfunctionf2K,letRf=f2M:f=Kgbethesetofmeasuresthatrepresentf.In[ 5 ,p.42]itisproventhatRfisaninnitesetandthatif,2Rfthend)]TJ /F3 11.955 Tf 12.11 0 Td[(d=dm,where2 H10:=f2H1(T):(0)=0g,whichisaclosedsubspaceofM(mistheLebesguemeasure).WiththisidenticationofthemeasuresinMwecanseethatthemapf7![]isanisomorphism,thus K=M= H10.(2)IfweequipKwiththequotientspacenormjjKjj:=jj[]jj,thenthemapbecomesanisometricisomorphismanditmakesKintoaBanachspace.WewillseethatthisBanachspaceisjustthedualofthediskalgebra.RecallthatthediskalgebrawhichwewilldenotebyAconsistsofallthecontinuousfunctionsonthecloseddisksuchthattheyarealsoanalyticontheopendisk(A=C( D)\O(D)). Denition5. IfAisthediskalgebra,thenA?:=f2M:Rf d=0forallf2Ag.NotethatinDenition 5 iff()=nfornanynonnegativeinteger,byatheoremofRiesz[ 5 ,1.9.7],wegetthatexactlyd=dm,with2H10,sothatinfactA?= H10.OntheotherhandrecallthatthespaceofmeasuresMcanbeidentiedwiththedualofthecontinuousfunctionsonthecircleC(T).WiththeaboveidenticationswehavethatthefollowingbasicfunctionalanalysistheoremimpliesthatK=A Theorem2.1. [ 5 ,1.4.6]ForaclosedsubspaceWofaBanachspaceX,thequotientspaceX=W?isisometricallyisomorphictoW.ToseethatindeedTheorem 2.1 impliesK=A,inEquation 2 letX=C(T)=MandW?=A?= H10. 2.1.2TheDualofO(D)TheCauchytransformplaysanimportantroleinthefollowingclassicalresult. 14

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Theorem2.2. ThedualofthespaceofholomorphicfunctionsonthediskO(D)isO( D):=ff:fisholomorphiconaneighborhoodof Dg. Proof. Weprovethatthefollowingpairingestablishestheduality.Forf2O(D)andg2O( D)(f,g):=1 2iZrTf(z)g(z)dzforsomer<1.Firstweshowthatthepairing(,g)denesaboundedlinearfunctionalonO(D)foreachg2O( D).Indeedforgiveng2O( D),wehavej(f,g)j1 2ZrTjf(z)jjg(z)jdzrkfkrTkgkrT<1Ontheotherhandgivenafunctional'2O(D)weprovethereisafunctiong2O( D),suchthat(,g)='.ByRieszrepresentationtheoremthereexistsuniquemeasuresupportedonacompactsetKD,suchthat'(f)=RKfdforallf2O(D).Nowletr2(0,1),suchthatjwj
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Aclassicalcomplexanalysisresult(forareferencesee[ 6 ])statesthatpreciselytheintegral(?)isthenthcoefcientinthepowerseriesexpansionoftheholomorphicfunctionfaboutzero,thus(f,g)=ZK0 1Xn=1(r2w)nan!r2d(w)=ZK0f(r2w)r2d(w)=ZKf(w)d(w)='(f) WementionedabovethattheCauchytransformplaysanimportantroleintheproofofthistheorem,indeedwecanseethatitappearsinthedenitionofthefunctiong(z):=zK().WewillseeinSection 2.3 thatageneralizationoftheCauchytransformcanbeusedtoproveasimilar,butmoregeneral,dualitytheorem. 2.2Riesz-HerglotzTheoremThemainreasonwehavediscussedtheCauchytransformintheprevioussection,istopointouthowcloselyrelateditistotheHerglotzrepresentation,whichweintroduceinthissection,aswellasthefollowingclassicalresultinanalysis,thecombinationoftheRieszRepresentationTheoremandHerglotzIntegralRepresentationTheorem,forareferencesee[ 5 ,Theorems1.3.6,1.8.9]. Theorem2.3(Riesz-Herglotz). Letf2O(D).Then
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Distributingtheproductsinthenumeratorandsimplifyingweobtain2<1+z 1)]TJ /F3 11.955 Tf 11.95 0 Td[(z =2)]TJ /F4 11.955 Tf 11.95 0 Td[(2zz (1)]TJ /F3 11.955 Tf 11.96 0 Td[(z )(1)]TJ /F4 11.955 Tf 12.1 0 Td[(z)andsince=1because2@Dwehave: <1+z 1)]TJ /F3 11.955 Tf 11.95 0 Td[(z =1)-222(jzj2 j1)]TJ /F3 11.955 Tf 11.96 0 Td[(zj20,forallz2D(2)WealsonotetheconnectionbetweentheCauchytransformandtheHerglotzrepresentationsincewecanwriteH(z,)=21 1)]TJ /F3 11.955 Tf 11.96 0 Td[(z)]TJ /F4 11.955 Tf 11.95 0 Td[(1. Denition6. BytherealpartofanoperatorTwemean<(T)=1 2(T+T) Denition7. WesayanoperatorToversomeHilbertspaceHispositiveifhTx,yiH0forallx,y2H.AndwewriteT0. Lemma1. Iftheoperator1)]TJ /F3 11.955 Tf 12.27 0 Td[(Tisinvertiblethen<(1+T)(1)]TJ /F3 11.955 Tf 12.27 0 Td[(T))]TJ /F6 7.97 Tf 6.59 0 Td[(10ifandonlyif1)]TJ /F3 11.955 Tf 11.96 0 Td[(TT0,orkTk1.Inparticular,ifkTk<1,then<(1+T)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(T))]TJ /F6 7.97 Tf 6.59 0 Td[(10. Proof. Supposethat<(1+T)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(T))]TJ /F6 7.97 Tf 6.58 0 Td[(10,thatis:0(1+T)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(T))]TJ /F6 7.97 Tf 6.59 0 Td[(1+(1+T)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(T))]TJ /F6 7.97 Tf 6.59 0 Td[(1=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(T))]TJ /F6 7.97 Tf 6.59 0 Td[(1[(1+T)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(T)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(T)(1+T)](1)]TJ /F3 11.955 Tf 11.95 0 Td[(T))]TJ /F6 7.97 Tf 6.59 0 Td[(1 )(1)]TJ /F3 11.955 Tf 11.95 0 Td[(T))]TJ /F6 7.97 Tf 6.59 0 Td[(1[2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(TT)](1)]TJ /F3 11.955 Tf 11.96 0 Td[(T))]TJ /F6 7.97 Tf 6.59 0 Td[(10(NotetheanalogywithEquation 2 )(2)Multiplying 2 ontheleftyby(1)]TJ /F3 11.955 Tf 13.1 0 Td[(T)andontherightby(1)]TJ /F3 11.955 Tf 13.09 0 Td[(T)wehavethat1)]TJ /F3 11.955 Tf 11.96 0 Td[(TT0,orkTk1. Theorem 2.3 hasimportantconsequencesinanalysis,amongotherswepointoutonethatpertainstoourstudy.Thestatementofthetheoremimpliesthattheclassesoffunctionsthatwedenebelow,inanapparentverydifferentway,areinfactequivalent.FirstconsidertheclassofallholomorphicfunctionsonDwithpositiverealpartO+(D)andthefollowingtwoclasses. Denition8. M+=f2O(D):f(z)=R@DH(z,)d()+it,t2R;>0 17

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Denition9. S+=nf2O(D):f(z)+ f(w) 1)]TJ /F7 7.97 Tf 6.58 0 Td[(z wispositivesemideniteoRecallthatasesquilienarfunctionk(z,w)denedonKKissaidtobepositivesemideniteif nXi,jci cjk(wi,wj)0,(2)foranyintegern,anycomplexnumbersc1,...,cnandanyelementsw1,...,wnbelongingtoK(InoursettingweconsiderK=D).Asimplealgebraiccalculationshowsthatwecanwrite H(z,)+ H(w,) 1)]TJ /F3 11.955 Tf 11.95 0 Td[(z w=21 1)]TJ /F3 11.955 Tf 11.96 0 Td[(z 1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(w (2)Thus,forf2M+andthekernelk(w,z):=f(z)+ f(w) 1)]TJ /F7 7.97 Tf 6.59 0 Td[(z w,lettingg(z)=p 21 1)]TJ /F3 11.955 Tf 11.95 0 Td[(ztheinequality 2 takestheform:Pci cjg(wi) g(wj)=jPcig(wi)j20.IntegratingeachsideofEquation 2 ,overtheboundaryoftheunitdiskwithrespecttosomepositivemeasure,bylinearityoftheintegralwecanseethatiff2M+thenf2S+aswell,thusM+S+.Converselyiff2S+,wecanseethattherealpartoffisnon-negativeandbyRiesz-HerglotzTheorem 2.3 ,f2M+.ThuswegetinfactthatO+=M+=S+.IntheintroductionwepointedoutthatthereisaconnectionbetweenthisresultandvonNeumann'sinequality,indeed,wecangetasimpleproofofthefamousinequalityusingthefactthatO+=S+. Theorem2.4(VonNeumann'sInequality[ 16 ]). LetTbeanoperatoronsomeHilbertspaceHwithkTk1.Thenforanycomplexpolynomialp,kp(T)kkpk1=supfjp(z)j:jzj1g SketchofProof. Letq(z)=p(z) kpk,thenkqk1,considertheinverseCayleytransformofq,sayf(z)=1+q(z) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(q(z) 18

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whichwillyield
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istakenoverallthepermutationsof11's,22's,etc.Ashortexampletoclarifythisdenitionfollows.Letd=2,=(2,1)thenforT=(T1,T2)wehave:(z)sym(T)=(z21z2)sym(T)=1 3(T21T2+T1T2T1+T2T21)Notethatwhentheoperatorscommutewewillindeedhave(z)sym(T)=T.JustlikewenoticedtheconnectionbetweentheHerglotzrepresentationandtheCauchytransforminonedimension,wewillseethatinhigherdimensionsaHerglotz-typerepresentationiscloselyrelatedtothesocalledFantappietransformwhich,aswediscussinthenextsection,isageneralizationtoseveralvariablesoftheCauchytransform. 2.3FantappieTransformInthissectionwedenetheFantappietransformanddiscussitsroleintheproofsthatweintroduceinChapter 3 .Wewillseeitprovidesthemeanstoestablishcertaindualityrelationshipbetweendomains,butrstwedenewhatwemeanbyadualdomain. Denition10. ThedualofadomainGisdenedas G=fzjhz,wi6=1forallw2Gg(2)Fromnowonweconsiderdomains,Gand=GinCdsuchthatBG,recallthatBistheunitballinthel2normind)]TJ /F1 11.955 Tf 9.29 0 Td[(dimensions.OneofthemainingredientstogetstartedwiththedualityargumentsisthefactthattheCauchy-Fantappiepairingsarewelldened,whichfollowsfromtheclassicalresultthatthedualofO(B)isO( B)(Martineau-AizenbergdualityTheorem).Whenwerestricttotheonedimensionalcase,thisisjustthecaseoftheunitdisk(Theorem 2.2 )mentionedinSection 2.1 wheretheCauchytransformappearsintheproof.Similarly, 20

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inthed-dimensionalcaseaproofthatusesageneralizationoftheCauchytransformisfoundin[ 14 ].ThisgeneralizationistheFantappietransform,whosedenitionfollows. Denition11. TheFantappietransformofacomplexmeasureisF(z)=ZCdd(w) 1)-222(hz,wi,withhz,wi6=1,forallw2supp()WewillfocusonlyonmeasureswithsupportcontainedintheclosureoftheunitballB,thus(F)(z)willbewelldenedforz2B. Denition12. IffisafunctionintheBergmannspaceA2()forsomedomainB,byFfwemeantheFantappietransformofthefunctionfdVj,thatis(Ff)(z)=ZfdV(w) 1)-222(hz,wi,withz2wheredVisthelebesguemeasureinCdnormalizedsothatV(B)=1.ThemonomialszdiagonalizetheFantappietransform,forinstanceiff(z)=PczisinA2(B)then (Ff)(z)=Xc (jj+1)(jj+n)z[ 14 ](2)InfactthemonomialszdiagonalizetheFantappietransforminanyA2()aslongasisaReinhardtdomain(seeDenition 2 ).Toseethiswerstprovethefollowingproposition. Proposition2.1. SupposethatCdisaReinhardtdomain,thenthemonomialszareorthogonalwithrespecttotheBergmaninnerproduct,thatis,hz,ziA2()=0whenever6=. Proof. WewanttoshowthattheintegralRzzdV(z)=0when6=.Notethat,bytheReinhardthypothesis,themeasuredVonisTd-invariant,thatis,forany=(1,...,d)2TdandsubsetFof,thecorrespondingsetF=f(1z1,...,dzd):z2Fg 21

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issuchthatVol(F)=Vol(F).Thusforall2TdwehaveZzzdV(z)=ZzzdV(z).NowweintegratethisexpressionoverTdZTdZzzdV(z)d=ZTdZzzdV(z)d,notethatbyFubini'stheoremwemayswitchtheorderofintegrationandthattheiteratedintegralsareproductofintegrals,thenweobtain: ZzzdV(z)=ZTdZzzdV(z)d=ZzzZTdddV(z),(2)andsince ZTdd=ZC1ZC2ZCdddd2d1(2)whereeachCiisaunitcirclewhichcanbeeasilyparametrized,andsincetheintegralisjustaproductofintegralsineachvariablei,weonlyneedtoshowthatforsomeCi,wehaveRCiiiiidi=0.Indeed,RCiiiiidi=R20ei(i)]TJ /F11 7.97 Tf 6.59 0 Td[(i)d=2ii.Nowbecause6=atleastforoneiitistruethati6=i,sothatatleasttheintegralalongthecorrespondingcircleCiiszero.ThustheintegralsinEquations 2 and 2 arezeroandtheresultfollows. ThefactthatthemonomialszareorthogonalunderthevolumemeasureforaReinhardtdomain,impliesthatthezareeigenvectorsfortheFantappietransform.ToseethisconsidertheFantappietransformofthefunctionf(z)=z: (Ff)(z)=Zw 1)-222(hz,widV(w)=Z Xjj! !zw!wdV(w)withz2,(2)sinceforsuchxedz,wehavethatjhz,wij<1,infactforallw2,wehavejhz,wij
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uniformlyboundedandwecanapplyDominatedConvergenceTheoremandProposition 2.1 toget,(Ff)(z)=Z Xjj! !zw!wdV(w)=XZjj! !zwwdV(w)=jj! !zZwwdV(w)thus,wehaveprovedthefollowinglemma. Lemma2. ThemonomialszdiagonalizetheFantappietransformF.ThroughoutthisdissertationweconsiderdomainsandGsuchthatBG,welistthefollowingdenitionsandresultsneededfrom[ 14 ]. Proposition2.2. LetR:Hd2!A2()betherestrictionoperatorRf=fj.ThentheoperatorRR:Hd2!Hd2haseigenvalues01and kfk(z)=Zfk(w)dV(w) 1)-222(hz,wi,(2)wherethefkarethecorrespondingeigenvectorsallnormalizedtohavelengthone. Proof. SinceB,RRisacompactoperator,alsoitisselfadjoint,thusithascountablemanyrealeigenvaluesthatconvergetozero.NownotethatRRfk=kfk,hRRfk,giH2d=khfk,giH2dforallg2H2d,hfk,giA2()=khfk,giH2dforallg2H2dLettinggbethereproducingkernelatzforH2d,weobtaintheidentityinEquation 2 Fromthisidentity(Equation 2 )wecanseethateacheigenfunctionfkextendsanalyticallytotheconnectedcomponentoftheoriginin. Denition13. LetHF()=fP1k=0akp kfk:Pjakj2<1gbeanewHilbertspaceofholomorphicfunctionsonbyrequiringthatfp kfkgtobeanorthonormalbasis. Proposition2.3. [ 14 ,Proposition3.5]TheFantappietransformisanisometricisomor-phismfromA2()ontoHF() 23

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Inotherwords,Proposition 2.3 saysthatthisnewHilbertspace,istheimageofA2()undertheFantappietransform,thus,wewillbeworkingbackandforthonbothspacesA2()andFA2()O(),whichhaveorthonormalbasesn1 p kfkoandp kfkrespectively. Theorem2.5. [ 14 ,Theorem4.9]ConsiderasequenceofsmoothlyboundedconvexdomainsGm,suchthatBGmGm+1and[mGm=G,thenthesequenceobtainedbytakingthedualoftheGm,thatism=Gm,isadecreasingsequencesuchthat\mm= .ThedualofO( )isO(G).Thedualityisimplementedforf2O( )andg2O(G),bychoosingmsufcientlylargesothatf2A2(m)andletting(f,g)=hFf,giHF(Gm);therighthandsideisindependentofm.NotethatbyProposition 2.3 thepairinginTheorem 2.5 canbealsowrittenas (f,g)=hFf,giHF(Gm)=hf,F)]TJ /F6 7.97 Tf 6.58 0 Td[(1giA2(m)(2)Theworkfoundin[ 14 ]and[ 12 ]usessuchpairing,however,itisbeingusedinthespecialcasewhenG=B=Gandunderthesecircumstances,wehave,likeitispointedoutin[ 14 ,pg.64],thatthepairinggiveninEquation 2 isinfacttheinnerproductintheDrury-ArvesonspaceH2d. 2.4OperatorSpacesFortheproofofourmainresultwewilltakeamoregeneralapproachandtourproblemintoanoperatorspaceframe.Inthissectionwereviewdenitionsandconceptsregardingoperatorspacesaswellassomeextensiontheorems.Mostofthisdiscussionisbasedon[ 16 ].FirstweconsideravectorspaceVanddenotebyMm,n(V)thesetofmnmatriceswithentriesfromV,weuseMm,nforMm,n(C)andMn,whenn=m. 24

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Denition14. AvectorspaceVisamatrixnormedspaceifthefollowingconditionholds: (R1): Therearenormskkm,nonMm,n(V)suchthatkAXBkp,qkAkkXkm,nkBkforallmatricesA2Mp,m,X2Mm,n(V)andB2Mn,q,wherekkistheoperatornorm. Denition15. AmatrixnormedspaceVisanabstractoperatorspaceifthefollowingconditionholds: (R2): kXYk=maxfkXkm,n,kYkp,qgforallm,n,p,q,X,andY,whereXY:=X00Y2Mm+p,n+p(V)NoticethatonecaneasilycheckthatanyclosedsubspaceofB(H)satises(R1)and(R2),inthiscasewesaywehaveaconcreteoperatorspace.Infactwewillseethatanyabstractoperatorspacecanbeviewedasaconcreteoperatorspaceandviceversa,thisresultisduetoZ.J.Ruan(1988),infactconditions(R1)and(R2)areknownasRuan'saxioms. Theorem2.6(Ruan'sTheorem). LetVbeamatrix-normedspace.ThenthereexistsaHilbertspaceHandacompleteisometry':V)166(!B(H)ifandonlyifVisanabstractoperatorspace.WecanalwaysputanoperatorspacestructureonsomevectorspaceVbyspecifyingthematrixnorms,thatis,foreachdifferentchoiceoffamilyofnormswegetacertainoperatorspacestructure,moreprecisely,byRuan'sTheorem 2.6 ,wemayobtainexactlyasmanyoperatorspacestructuresoverVastherearelinearisometries':V)167(!B(H),bydeningtheoperatorspacenormsthroughk(vij)k'=k('(vij))kB(H(n)),for(vij)2Mn(V).Matricesoftheform(vij)liketheonesmentionedintheabovedenitions,whereeachvijisavectorinCd,areinMn(Cd)=MnCd,theisomorphismbetweenthese 25

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twospacesisknownasthecanonicalshufe.Thustoeachmatrix(vij)2Mn(Cd)correspondsad-tupleofnnmatrices(A1,...,Ad)2MnCd(eachAi2Mn)andifweconsiderthecanonicalorthonormalbasis(e1,...,ed)ofVwenotethatwecanwrite(vij)asPdi=1Aiei.Similarlyfor('(vij))wecanwritePdi=1Ai'(ei). Denition16. Consideramap:V)166(!K,forsomeHilbertspaceK,thendenen:Mn(V))166(!Mn(K),viatheformulan(aij)=((aij)) Denition17. Amapiscalledcompletelyboundedifsupnknkisnite,wedenotebyCB(E,F)thespaceofcompletelyboundedmapsbetweentheoperatorspacesEandF,withnormkkcb=supnknk.RuanshowedthatthereisanaturalwaytoendowtheBanachspacedualofanoperatorspacewithamatrix-normedstructuresuchthatthedualspaceisanoperatorspaceagain.ThelinearboundedlinearfunctionalsoverEareinfactalltheboundedlinearfunctionals,sinceeverylinearmap':E)166(!Chasnormk'k=k'kcb[ 16 ],thenE:=CB(E,C)withmatrix-normedstructureasfollows.IfA=('ij)2Mm(E),weidentifyitwithamap:E)166(!Mm,givenby(e)=('ij(e))andletk('ij)k=kkcb.Wecanalsoviewthismatrix-normedstructureasfollows,thenormofAinMm(E)isgivenbykAkMm(E):=supkABkMml(C)=supdXk=1AkBkMml(C),wherethesupremumistakeoveralll1andallBintheunitballofMl(E)[ 13 ]. 2.4.1TheOperatorSpacesMIN(V)andMAX(V)WewilltalkabouttwospecialoperatorspacestructuresthatcanbeputonanyvectorspaceVcalledMIN(V)andMAX(V).Consideringalltheoperatorspace 26

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structuresthatcanbeputonV,likewedescribedabove,wedeneMAX(V)=supfjj('(vij))jjB(H(n)):':V)166(!B(H)isometricgTodeneMIN(V)rstconsiderthecontinuousfunctionsontheunitballofthedualspaceofV,C(V1)andthelinearisometryj:V)166(!C(V1)denedbyj(v)(f):=f(v),forf2V1,identifyingVwithj(V)inducesaparticularfamilyofnormsonMn(V)makingVtheoperatorspaceknownasMIN(V).MoreexplicitlythenormsinMIN(V)aredenedasfollows jj(vij)jjn:=jj(j(vij)jjn=supfjj(f(vij))jjMn:f2Vg,(2)wherejj(f(vij))jjMnindicatesthenormofthescalarmatrix(f(vij))viewedasalineartransformationfromann)]TJ /F1 11.955 Tf 9.3 0 Td[(dimensionalHilbertspaceontoitself.Infactwehavethefollowingresult,whichmakesapparentwhythechoiceofnameforMIN(V). Theorem2.7. [ 16 ,Theorem14.1]LetVbeanormedspace,HaHilbertspace,andlet':V)166(!B(H)beanisometricmap.Thenfor(vij)2Mn(V)wehavejj(vij)jjMIN(V)jj('(vij))jjB(H(n))ThefollowingtheoremshowsaninterestingrelationshipbetweentheMIN(V)andMAX(V)operatorspacestructures. Theorem2.8(Blecher). [ 4 ]LetVbeanormedspace.ThenMIN(V)=MAX(V)andMAX(V)=MIN(V)completelyisometrically. 2.4.2ExtensionTheorems Denition18. Amapiscalledcompletelycontractiveifforeachpositiveintegernthemapniscontractive. Denition19. Amapiscalledcompletelypositiveifforeachpositiveintegernthemapnispositive. Denition20. LetA,BbeC-algebras,amap:A)165(!B,iscalleda*-homomorphismifitsatisesthefollowing: 27

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(xy)=(x)(y) (x+y)=(x)+(y) (x)=(x) Denition21. IfSisasubsetofaC-algebraA,thenwedeneS=fa:a2Sg,whenS=SwesaythatSisself-adjoint. Denition22. IfAhasaunit1andSisaself-adjointsubspaceofAcontaining1,thenwesaySisanoperatorsystem.NoticethatforanysubspaceMofAsuchthatM31,S=M+Misanoperatorsystem.WementionsomeextensiontheoremsthatweuseinChapter 4 Theorem2.9(Arveson'sExtensionTheorem). [ 16 ]LetAbeaC-algebra,SanoperatorsystemcontainedinA,and:S)166(!B(H)acompletelypositivemap.Thenthereexistsacompletelypositivemap, :A)166(!B(H),extending. Theorem2.10(Stinespring'sDilationTheorem). [ 16 ]LetAbeaunitalC-algebra,andlet:A)166(!B(H)beacompletelypositivemap.ThenthereexistsaHilbertspaceK,aunital*-homomorphism:A)165(!B(K),andaboundedoperator:H)165(!K,withk(1)k=kk2suchthat(a)=(a). 28

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CHAPTER3MAINRESULTS 3.1DualityInthissectionweestablishanalogousdualityresultstothosefoundin[ 14 ]and[ 12 ],tothisendweneedanewwaytopairfunctionsinordertobeabletodenedualitybetweenspacesofholomorphicfunctionsdifferentfromH2d.Recallthefollowingdenitionsfrom[ 14 ]and[ 12 ]respectively,topairholomorphicfunctionsonB Denition23. [ 14 ]Letf=Pcz,g=PdzbelongtoO(B).DenethepairingQ(f,g)=Xc d! jj!+ f(0)g(0)whenevertheseriesconvergesabsolutely.NoteintheabovedenitionthatifbothfandgareinH2d,thenQ(f,g)=hf,giH2d+ f(0)g(0).In[ 12 ]afamilyofpairingsisdenedslightlydifferentthanthepairingin[ 14 ]toarrangeabsoluteconvergenceoftheseriesinthedenition.[ 12 ,Lemma2.2]. Denition24. [ 12 ]Letf=Pcz,g=PdzinO(B)thenletQr(f,g)=Xc dr! jj!+f(0) g(0)Againwenotethat,forgivenr2[0,1),thepairingQr(,)denedin[ 12 ],exceptforthetermf(0) g(0),isjusttheH2dinnerproductofthedilatedfunctionsfp randgp r;recallthatfr(z):=f(rz).InviewofthefactthatthedenitionsofthesepairingsarecloselyrelatedtotheH2dinnerproduct,wedeneanewpairing,forholomorphicfunctionsdenedondualdomains,throughtheinnerproductofanauxiliaryHilbertspace(aBergmanspace)asfollows. 29

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Denition25. SupposethatBG,that=G,andthatf2O()andg2O(G)thendenethepairing ~Qr(f,g)=hfp r,F)]TJ /F6 7.97 Tf 6.59 0 Td[(1gp riA2(m)+f(0) g(0),(3)wherethedomainmsatisesmB.NotethatthispairingisbasicallytheonegivenbyEquation 2 thatwasintroducedin[ 14 ]toprovethedualityofMartineau-AizenbergviatheFantappietransform.WhenthedomainsandGareReinhardt,wherethemonomialszdiagonalizetheFantappietransform,thefunctionsfandgcanbeexpandedaspowerseriesasinDenitions 23 and 24 ,andthentheexpressionfor~Qr,inthiscase,couldbewrittenexactlyasinDenition 24 ,thatis,~Qr(f,g)=Pc dr! jj!+f(0) g(0).Indeediff=Pczandg=PdzbyDenition 25 wehavethat: ~Qr(f,g)=hfp r,F)]TJ /F6 7.97 Tf 6.59 0 Td[(1gp riA2(m)+f(0) g(0)=Xcdrz,F)]TJ /F6 7.97 Tf 6.59 0 Td[(1zA2(m)+f(0) g(0)=Xcdrhz,ziH2d+f(0) g(0)=Xcdr jj!+f(0) g(0)(3)wherethesecondtolastequalityisduetothefactthatthefunctionz2H2d[ 14 ,pg.64].Wemustkeepinmind,however,thattheinnitesumthatappearsheredoesnotrepresentaninnerproductoffp randgp rsincetheydonotevenbelonginacommonHilbertspace.Weneedtoknowinwhatcircumstancesistheinniteseries,appearingin 3 ,convergent,orinotherwordsif~Qr(f,g)iswelldened.Thatis,weneedtoensurethatthedilatedfunctionsfp randF)]TJ /F6 7.97 Tf 6.58 0 Td[(1gp rastheyappearinDenition 25 indeedbelongtotheBergmanspaceA2(m).Thefollowinglemmaprovesthisfact. 30

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Lemma3. Thepairing~Qr(f,g)=hfp r,F)]TJ /F6 7.97 Tf 6.59 0 Td[(1gp riA2(m)+f(0) g(0)iswelldenedforf2O()andg2O(G)andallr2[0,1). Proof. Notethatfp r2O()andgp r2O)]TJ ET q .478 w 225.73 -55.77 m 235.52 -55.77 l S Q BT /F3 11.955 Tf 225.73 -65.75 Td[(G,forallrsuchthat0m>)andm:=m=GmsothattheconditionsinTheorem 2.5 hold,thusobtainingthatfp r2A2(m)andgp r2HF(Gm)=FA2(m).Thisisallweneedtoassurethatforany0r<1,~Qr(f,g)iswelldenedforsuchfandg;thisalsoprovesthattheseriesappearingin 3 isabsolutelyconvergent. WeshouldkeepinmindthatwhenthedomainisnotReinhardt,thechainofequalitiesin 3 doesnotholdentirelysincewedonothavethepowerseriesrepresentationforfandgwithrespecttothemonomialsz.HoweverbyLemma 3 wehavethatfp r2A2(m)=F)]TJ /F6 7.97 Tf 6.59 0 Td[(1HF(Gm)andgp r2HF(Gm),sothatwecanrepresentFfp randgp raselementsofHF(Gm),thatis,Ffp r(z)=Ff(p rz)=Pakp kfk(p rz)forsomeak'sandgp r(z)=g(p rz)=Pbkp kfk(p rz)forsomebk's.Inthissituation,amoreexplicitdenitionforthepairing~Qr(f,g)canbegivenbythecorrespondinginnerproductinHF(Gm)asfollows ~Qr(f,g)=Xkak bk+f(0) g(0),(3)wheretheakandbkarethecoefcientsintheseriesforFfp randgp rasabove.Asweareinterestedinestablishingdualityrelationships,wewouldliketobeabletoidentifythecontinuouslinearfunctionalsactingononeofthegivenspacesofholomorphicfunctionswewillbefocusingon.TheMartineau-AizenbergTheorem(speciesthatthespacesofholomorphicfunctionsofcertaindomainisthedualspaceoftheholomorphicfunctionsovertheclosureofthedualdomain)isthemaintoolweusetodothis.Toallowustoidentifythedualspacesweareinterestedin, 31

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consistingofelementspairedthrough~Qr(f,g),weneedthefollowingversionoftheMartineau-AizenbergTheorem,alsoanextendedversionof[ 12 ,Theorem2.5]. Theorem3.1. Alinearmap':O()!Ciscontinuousifandonlyifthereexistsg2O( G)suchthat '(f)=~Q(f,g)8f2O()(3) Proof. FirstnotethatfromtheproofofLemma 3 ,~Qr(f,g)iswelldenedforr=1aslongasg2O( G),andweshowthat,forsuchg,~Q(,g)inducesacontinuouslinearfunctional,tothatenditsufcestoprovecontinuityatzero.Weconsiderasequenceoffunctionsfn2O()tendingtozerouniformlyoncompactsubsets.Alsonotethatsinceg2O( G)bytheproofofLemma 3 thereexistssomeGmsuchthatGmGandg2O(Gm)thenbylettingm=Gm,weagainhavemsothatclearlyforeachn,fn2O(m).InparticularwecanalsoapplytheproofofLemma 3 tofNforsomesufcientlylargeNtoobtainfn2A2(m)forallnNandg2FA2(m),sothatj~Q(fn,g)jjfn(0) g(0)j+hfn,F)]TJ /F6 7.97 Tf 6.59 0 Td[(1giA2(m)jfn(0) g(0)j+jjfnjjA2(m)jjF)]TJ /F6 7.97 Tf 6.58 0 Td[(1gjjA2(m).Sincefnconvergestozerouniformlyon m,fnconvergestozerointheA2(m)norm,thus~Q(fn,g)!0.Fortheconversewenotethatwecanadapttheprooffoundin[ 12 ],sowelet'beacontinuouslinearfunctionalonO()andC()bethespaceofcontinuousfunctionsonequippedwiththetopologyoflocaluniformconvergence.ThedualofC()isthespaceofniteBorelmeasureswithcompactsupportin[ 7 ,Proposition4.1]andthespaceO()isclosedinC(),sobyHahnBanachwecanextend'toacontinuousfunctionalonC().Thus,thereexistsameasurewithcompactsupportMsuchthat '(f)=ZMfd8f2O().(3) 32

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Nextwedenethefunctiong(z)=1 2ZM1+hz,wi 1)-221(hz,wid(w)=ZM1 1)]TJ ET q .478 w 291.05 -47.22 m 322.04 -47.22 l S Q BT /F2 11.955 Tf 291.05 -57.86 Td[(hw,zi)]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2d(w).Wehavethatg(z)isholomorphicifzissuchthathz,wi6=1,forallw2M,thatis,itisholomorphicforeveryz2MandsinceMandG=M[ 14 ,Lemma4.1]thefunctiong(z)isalsoholomorphicinG.Toverifythatthefunctiongdenedthiswayandanygivenfunctionf2O()satisfytheconditioninEquation 3 ,rst,weexpressthesefunctionsaselementsofHF(m)andA2(m),respectively,intermsofthecorrespondingorthonormalbasesfp kfkgand1 p kfk.Thatis,considerf=Xkak1 p kfkandg(z)=Xkbkp kfk,forsomeconstantsakandbkTocomputethepairingweneedtorewritetheexpansionforgmoreexplicitlybyndingthecoefcientsbk.Firstnotethatg(z)=Xk(bkk)1 p kfk,andsincen1 p kfkoisanorthonormalbasisforA2(m),wecancomputethecoefcientsbkkasfollows,foramoreconvenientnotationwecompute bkkinstead. bkk= g(z),1 p kfkA2(m)=ZmZM1 1)-222(hw,zi)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2d(w)1 p kfk(z)dV(z)=1 p kZmZMfk(z) 1)-222(hw,zi)]TJ /F3 11.955 Tf 13.15 8.08 Td[(fk(z) 2d(w)dV(z)ThenweapplyFubini'sTheoremtoobtain: bkk=1 p kZMZmfk(z) 1)-222(hw,zidV(z))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2Zmfk(z)dV(z)d(w)=1 p kZMkfk(w))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2kfk(0)d(w)(byProposition 2.2 )=1 p k'kfk(w))]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2kfk(0), 33

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thenwecansolveforbk, bk=1 k1 p k'kfk(w))]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2kfk(0)=1 p k'fk(w))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2fk(0)(recalleachkisreal)='1 p kfk(w))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 21 p kfk(0)Thusg(z)=Xk '1 p kfk(w))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 21 p kfk(0)p kfk.NowwemayverifythatgdenedthiswaysatisestheconditioninEquation 3 bycomputing~Q(f,g)usingitsdenitiongivenbyEquation 3 .~Q(f,g)=Xkak'1 p kfk(w))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 21 p kfk(0)+f(0) g(0)=' Xkak1 p kfk(w)!)]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2' Xkak p kfk(0)!+f(0) g(0)='(f))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2'(f(0))+f(0) g(0)='(f))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2f(0)'(1)+f(0) g(0).Wecompletetheproofbynotingthat g(0)=1 2'(1),whichfollowsformtheoriginaldenitionofg. Wenowusethispairingtoestablishdualitybetweenclassesofholomorphicfunctions.Beforewegivethecorrespondingdenitionforduality,weneedtointroduceadenitionofaparticularsubclassofanygivenclassofholomorphicfunctions. Denition26. IfCisaclassoffunctions,wedenotebyC0thesubclassofallfunctionsf2Csuchthat=f(0)=0(Theimaginarypartoff(0)iszero).Thatmeansthat,forinstance,functionsf2M+0M+(seeDenition 8 ),oftheformf(z)=R@DH(z,)d()+it,willinfactbesuchthatt=0. 34

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Denition27. ForaclassCO()dene,thedualofC,C=ng2O()j<~Qr(f,g)0forallf2C0andallr2[0,1)oUsingthesenewdenitionsweextendsomeoftheresultsfrom[ 12 ],butrstweintroduceapositiveclassofholomorphicfunctionsover,analogoustothepositiveclassM+. Denition28. Let@distdenotethedistinguishedboundaryofthesetCd,whichconsistsofthesmallestsubsetof,forwhichthemaximumprincipleholds. Denition29. WedenetheclassM+ofholomorphicfunctionsonadomainasfollows,M+()=f2O()jf(z)=Z@dist1+hz,wi 1)-222(hz,wid(w),forsome0 Theorem3.2. [ 12 ,Theorem2.8]IfapositiveclassPcontainsM+,thenPisapositiveclassandP=P.Note:In[ 12 ]theclassesPandM+consistofholomorphicfunctionsontheballinonedimension.Thistheoremholdsandtheargumentin[ 12 ]carrieson,inexactlythesameway,ifwesubstituteM+withM+()andassumePconsistsofholomorphicfunctionson.Likeinthecaseoftheball,oncewemoveontoworkindimensionsgreaterthanone,wehavethattheclassM+()isproperlycontainedinO+()andwearestillabletondsomedualityrelationsthatwecancomparetothosein[ 12 ,Theorem2.7]. Theorem3.3. M+()=O+()andO+()=M+() Proof. (Toprovetherststatement)Letg2M+0()andf2O().Then,bytheformofg,computationssimilartothoseintheproofofTheorem 3.1 yield,~Qr(f,g)=R@distfp rd,thusRe~Qr(f,g)0forallandallr>1,ifandonlyiff2O+().ToobtaintheoppositedualityweapplyTheorem 3.2 Corollary1. M+()=O+()andO+()=M+() 35

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Proof. WesimplyapplytheproofofTheorem 3.3 ,sinceweknow()=[ 14 ,Lemma4.1] 3.2Operator-ValuedHerglotzKernelsInthissectionwedeneatypeofpositiveSchurclassforholomorphicfunctionsdenedonthedomainandgiveacharacterizationofthisclassviaanoperator-valuedHerglotzKernel.Throughthisrepresentationandafactorizationtheoremfoundin[ 12 ]weidentifythedualofthisSchur-typeclass.LikewecanseeintherstpartofExample1inChapter 4 ,inthecasewhenwearedealingwithholomorphicfunctionsonthePolydisk,thatiswhen=D,therealizationofthegeneralizedSchurclassfollowsfromafactorizationtheoremduetoJ.Agler[ 1 ].Followingthattrendweuse[ 13 ,Theorem6.1],duetoM.T.Jury,whichextendsAgler'sfactorizationtoamoregeneralsetting.Asin[ 13 ]wesituateourproblemunderthefollowingconsiderations.WeletVbeathevectorspaceV=(Cd,jjjj),forsomenormjjjj.NextweletourdomainbetheunitballofV,=Ball(V)(Note=Ball(V)).WealsoconsideranoperatorspacestructureEoverVandthecorrespondingdualoperatorspaceE.FinallyweletT:E!B(H),bealinearmap,whereTisacommutingd)]TJ /F1 11.955 Tf 9.29 0 Td[(tupleofoperatorsandHissomeHilbertspace,denedasT(z)=dXj=1zjTj.Infacteverylinearmap:E)166(!B(H)hasthisform.NowwecandeneourSchur-typepositiveclassforholomorphicfunctionson.WedosoinanalogytothecharacterizationofthepositiveSchurclassinonedimensionthroughvonNeuman'sinequality.ItisworthnoticingthattheclassobtainedwilldependontheoperatorspacestructurethatwechoosetoputonV,hencethenotationbelow. Denition30. LetTEdenotethesetofallcommutatived)]TJ /F14 11.955 Tf 9.3 0 Td[(tupleofoperatorsT=(T1,...,Td)inB(H),suchthatTiscompletelycontractiveforE,thenwedenethe 36

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followingclassofholomorphicfunctions. S+E()=ff2O()j<(fr(T))0,forallT2TEandall0r<1g(3) 3.2.1RealizationofS+E()Withthisoperatorspacesettingweseethatpart3)of[ 13 ,Theorem6.1]holdsforf2S+E()sothatthereexistssomeHilbertspaceK,ananalyticfunctionF:!B(K,C)andacompletelycontractivelinearmap:E!B(K),suchthatwehavethefollowingfactorization f(z)+f(w)=F(z)(IK)]TJ /F13 11.955 Tf 11.95 0 Td[((z)(w))F(w).(3)Nowconsiderthefollowingequations: f(z)+ f(0)=F(z)F(0) (3) f(0)+ f(w)=F(0)F(w) (3) f(0)+ f(0)=F(0)F(0); (3) addEquations 3 3 andsubstractEquation 3 ,toobtain f(z)+f(w)=F(z)F(0)+F(0)F(w))]TJ /F3 11.955 Tf 11.95 0 Td[(F(0)F(0).(3)Plugginginz=wandputtingEquations 3 and 3 together,weobtain:F(z)F(z))]TJ /F3 11.955 Tf 11.96 0 Td[(F(z)(z)(z)F(z)=F(z)F(0)+F(0)F(z))]TJ /F3 11.955 Tf 11.96 0 Td[(F(0)F(0),F(z)(z)(z)F(z)=(F(z))]TJ /F3 11.955 Tf 11.95 0 Td[(F(0))(F(z))]TJ /F3 11.955 Tf 11.96 0 Td[(F(0)),jj(z)F(z)jj2=jj(F(z))]TJ /F3 11.955 Tf 11.95 0 Td[(F(0))jj2ThisindicatesthatthemapfromthespaceM=spanf(z)F(z):z2gKtothespaceN=spanf(F(z))]TJ /F3 11.955 Tf 12 0 Td[(F(0)):z2g,thatsendseach(z)F(z)7!(F(z))]TJ /F3 11.955 Tf 11.99 0 Td[(F(0))isisometricandhencecanbeextendedtoaunitary,sayU,inKifthecodimensionsofMandNmatchup,orotherwisetosomeotherHilbertspace~K(obtainedbyenlarging 37

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K).ThisunitarywillsatisfyU(z)F(z)=F(z))]TJ /F3 11.955 Tf 11.74 0 Td[(F(0).SolvingforF(z),wecanthenwriteF(z)=F(0)(1)]TJ /F13 11.955 Tf 11.96 0 Td[((z)U))]TJ /F6 7.97 Tf 6.59 0 Td[(1,wheretheinverseof(1)]TJ /F13 11.955 Tf 12.14 0 Td[((z)U)exists,sinceUisunitaryandbyhypothesisk(z)k<1,forz2.PlugginginthisexpressionforF(z)intoEquation 3 withw=z,wegetf(z)+f(z)=F(0)(1)]TJ /F13 11.955 Tf 11.96 0 Td[((z)U))]TJ /F6 7.97 Tf 6.59 0 Td[(1F(0)+F(0)(F(0)(1)]TJ /F13 11.955 Tf 11.96 0 Td[((z)U))]TJ /F6 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(F(0)F(0)=F(0)(1)]TJ /F13 11.955 Tf 11.96 0 Td[((z)U))]TJ /F6 7.97 Tf 6.59 0 Td[(1F(0)+F(0)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(U(z)))]TJ /F6 7.97 Tf 6.59 0 Td[(1)F(0))]TJ /F3 11.955 Tf 11.95 0 Td[(F(0)F(0)=F(0)[(1)]TJ /F13 11.955 Tf 11.96 0 Td[((z)U))]TJ /F6 7.97 Tf 6.59 0 Td[(1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(U(z)))]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(1]F(0)Sinceislinear,itisoftheform~Sforsome~S2B(K),suchthatdXj=1~SjTj1,foralld)]TJ /F1 11.955 Tf 9.3 0 Td[(tuplesTofoperators,suchthatTiscompletelycontractiveforE.[ 13 ,Proposition2.1]Noticethat~S(z)U= dXi=1zi~Si!U=dXi=1(zi~SiU)=hz,U~Si,thenwecanwriteH(z,U~S):=2(1)]TJ /F13 11.955 Tf 12.39 0 Td[((z)U))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 12.39 0 Td[(1,andsince<1+(z)U 1)]TJ /F11 7.97 Tf 6.59 0 Td[((z)U0,wehavethat<(H(z,U~S))=(1)]TJ /F13 11.955 Tf 11.95 0 Td[((z)U))]TJ /F6 7.97 Tf 6.58 0 Td[(1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(U(z)))]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(10.Hencewehaveshownthat f2S+E()ifandonlyifithastheformf(z)=DH(z,U~S),E+it,t2R.(3)SinceU~SisalsosuchthatjjP(U~S)jTjjj1wecanreformulatethisresultasfollows. Theorem3.4. Afunctionf2S+E()ifandonlyifitisoftheform f(z)=hH(z,S),i+it,t2R(3) 38

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whereS2B(~K),suchthatdXj=1SjTj1,foralld)]TJ /F14 11.955 Tf 9.3 0 Td[(tupleTofoperators,suchthatTiscompletelycontractiveforEorequivalently,SissuchthatSiscompletelycontractiveforE. 3.2.2DualoftheClassS+E()Below,wedeneaclassofholomorphicfunctionsoverthedualdomainwhich,underDenition 27 ofadualclass,weproveisthedualofS+E().TherealizationinEquation 3 obtainedinthelastsectioniscrucialintheproof. Denition31. LetTEdenotethesetofallcommutatived)]TJ /F14 11.955 Tf 9.3 0 Td[(tupleofoperatorsT=(T1,...,Td)inB(H),suchthatTiscompletelycontractiveforE,thenwedenethefollowingclassofholomorphicfunctions. R+E()=ff2O()jf(z)=hH(z,T),i+it,witht2RforallT2TEg(3)where,werecallthatH(z,T)=2(1)-222(hz,Ti))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(1=2(1)]TJ /F7 7.97 Tf 18.02 14.95 Td[(dXi=1ziTi))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(1istheHerglotzKernelevaluatedintheoperatorT.FirstweprovethatthisclassR+E()isthedualofS+E(),inthecasewhenthedomainisinfactaReinhardtDomain. Theorem3.5. R+E()=S+E()andR+E()=S+E() Proof. AssumeisaReinhardtdomainandletg2(R+E)0().Bydenitionofthisclasswehave:1 2[g(z)+ g(0)]=h(1)-221(hz,Ti))]TJ /F6 7.97 Tf 6.58 0 Td[(1,i=1Xj=1hhz,Tij,i=XzhT,ijj! 39

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sog(z)=2XzhT,ijj! !)]TJ ET q .478 w 185.86 -6.41 m 208.67 -6.41 l S Q BT /F3 11.955 Tf 185.86 -17.05 Td[(g(0).Butsince=g(0)=0,wehaveg(z)=2XzhT,ijj! !h,i,orwritteninamoreconvenientwayg(z)=2X6=0zhT,ijj! !+h,i.Nowletf2O(),andsaythatf(z)=Pcz,thenwecompute~Qr(f,g).SinceforthisrstproofweareassumingisaReinhardtdomain,theDenitions 25 and 24 coincidebythecalculationsin 3 ,then~Qr(f,g)=2X6=0chT,ijj! !! jj!r+c0h,i+f(0) g(0)=2X6=0chT.ir+2c0h,i=2*XcrT,+=2*Xfr(T),+,andnoticethat<~Qr(f,g)0ifandonlyif
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kernelofH2datzwehave(RR1)(z)=hRR1,1 1hw,ziiH2d=hR1,R1 1hw,ziiA2()=h1,1 1hw,ziiA2()=R1 1hz,widV(w)ThusweneedtoshowthattheintegraltheintegralR1 1hz,widV(w)isconstant.ToseethiswemayexpandtheintegrandasusualthroughageometricseriessoR1 1hz,widV(w)=RPjj! !zwdV(w)towhich,bythesameargumentintheproofofLemma 2 ,wecanapplyDominatedConvergenceTheoremtointerchangetheintegralandtheseries.Afterdoingthisweseethatweonlyneedtoshowthatforeach6=0,RwdV(w)=0.Toseethis,anargumentsimilartotheproofofProposition 2.1 ,yieldsthefollowing:RwdV(w)=RwdV(w)for2TdRTdRwdV(w)d=RTdRwdV(w)dRwdV(w)=RwdV(w)RTddWhereweput=(ei,ei,...,ei)withnotcoterminaltozero,sothatsinceatleastoneofj6=0thenoneoftheiteratedintegralsR20eiidofRTddiszero,thenRwdV(w)=0.Whichnishestheproofthatf01isaneigenfunction.Theconstanttermwhen=0remainsintheseriesexpansionofR1 1hz,widV(w)=RPjj! !zwdV(w),sothatR1 1hz,widV(w)=Rz0w0dV(w)=RdV(w)=Vol(),thustheeigenvaluecorrespondingtothefunctionf01iswhatweclaimed.NowrecalltheidentityfromProposition 2.2 kfk(z)=Zfk(w) 1)-222(hz,widV(w)(3)Toshowfk(0)=0forallk1,supposethatk6=0thenfromEquation 3 ,theorthogonalityofthefunctionsfkandthefactthatf01isoneoftheeigenfunctionswe 41

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havethatfk(0)=1 kZfk(w)dV(w)=1 khfk,1iA2()=1 khfk,f0iA2()=0 Lemma5. Letz2andw2,then,wehavethefollowingexpansionintermsoftheeigenfunctionsfkofRR,(1)-222(hz,wi))]TJ /F6 7.97 Tf 6.59 0 Td[(1=Xkfk(z) fk(w) Proof. Wecanwrite(1)-221(hz,wi))]TJ /F6 7.97 Tf 6.59 0 Td[(1=XkCkfk(z) p k2A2(),andcomputeCkas:Ck=(1)-222(hz,wi))]TJ /F6 7.97 Tf 6.59 0 Td[(1,fk p kA2()=1 p kZ fk(z) 1)-222(hz,widV(z)=1 p k Zfk(z) 1)-221(hw,zidV(z)=1 p k kfk(w)=p k fk(w)Thus,(1)-222(hz,wi))]TJ /F6 7.97 Tf 6.58 0 Td[(1=XkCkfk(z) p k=Xkp k fk(w)fk(z) p k=Xk fk(w)fk(z) Giventhisexpansionandthefunctionalcalculusforoperators,thefollowingcorollaryisimmediate. Corollary2. Foranyd)]TJ /F14 11.955 Tf 9.3 0 Td[(tupleT=(T1,...,Td)ofcommutingoperatorsinTE(thatis,suchthatTiscompletelycontractiveforE),wehavetheidentity (1)-222(hz,rTi))]TJ /F6 7.97 Tf 6.58 0 Td[(1=Xkfk(z)fk(rT)forallz2andall0r<1,(3)wherethefunctionsfkareasinLemma 5 ProofofTheorem 3.5 inthecasewhenisanonReinhardtdomain. Letg2(R+E)0(),thenweknowthereexistssomemandsomer<1,suchthatgp r2 42

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HF(m)A2(m),andithasanexpansiong(p rz)=Pkbkp kfk(p rz)=Pkbkkfk(p rz) p kontheotherhand,bydenition,weknowthatg(z)=hH(z,T),i(withnullimaginarypartatzerobecauseg2(R+E)0()),whichwecanrewriteasfollowsusingCorollary 2 gp r(z)=hH(p rz,T),i=h2(1)-222(hp rz,Ti))]TJ /F6 7.97 Tf 6.59 0 Td[(1,i)-222(h,i=h2(1)-222(hz,p rTi))]TJ /F6 7.97 Tf 6.59 0 Td[(1,i)-222(h,i=h2Pkfk(z)fk(p rT),i)-223(h,iUsingthisexpressionforgp r(z),wecomputethecoefcientbkkbkk=Dgp r,fk p kEA2()=D[h2Pjfj(z)fj(p rT),i)-222(h,i],fk(z) p kEA2()=Dh2Pjfj(z)fj(p rT),i,fk(z) p kEA2())]TJ /F10 11.955 Tf 11.96 13.27 Td[(Dh,i,fk(z) p kEA2()=2ZhPjfj(z)fj(p rT),i fk(z) p kdV(z))]TJ /F10 11.955 Tf 11.96 9.63 Td[(Rh,i fk(z) p kdV(z)=2*Xjfj(p rT)Zfj(z) fk(z) p kdV(z),+)]TJ /F10 11.955 Tf 11.95 20.44 Td[(*Z fk(z) p kdV(z),+=2*fk(p rT)p kZfk(z) p k fk(z) p kdV(z),+)]TJ /F10 11.955 Tf 11.96 20.44 Td[(*Z fk(z) p kdV(z),+=2Dp kfk(p rT),E)]TJ /F10 11.955 Tf 11.95 13.27 Td[(Dp k fk(0),ESincefk(0)=08k6=0andf01,byLemma 4 ,wehavethatb0=1 p 0,andfork6=0,bk=1 p 0fk(p rT),.Nowwecompute~Qr(f,g),foranyg2(R+E)0()andf2O().Recallfp r(z)=Xkakfk(p rz) p k2A2(m)andnotethatfp r(0)=a0 p 0and 43

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gp r(0)=b0p k,then~Qr(f,g)=Xkak bk+f(0) g(0)=2*Xk6=0ak p kfk(p rT),++2a0 b0=2*Xk6=0ak p kfk(p rT),++2a01 p 0,=2*Xkak p kfk(p rT),+=2fp r(T),Thus<~Qr(f,g)0ifandonlyif<(fp r(T))08r<1ifandonlyiff2S+E() 3.2.3VonNeuman-typeInequalitiesInthissectionwepresentaresultanalogousto[ 12 ,Theorem3.4],whichcharacterizesthedualofthepositiveSchurclassontheball,throughavonNeuman-typeinequalitythatallowsallrowcontractions.Weextendtheresulttoourarbitraryunitballand,likeinthecaseoftheeuclideanballB,non-commutativeoperatorsplaythemainrolesowewillappealtothesymmetrizedfunctionalcalculusdiscussedattheendofSection 2.2 Theorem3.6. f2R+E()ifandonlyif
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g(z)=hH(z,S),i=h[2(1)-222(hz,Si))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(1],i=2*dXn=0(hz,Si)n,+)-222(h,i=2dXn=0* dXj=1zjSj!n,+)-222(h,i=2X6=0zh(z)sym(S),ijj! !+h,iNowletf2O+(),withf(z)=Pcz,andcompute~Qr(g,f):~Qr(g,f)=2X6=0rh(z)sym(S),i cjj! !! jj!+ c0h,i+g(0) f(0)=2X6=0rh,(z)sym(S)i c+2 c0h,i=2*,Xrc(z)sym(S)+=2h,fsymr(S)iThus,bydenition,wehavethatforarbitraryg(z)2(S+E)0(),Re~Qr(g,f)08r2[0,1)ifandonlyiff2S+E()=R+E()(Theorem 3.5 ),thusf2R+E()ifandonlyif
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Proof. Firstwedenewhatwemeanbysymmetriccalculusinthiscase(wherenopowerseriesrepresentationisavailable).Letfsymk(rS)=1 kZfk(w)(1)-221(hrS,wi))]TJ /F6 7.97 Tf 6.59 0 Td[(1dV(w)Sinceffk p kgisanorthonormalbasisofA2(),assumingthatk(z,w)isthereproducingkernelforA2()weget:Pfk(z)fk(rS)=ZXkfk(z) p k fk(w) p k(1)-222(hw,rSi))]TJ /F6 7.97 Tf 6.59 0 Td[(1dV(w)=R k(z,w)(1)-222(hw,rSi))]TJ /F6 7.97 Tf 6.59 0 Td[(1dV(w)=(1)-221(hz,rSi))]TJ /F6 7.97 Tf 6.58 0 Td[(1 ProofofTheorem 3.6 Non-ReinhardtCase. Again,letg2(S+E)0(),whichhastheformg(z)=hH(z,S),iandletf2O(),thenfp r(z)=Pkakp kfk(p rz)andgp r(z)=Pkbkfk(p rz) p k(sincefp r2HF(m)andgp r2A2(m))tocomputethepairing~Qweneedanexplicitrealizationofthecoefcientsbk,tondthemwerewritegp r(z)usingCorollary 3 g(p rz)=h2(1)-222(hp rz,Si))]TJ /F6 7.97 Tf 6.59 0 Td[(1,i)-222(h,i=h2(1)-222(hz,p rSi))]TJ /F6 7.97 Tf 6.59 0 Td[(1,i)-222(h,i=2Pk6=0fk(z)hfsymk(p rS),i+h,iThenfork6=0bk=2p khfsymk(p rS),iandb0=p 0h,i.Nowwemaycompute~Qr(g,f)~Qr(g,f)=Pkbk ak+g(0) f(0)=2Pk6=0 akp khfsymk(p rS),i+ a0p 0h,i+g(0) f(0)=2Pk6=0h,akp kfsymk(p rS)i+2 a0p 0h,i=2h,Pkakp kfsymk(p rS)i=2h,fsymp r(S)i 46

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Thuswehavethat
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CHAPTER4EXAMPLESInalltheexamplesthatweaddressinthischapterwewilldroptheE-subscriptinthenotationS+E()andR+E(),sincetheoperatorspacestructureinplaywillbeclearineachseparateexample. 4.1ThePolydiskOurrstexampleisthecaseofthepolydisk,thatis,weconsiderthemutuallydualdomains=D=fz2Cdjjzij<1,8i=1,...,d.gand=:=fw2CdjPdi=1jwij<1g.Inthiscase,wherecorrespondstotheunitballofV=(Cd,kk1)=l1(Cd),weputtheMINoperatorspacestructureonV,E=MIN(l1(Cd)),thenbyTheorem 2.8 ,weknowthatE=MAX(V)=MAX(l1(Cd)).UndertheseassumptionswehavethattheoperatorsTthatmakeTcompletelycontractiveforEareallthecontractions.Indeed,wehavethat,sinceEisaMAXoperatorspace,amapiscompletelycontractiveifandonlyifitiscontractive[ 16 ],thatis,wecanseethatad)]TJ /F1 11.955 Tf 9.3 0 Td[(tupleofoperatorssatisesjjT(z)jjjjzjj1,orequivalently, dXi=1ziTidXi=1jzijforallz2Cd,(4)whenevereachTiisacontraction.ConverselyifInequality 4 holds,itwillholdinparticularforanystandardbasisvectorei=(0....,1,...,0),thus,wegetthateachjjTijj1.KnowingnowwhichoperatorsmakeTcompletelycontractiveforE,wecangiveaconcretedenitionfortheclassS+()=S+(D)usingDenition 30 asfollows. Denition32. LetT=(T1,...,Td)beacommutingcontraction,thatis,jjTijj1foralli=1...,d,thendene S+(D)=ff2O(D)jRef(rT)0,withTasabove,andall0r<1g(4) 48

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Asweproceededinthegeneralcase,therealizationofS+(D)willprovidetheinformationtonditsdualandthecorrespondingvonNeuman-typeinequality.FollowingthestepsintherealizationforS+()fromSection 3.2.1 we,infact,observethatrstofallwewouldstartwithAgler'sfactorizationTheorem[ 1 ,Theorem2.6]toobtainthatf2S+(D)ifandonlyifithastheformf(z)=hH(z,~U),i,forsomeunitaryoperator~U,whichmatcheswiththeexpectedrepresentationgivenin[ 1 ,Theorem1.8],alsoduetoAgler.Indeed,translating[ 1 ,Theorem1.8]tooursettingforcomplexvaluedholomorphicfunctionswehavef2S+(D)ifandonlyifthereexistsomeHilbertspaceH,apartitionDofH(intodsummands),videinfra,anisometry:C)166(!H,andaunitaryoperatorU2Hsuchthat f(z)=(1+UD(z))(1)]TJ /F3 11.955 Tf 11.95 0 Td[(UD(z)))]TJ /F6 7.97 Tf 6.59 0 Td[(1,forallz2D(4) Denition33. [ 1 ]IfHisaHilbertspace,wesaythatD=(P1,...Pd)isapartitionofHifDisad-tupleoforthogonalprojectionsonHthatarepairwiseorthogonalandsumtoone(equivalently,H=Ldr=1ranPr). Denition34. [ 1 ]IfDisapartitionofM,thendenetheB(H)-valuedholomorphicmaponD D(z)=dXr=1zrPr(4)Toseethattherepresentationobtainedintermsoftheoperator-valuedHerglotzkernelH(z,~U)aboveispreciselythatofAgler's,wejustneedtoleti=PiUsothatclearly=UPiandconsiderthed)]TJ /F1 11.955 Tf 9.29 0 Td[(tuple~U=(1,...,d),where~U=(1,...,d).ThenweseethatUD(z)=Pdi=1ziUPi=Pdi=1zii=hz,~Ui.ThuswiththisrealizationforS+(D),wecangiveaconcretedescriptionofthedualoftheclassS+(D)usingTheorem 3.5 ,thatis,weidentifythefamilyofoperatorsthatcharacterizesitandwestatethevonNeumann-typeinequalitycorrespondingtothedual. 49

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Denition35. Let,T=(T1,...,Td),beacommutingcontraction,thendene R+()=ff2O()jf(z)=hH(z,T),i+it,withTasaboveandsomet2Rg(4)Where,asusual,H(z,T)=2(1)-221(hz,Ti))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(1=2(1)]TJ /F10 11.955 Tf 11.95 8.97 Td[(Pdi=1ziTi))]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(1Toshowthatthisclassispreciselythedualwearelookingfor,thatistoshowthatS+(D)=R+(),weappealtoTheorem 3.5 .FinallywiththerepresentationobtainedforS+(D)intermsoftheoperator-valuedHerglotzkernelH(z,~U),wegetthevonNeumann-typeinequalityasgivenbyTheorem 3.6 ,asfollows.Recall~U=(P1U,...,PdU),andUistheunitaryoperatorgivenbyAgler'srepresentationTheorem,thenwehavethefollowingcorollary. Corollary4. Afunctionf2R+()ifandonlyif
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saythatford=2,wehavetheoperatorsT1=2641000375andT2=2640001375.ThenjjP2i=1ziTijj=z100z2=jjzjj1,howeverP2i=1jjTijj=21.InthiscasewedonotknowmoreconcretedenitionsofS+()andR+(D)aswedoforthePolydisk.ForthisexamplewehaveS+()=ff2O()j<(f(T))0,forT2TgR+(D)=ff2O(D)jf(z)=hH(z,T),i+it,withT2Tandt2RgaswellasthefollowingvonNeuman-typeinequality: Corollary5. Afunctionf2R+(D)ifandonlyif
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Byhypothesiswehavethattheoperator:E)166(!B(H),denedonthebasisvectorszias(zi)=Tiandonanyelementby(Pdi=1aizi)=Pdi=1aiTi(=T),iscompletelycontractive.FirstweconsidertheunitizationofE,thatis,theoperatorspaceE0obtainedfromadjoiningaunittoE.In[ 17 ,Proposition8.19]weseethatE0canbeidentiedcompletelyisometricallywiththedirectsumC1E(wheretheunitis(1,0)).Thedirectsum1referstoadirectsuminthel1sense,thatis,C1Eisequippedwiththenormkk1,denedforanelement(,v)2C1Eas:k(,v)k1=jj+kvkE.Underthisidenticationitiseasytoshowthenextclaim. Claim1. Themap0:E0)166(!B(H),thatextendsdenedas0((,v))=I+(v),isunitalandcompletelycontractive. Proof. Bydenition0((1,0))=1I+(0)=I,thus0isunital.Nowlet(,v)2C1E=E0,then,sinceiscompletelycontractivewehave:k0((,v))k=kI+(v)kkIk+k(v)kjj+kvkE=k(,v)k1 Thenextstepistoconstructanoperatorsystemasfollows,S=E0+E0,since0:E0)166(!B(H)isaunitalcompletecontraction,by[ 16 ,Proposition3.5],wegetthatthemap~:S)166(!B(H),denedby~(v+v)=0(v)+0(v),iscompletelypositiveandcompletelycontractive.NowweareinthepositiontoapplyArveson'sExtensionTheorem(Theorem 2.9 onpage 28 )to~.Theconclusionisthatthereexistsacompletelypositiveuniqueextension of~tothewholeC-algebraC(Td).Thatis,wehaveamap :C(Td))166(!B(H),suchthat jS=~and iscompletelypositive.Themap justobtainedsatisestheconditionsofStinespring'sDilationTheorem(Theorem 2.10 onpage 28 ),whichguarantiestheexistenceofaHilbertspaceKH,aunital*-homomorphism:C(Tn))166(!B(K)andaboundedoperator:H)166(!Kwith 52

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k (1)k=kk2,suchthat (a)=(a).Notethatsince isunital,inthiscase,wehavethatisanisometry.AlsonotethatforanyoftheunitaryelementsziinE,wehave(zizi)=(1)=1andontheotherhand,sinceisa*-homomorphism,1=(zizi)=(zi)(zi),thus,thereexistunitaryoperatorsinB(K),U1,...,UdsuchthatUi=(zi).Finallywehave,Tj=(zi)=0(zi)=~(zi)= (zi)=(zi)=Ui 4.2.2E=MAX(l1d)AgainwithV=l1dbutnowassigningtheMAXoperatorspacestructuretoVwewillseethatndingtherequiredfamiliesofoperatorstodescribeS+()andR+(D)willfollowfromapreviousexample.ToestablishthevonNeuman-typeinequalityfortheclassR+(D),inthisexample,weneedtoidentifythefamilyS=fS:SiscompletelycontractiveforE=MAX(l1d)g,butthisisnothingbutallthecontractions,whichwecanseefromourrstexampleinSection 4.1 .NoticethatthedifferenceisthatinExample1werequiredtheoperatorstocommute,butthefortheclassSwedonotrequirecommutativity.Similarly,theclassT=fTcommutative:TiscompletelycontractiveforE=MIN(l1d)g, 53

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correspondstothoseunitaryoperatorsgivenbyAgler'sdecompositionthatwefoundtogivethevonNeuman-typeinequalityforR+()inSection 4.1 ,exceptnowweonlylookatthesubsetofthosethatcommute.ThuswehavethefollowingcharacterizationoftheclassesofinterestaswellasthecorrespondingvonNeuman-typeinequalities. Denition36. LetT=(T1,...,Td)beacommutatived-tupleofoperatorsinB(H)suchthatTi=PiU,foralli=1,...,d,whereU2B(H)isunitaryand(P1,...,Pd)formapartition(seeDenition 33 )ofH.ThenS+()=ff2O()j<(f(T))0,withTasabovegandR+(D)=ff2O(D)jf=hH(z,T),i+it,forsomet2RandTasabovegThusapplyingourresultswegetarepresentationforS+()andavonNeuman-typeinequalityforR+(D)asfollows:S+()=ff2O()jf=hH(z,S),i+it,forsomet2RandS2B(H)anycontractiongFinally,wehavethatf2R+(D)ifandonlyif<(fsym(S))0forallstrictcontractionsS2B(H). 4.3Anon-ReinhardtDomainExampleInthissectionwewillworkoutanexamplewithanormedspacewhichhasanon-Reinhardunitball.TobeginwestartbyintroducingthenormthatwewillputonCdwhichmakesnon-Reinhard.ThenormwewilluseisknownastheLienorm. Denition37. [ 15 ]TheLienormonCdisdenedbyL(z)=q kzk2+p kzk4)-222(jz2j2,wherekzk2=jz1j2+jz2j2++jzdj2,andz2=z21+z22++z2d. 54

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Denition38. [ 15 ]ThedualoftheLienormisdenedbyL(z)=r kzk2+jz2j 2 Proposition4.2. LetbetheunitballoftheLienormL(z)andtheunitballofthedualLienormL(z),thenneithernorareReinhardtDomains. Proof. NotethatkzkL(z),andinfactifz2Rd,wecanseethatkzk2=jz2j2,sowestartwithapointz2RdwhichwillmakeL(z)=kzk.Weshowtheresultforthecasewhend=2,andhencetrueforanyd,byconsideringtheprojectionontoC2.Takez=p 6 4,p 6 4,thenL(z)=kzk=p 3 20.87<1,sothatz2.Now,rotatethecoordinatesofzindependentlytogetanewelement~z=p 6 4,ip 6 4,toobtain~z2=6 16)]TJ /F6 7.97 Tf 15.36 4.7 Td[(6 16=0.ThusL(~z)=p 2k~zk=p 6 21.22>1,and~z=2.Toshowthatisnon-Reinhardtweconsiderz=p 3 2,ip 3 2,thenjz2j=0andL(z)=kzk p 2=p 3 20.87<1,butrotatingeachcoordinateofz,say~z=p 3 2,p 3 2,wegetL(~z)=k~zk=q 3 21.2>1,sothat~z=2. ForthisexampleournormedspaceisV=(Cd,L()),andasusualthecorrespondingunitball.IttakesashortalgebraiccalculationtoseethattheconditionL(z)2<1,isequivalenttoanyofthefollowing,dependingonthenotationweprefer, 1)]TJ /F4 11.955 Tf 11.96 0 Td[(2kzk2+jz2j2>01)]TJ /F4 11.955 Tf 11.96 0 Td[(2Pdi=1zizi+Pdi)]TJ /F6 7.97 Tf 6.59 0 Td[(1z21Pdi=1z2i>01)]TJ /F4 11.955 Tf 11.95 0 Td[(2zz+z2z2>09>>>>=>>>>;(4)InfactthisisthedescriptionofthedomainsknownasCartandomainsoftypeIV.[ 11 ]InordertodescribeS+E()weneedtodeterminerstwhatoperatorspaceEwewillconsider.FromhereonwecanjustverygenerallygivethedenitionasseeninChaper 3 ,butwewouldliketodescribetheoperatorspacestructuretogetherwithamore 55

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concretelydenedfamilyofoperators.TopursuitthisweconsiderafamilyofoperatorswhichwillsatisfycertainconditionsrelatedtotheBergmanKernelofthisball,whichwedenebelow,noticeinthedenitionthecloserelationshiptotheconditionsinEquations 4 Denition39. [ 10 ]TheBergmanKernelfortheunitballoftheLienormisgivenbyB(z,w)=1 (1)]TJ /F4 11.955 Tf 11.96 0 Td[(2zw+z2w2)d+1WeconsiderthefollowingfamilyofoperatorsTB=Tcommutingd-tupleinB(H)j1 B(T,T)0WewishtoobtainoperatorspacesEandEforwhichtheclassTBisexactlytheclassofalloperatorssuchthatTiscompletelycontractiveforE.ThoughwemaydeneanoperatorspacesuchthatallT(T2TB)arecompletelycontractiveforE,thisclassmightnotcontainallthoseoperatorsthatgiverisetocompletecontractionsfortheoperatorspacestructureEweputonV.Moreexplicitly,rstwemayputonVanoperatorspacestructurebyaskingthatallthemapsT(T2TB)arecompletelycontractiveforE,thatis,byletting':V)166(!B(H)bethelinearisometrythatdeterminestheoperatorspacestructureonE,insuchawaythat,k(T(vij))kk('(vij))k,forallmatrices(vij)2Mn(Cd).Equivalently,likewesawinSection 2.4 wecanexpandamatrix(vij)withrespecttotheorthonormalbasis(e1,...,ed)andad-tupleofnnmatrices,sothattheconditionon'isequivalenttokPdi=1AiTikkPdi=1Ai'(ei)k,foralld-tuples(A1,...,Ad)andallT2TB.AndsoanycompletecontractionontheoperatorspaceobtainedE,correspondstosomelinearmapping suchthatkPdi=1Ai (ei)kkPdi=1Ai'(ei)k,andthequestionisnowwhetherthed-tuple( (e1),..., (ed))2TB.SofarfortheoperatorspaceEconstructedabove,wecanonlysaythatTBTE:=fallcommutatived)]TJ /F1 11.955 Tf 12.62 0 Td[(tuplesT:TiscompletelycontractiveforEg.(whichistheclassthatdeterminesS+E())To 56

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decidewhetherthecontainmentjustmentionedisproperornot,thatis,todecidewhethertheoperatorsinTBdetermineanoperatorspacestructure,wecouldcheckwhether1 B(T,T)0determinesanormthatsatisesRuansaxioms(R1)and(R2)(seeDenitions 14 and 15 ).Itisnottoohardtoshowthat(R2)holds.Howeverwehavenotbeenabletoshowwhether(R1)holdsornot. 57

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CHAPTER5FUTURERESEARCH 5.1OperatorSpacesAspointedoutinsomeoftheexamplesinChapter 4 aconcretefulldescriptionoftheoperatorsthatarecompletelycontractiveforagivenoperatorspaceisnotobvious,eveninthewellknownsimplercasesoftheMINandMAXoperatorspacestructures.Itisaninterestingquestiontounderstandwhataresufcientconditionsforacertainclassofoperatorstodeterminecompletelyanoperatorspace.Forinstanceconsiderinnon-ReinhardtExample 4.3 ,wherewedeclaredoperatorspacestructuresEandE,byaskingthatallT(T2T)becompletelycontractiveforE,consideracompletelycontractivelinearmaponE,whichcanbewrittenasP,thequestionisstillopentoseeif1 B(P,P)0,thatis,ifP2T. 5.2KernelsontheLieBallInExample 4.3 wesawthattheBergmankernelisgivenbyB(z,w)=1 (1)]TJ /F4 11.955 Tf 11.96 0 Td[(2zw+z2w2)d+1andnoticethatwithB(z,z)theexpressioninthedenominatorappearsintheconditionforztobelongtotheunitLieball(seeEquations 4 ).UndertheseobservationswehaveastronganalogytothecaseoftheeuclideanballB,wheretheBergmankernelisgivenby1 (1)-221(hz,wi)d+1andwecanseeaswellthedenominatoriscloselyrelatedtotheconditionforztobeinB.InanalogytothekerneloftheDrury-ArvesonspaceH2d,k(z,w)=1 1)-222(hz,wi,wewouldliketoshowthattheexpression1 1)]TJ /F4 11.955 Tf 11.96 0 Td[(2zw+z2w2isapositivedenitekernelaswell. 58

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5.3TheSpaceofFantappieTransformsFInSection 2.1 wementionedanimportantandwellknownresultinvolvingtheCauchytransform,thatthespaceofCauchytransformsK:=fK:isaborelmeasureonTgisisometricallyisomorphictothedualofthediskalgebra.WhilestudyingcloselytheFantappietransform,whichgeneralizestheCauchytransform,thenaturalquestionaroseifthereisananalogousresult,ifthespaceofFantappietransformsisisomorphictoadualspace,andidentifysuchspace.AgoodstartingpointwillbetoconsidertheFockspacesetting.WeintroducethetermsandnotationnecessarytoworkontheFockSpacesetting.Mostofthefollowingcanbefoundin[ 8 ]. Denition40. Considerthesetofdsymbols=f1,...,dganddeneawordonthissetofsymbolsbythestringw=i1im,withij2.Thesetofsuch(non-commutative)wordsisdenotedbyFd+=fw=i1im:m2Nandij2gandtheemptywordw=;isallowed.ThesetofwordsFd+isafreesemigroupwiththeemptywordastheidentityandtheoperationisconcatenation.Wealsousejwjtodenotethelengthofthewordw,thatisthenumberofsymbolsinthestring. Denition41. TheFockSpaceisgivenbyl2(Fd+)withsomeorthonormalbasisfw:w2Fd+g,sothatl2(Fd+)=nXaww:aw2CandXjawj2<1o.ThenormofanelementPawwisgivenbyPjawj2 Denition42. Sincefw:w2Fd+gisanorthonormalbasisforl2(Fd+),foreachi2,wedenetheoperatorLionl2(Fd+)byLiw:=iw.Thisoperatorismentionedin[ 8 ]anditisinfactaspecialcaseoftheleftregularrepresentationofFd+whichisdenedas(v)w:=vw,sothatLi=(i)(visaword) 59

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Notethattheinnerproductoftwoelementsinl2(Fd+)isgivenbyhPaww,Pbvvi=Paw bwandsothenormofPawwisPjawj2.SinceLi(Paww)=Pawiw,wecanseethatLiisisometricandwecancomputeLiasfollows:hLi(Pavv),Pbwwi=hPaviv,Pbwwi=Pw=ivav biv=Pw=ivav bw=hPavv,Pw=ivbwv)i=hPavv,Li(Pbww)iThus,wehavethatLi(w)=8><>:vifw=iv0otherwiseLiLi=Id,ontheotherhandL=(L1,...,Ld)isarowcontraction,thatis,I)]TJ /F10 11.955 Tf -424.22 -14.94 Td[(Pi2LiLi0,indeedwecanseethatI)]TJ /F10 11.955 Tf 12.01 8.96 Td[(Pi2LiLiisinfacttheprojectionP;andthuspositive. Denition43. ConsidertheCalgebrageneratedbyL,(withnotationasin[ 8 ])thisalgebraisknownastheCuntz-ToeplitzalgebraEdandletF=f(z)=((1)-222(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1):z2Band2(Ed).Notethat,sinceLisarowcontraction,andz2B,(1)-226(hz,Li)doeshaveaninverse,infactthistheoperatorvaluedFantappiekernel,andthesetFconsistsofanalyticfunctionsonB. Claim2. Inthecasewhend=1,theFockspaceisjustH2,L1S(theregularshiftonH2),E1=C(T)andtheFantappiekernelisjusttheCauchykernel,sothatindimensionone,F=K. Denition44. Rf=f2(Ed):f(z)=((1)-221(hz,Li))]TJ /F6 7.97 Tf 6.58 0 Td[(1)g Denition45. Lsym= spannPN(w)=Lw:w2Fd+and2Ndowhere,Lw=Li1Li2Linifw=i1i2in,andN(w)=(N1(w),...,Nd(w)),withNi(w)=numberoftimesthesymboliappearsinthewordw Denition46. L?sym=f2(Ed):(f)=0,8f2Lsymg 60

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Itiswellknown,thatKisthedualofthediskalgebra,(aswementionedintheSection 2.1 )wewouldliketoidentifyFasthedualofsomealgebraLn.ThisalgebraLnisthenormedclosedsubalgebraofAd(thenoncommutativediskalgebra[ 8 ])generatedbyL.WesketchaproofoftheidenticationF=Ln,emulatingtheproofinSection 2.1 thatthespaceofCauchyTransformsKisthedualofthediskalgebra.FirstweequipFwiththequotientnormonEd=L?sym,iff(z)=((1)-283(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1),dene jjfjj:=jj[]jj=infl2L?symjj)]TJ /F3 11.955 Tf 11.95 0 Td[(ljjEd(5) Proposition5.1. Forf(z)=((1)-221(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)2F,kfk=inf2Rfkk. Proof. Notethatifl2L?symthen()]TJ /F3 11.955 Tf 12.6 0 Td[(l)((1)-276(hz,Li))]TJ /F6 7.97 Tf 6.58 0 Td[(1)=((1)-276(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 12.61 0 Td[(l((1)]TJ -435.83 -23.91 Td[(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)=((1)-250(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)=f(z),sothat)]TJ /F3 11.955 Tf 12.3 0 Td[(l2Rf,thenfromthedenitionofthenorminEquation 5 wehavejjfjj=infjj)]TJ /F3 11.955 Tf 11.95 0 Td[(ljj=inf2Rfjjjj Proposition5.2. Foreachf2F,thereisa2Rfsuchthatkfk=kk. Proof. Bydenitionofkfk,foreach"=1 nthereexistan2Rf,suchthatjjnjjjjfjj+1 n,thatisfngn1isauniformlyboundedsequenceinEd,sothatbyAlaoglu'sTheorem,thereexist=limn.Notethattheconvergenceisweak-,sothatinparticularwehavejjf(z))]TJ /F13 11.955 Tf 11.77 0 Td[(((1)-207(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)jj=jjn((1)-207(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1))]TJ /F13 11.955 Tf 11.78 0 Td[(((1)-207(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)jj)165(!0,sothat2Rf Theorem5.1. FisthedualofLsym= spannPN(w)=Lw:w2Fd+and2Ndowhere,Lw=Li1Li2Linifw=i1i2in,andN(w)=(N1(w),...,Nd(w)),withNi(w)=numberoftimesthesymboliappearsinthewordw. 61

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Proof. Letf2FandconsiderthefollowingsetRf=f2(Ed):f(z)=((1)]TJ -429.39 -23.9 Td[(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)g.Notethatif1and22Rf,then1((1)-222(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)=2((1)-222(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)1((1)-222(hz,Li))]TJ /F6 7.97 Tf 6.58 0 Td[(1))]TJ /F13 11.955 Tf 11.96 0 Td[(2((1)-221(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)=0(1)]TJ /F13 11.955 Tf 11.96 0 Td[(2)(1)-222(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)=0Nowobservethat1)-221(hz,Li))]TJ /F6 7.97 Tf 6.58 0 Td[(1=1Xn=0(hz,Li)n=1Xn=0Xjwj=nzN(w)Lw=XzXN(w)=LwThusfor=1)]TJ /F13 11.955 Tf 11.96 0 Td[(2)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)-222(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1=0@1Xn=0Xjwj=nzN(w)Lw1A=0@XzXN(w)=Lw1A=Xz0@XN(w)=Lw1A=0ifandonlyifPN(w)=Lw=0,forallmultindices,thismeansthat2L?sym.Hencethemapthatidentiesagivenf(z)=((1)-312(hz,Li))]TJ /F6 7.97 Tf 6.59 0 Td[(1)2Fwiththecoset[]:=+L?sym2(Ed)=L?symiswelldened.Notethatbydenitionthismapisisometric,andbylinearityofthefunctionals,wecaneasilyseethatitisalsoanisomorphism.ThusF=(Ed)=L?sym.FinallyweonlywouldneedtocheckthatLsymisaclosedsubspaceofEdandthatEdisaBanachspace,toapplyTheorem 2.1 foundinSection 2.1 toget(Ed)=L?sym=Lsym)F=Lsym. 62

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REFERENCES [1] JimAgler,Ontherepresentationofcertainholomorphicfunctionsdenedonapolydisc,Topicsinoperatortheory:ErnstD.Hellingermemorialvolume,Oper.TheoryAdv.Appl.,vol.48,Birkhauser,Basel,1990,pp.47.MR1207393(93m:47013) [2] JimAglerandJohnE.McCarthy,Pickinterpolationandhilbertfunctionspaces.,GraduateStudiesinMathematics,vol.44,AmericanMathematicalSociety,2002. [3] WilliamArveson,SubalgebrasofC-algebras.III.Multivariableoperatortheory,ActaMath.181(1998),no.2,159.MR1668582(2000e:47013) [4] DavidP.Blecher,Thestandarddualofanoperatorspace,PacicJ.Math.153(1992),no.1,15.MR1145913(93d:47083) [5] JosephA.Cima,AlecL.Matheson,andWilliamT.Ross,TheCauchytransform,MathematicalSurveysandMonographs,vol.125,AmericanMathematicalSociety,Providence,RI,2006.MR2215991(2006m:30003) [6] JohnB.Conway,Functionsofonecomplexvariable,seconded.,GraduateTextsinMathematics,vol.11,Springer-Verlag,NewYork,1978.MR503901(80c:30003) [7] ,Acourseinfunctionalanalysis,seconded.,GraduateTextsinMathematics,vol.96,Springer-Verlag,NewYork,1990.MR1070713(91e:46001) [8] KennethR.Davidson,EliasKatsoulis,andDavidR.Pitts,Thestructureoffreesemigroupalgebras,J.ReineAngew.Math.533(2001),99.MR1823866(2002a:47107) [9] S.W.Drury,AgeneralizationofvonNeumann'sinequalitytothecomplexball,Proc.Amer.Math.Soc.68(1978),no.3,300.MR480362(80c:47010) [10] KeikoFujita,HarmonicBergmankernelforsomeballs,Univ.Iagel.ActaMath.(2003),no.41,225.MR2084765(2005e:32003) [11] L.K.Hua,Harmonicanalysisoffunctionsofseveralcomplexvariablesintheclassicaldomains,TranslationsofMathematicalMonographs,vol.6,AmericanMathematicalSociety,Providence,R.I.,1979,TranslatedfromtheRussian,whichwasatranslationoftheChineseoriginal,byLeoEbnerandAdamKoranyi,WithaforewordbyM.I.Graev,Reprintofthe1963edition.MR598469(82c:32032) [12] MichaelJury,Operator-valuedherglotzkernelsandfunctionsofpositiverealpartontheball,ComplexAnalysisandOperatorTheory4(2010),301,10.1007/s11785-009-0012-6. [13] ,Universalcommutativeoperatoralgebrasandtransferfunctionrealizationsofpolynomials,IntegralEquationsandOperatorTheory73(2012),305,10.1007/s00020-012-1973-9. 63

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[14] JohnE.McCarthyandMihaiPutinar,Positivityaspectsofthefantappietransform.,J.Anal.Math.97(2005),57. [15] MitsuoMorimotoandKeikoFujita,BetweenLienormanddualLienorm,TokyoJ.Math.24(2001),no.2,499.MR1874986(2003e:32012) [16] VernPaulsen,Completelyboundedmapsandoperatoralgebras,CambridgeUniversityPress,Cambridge,2002. [17] GillesPisier,Introductiontooperatorspacetheory,LondonMathematicalSocietyLectureNoteSeries,vol.294,CambridgeUniversityPress,Cambridge,2003.MR2006539(2004k:46097) 64

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BIOGRAPHICALSKETCH Intheyear2001MiriamSalomeCastilloGilobtainedherbachelor'sdegreeinmathematicsfromtheUniversityoftheAmericas-Puebla(UDLA-P)locatedinherhometownPueblainMexico.Later,in2003shecompletedamaster'sdegreeinmathematicsattheUniversityofToronto,inCanada.Since2005shelivesinGainesville,Floridaandreceivedamaster'sdegreeinmathematicsfromtheUniversityofFloridain2007.ShecompletedaPhDinmathematicsattheUniversityofFloridainAugust2012. 65