Matrix Spaces And Schur Multipliers Matriceal Harmonic Analysis

《Matrix Spaces And Schur Multipliers Matriceal Harmonic Analysis》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1MATRIXSPACESANDSCHURMULTIPLIERSMatricealHarmonicAnalysis

2May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

3MATRIXSPACESANDSCHURMULTIPLIERSMatricealHarmonicAnalysisLars-ErikPerssonLuleåUniversityofTechnology,Sweden&NarvikUniversityCollege,NorwayNicolaePopa“SimionStoilov”InstituteofMathematics,RomanianAcademy,Romania&TechnicalUniversity“PetrolsiGaze”,RomaniaWorldScientificNEWJERSEY•LONDON•SINGAPORE•BEIJING•SHANGHAI•HONGKONG•TAIPEI•CHENNAI

4PublishedbyWorldScientificPublishingCo.Pte.Ltd.5TohTuckLink,Singapore596224USAoffice:27WarrenStreet,Suite401-402,Hackensack,NJ07601UKoffice:57SheltonStreet,CoventGarden,LondonWC2H9HELibraryofCongressCataloging-in-PublicationDataPersson,Lars-Erik,1944–author.MatrixspacesandSchurmultipliers:matricealharmonicanalysis/byLars-ErikPersson(LuleåUniversityofTechnology,Sweden&NarvikUniversityCollege,Norway)&NicolaePopa(“SimionStoilov”InstituteofMathematics,RomanianAcademy,Romania&TechnicalUniversity“PetrolsiGaze”,Romania).pagescmIncludesbibliographicalreferencesandindex.ISBN978-9814546775(alk.paper)1.Matrices.2.Algebraicspaces.3.Schurmultiplier.I.Popa,Nicolae,author.II.Title.QA188.P432014512.9'434--dc232013037182BritishLibraryCataloguing-in-PublicationDataAcataloguerecordforthisbookisavailablefromtheBritishLibrary.Copyright©2014byWorldScientificPublishingCo.Pte.Ltd.Allrightsreserved.Thisbook,orpartsthereof,maynotbereproducedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recordingoranyinformationstorageandretrievalsystemnowknownortobeinvented,withoutwrittenpermissionfromthepublisher.Forphotocopyingofmaterialinthisvolume,pleasepayacopyingfeethroughtheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,USA.Inthiscasepermissiontophotocopyisnotrequiredfromthepublisher.PrintedinSingapore

5October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013ToIrina,Andrei,ManuelaandAlexandrav

6May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

7October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013PrefaceInthelasttwocenturiestheFourieranalysis,knownalsoasharmonicanalysis,experiencedastrongdevelopment.RoughlyspeakingitconsistsmainlyinthestudyofpropertiesofperiodicalfunctionsconnectedtotheirFouriercoefficients.AsexamplesofsuchpropertieswemayconsiderthebeautifulFejer’stheory[31]abouttheconvergenceoftheFourierseriestoafunctionwithrespecttoitsCesaromeans,orthewell-knownFej´er-Hardy-Littlewoodinequality[96].OntheotherhandmanymathematicianshaveobservedthatthereisalinearbijectivecorrespondencebetweentheperiodicalfunctionsfonthetorusTanditscorrespondingToeplitzmatrix,thatistheinfinitematrixhavingthenthFouriercoefficientoffonthenthdiagonalsubmatrixpar-alleltothemaindiagonal,numberedby0.(Seeforinstance[94],[11].)ForinstancewebegantothinkaboutthistopicafterreadingthepaperofJ.Arazy[1],whereashortremarkabouttheanalogybetweenFouriercoefficientsanddiagonalsubmatriceswasmade.Infact,itappearsthatperiodicalfunctionsonthetorusareparticularcasesofinfinitematricesanditistemptingtodevelopamatrixversionoftheclassicalharmonicanalysis,whereinsteadofnthFouriercoefficientofaperiodicalfunctionyouhavetoconsiderthenthdiagonalofagiveninfinitematrix.Thepresentvolumeisdedicatedtothisgoal,namelytoformulateandprovesomestatementsinthisnewmatrixversionofclassicalharmonicanalysisintermsof“diagonal”submatricesofaninfinitematrix,whichareanalogousofthewell-knownstatementsinharmonicanalysis.Thecurrentknowledgeispresentedinaunifiedwayandalsosomenewresultsareincludedtocompletethepicturetoafairlynicetheoryweherebycallmatricealharmonicanalysis.Nowwebrieflydescribesomeofthemostmotivatingresultsofthisvii

8November18,201311:12WorldScientificBook-9inx6invers*11*oct*2013viiiMatrixspacesandSchurmultipliers:Matricealharmonicanalysisbook.Afteranintroductionandapresentationofprincipalnotions,inChapter2wepresentsomeresultsfromthemasterdissertationofVictorLie(see[50]).Themainideaofthisworkistoconsideraninfinitematrixasasequenceoffunctions(Lk)k≥1andtoexploitthisinterpretationinordertoobtainausefulformulafortheoperatornormofaninfinitematrix,namely:||A||B(2)=sup||VLB,h||∞,||h||2≤1where∞1/222VLB,h(x)=|(Lk∗h)(x)|,∀x∈[0,1]and∀h∈H0([0,1]).k=1Byusingthisformulawegivenewproofsofsomeclassicalresultsofinfinitematrixtheory.Forinstance,itispossibletoprove,withthesamemethod,somescatteredfactslike:•ForaToeplitzmatrixA,A∈B()ifandonlyiff∈L∞([0,1])2A(seeTheorem3.2),•(Bennett’sTheorem)ForaToeplitzmatrixA,A∈M(2)ifandonlyfAisaboundedBorelmeasureon[0,1](seeTheorem3.3),•(Nehari’sTheorem)ForaHankelmatrixA,A∈B(2)ifandonlyifgA∈BMO(seeTheorem3.5),•(TheoremofKwapien-Pelczynski)IfPnisthemaintriangleprojectionofordern,thensup||Pn(A)||B(2)=O(logn)n→∞||A||B(2)≤1(seeTheorem3.8).InChapter3wegiveamatrixversionofFejer’stheory,introducingasubclassofinfinitematricesrepresentingboundedlinearoperatorson2,namelytheclassofcontinuousmatricesC(2).Thisclassconsistsofthoseinfinitematrices,whichareapproximableintheoperatornormbymatricesoffinitelyband-type.Inparticular,wepresenttheresultsin[17],whichwereextendedlaterin[48]and[46].JustasonepossibleapplicationofthematrixversionofFourieranalysiswementiontheapproximationofaninfinitematrixinvariousways.For

9October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Prefaceixinstancethefollowingnaturalquestionarise:Whichinfinitematricescanbeapproximatedintheoperatornormbyitsfinite-typebandsubmatrices?Inparticular,ananswertothisquestion,whichextendsthewell-knownJordan’stheorem,isgiveninthischapter.AnothertypeofapproximationofinfinitematricesbyaspecialtypeofmatricealpolynomialsisalsogiveninChapter3anditextendsawell-knowntheoremofA.Haarfrom1910(seeTheoremConpage49):•LetA=(al)beamatrixbelongingtoC()suchthatallkl≥1,k∈Z2deflsequencesak=(ak)l≥1,k∈Z,locatedonthekthdiagonal,belongtotheclassms.Then,forany>0thereisann∈N∗andsequencesα∈ms,kk∈{0,...,n−1},suchthatn−1||A−αkHk||B(2)<.k=0Heremsisaspeciallinearsubspaceofthespaceofallboundedsequences∞,andmeansaproductbetweentwomatrices,whichextendstheusualproductofascalarandafunction.Nextwementionthatin1983A.Shields[84]statedandprovedthefollowingbeautifultheorem:•LetM∈C1haveuppertriangularformwithrespecttotheor-thonormalbasis{en}(n=1,2,...)of2.Then∞k|M(j,k)|≤π||M||T1,1+k−jk=1j=1withequalityonlywhenM=0.HereT1meansthesubspaceofalltraceclassinfiniteuppertriangularmatrices.Theaboveresultisanalogoustothewell-knownHardy-Littlewood-Fejerinequality.WepresentanddiscussthistheoremandsomeotherrelatedresultsinChapter4.

10October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013xMatrixspacesandSchurmultipliers:MatricealharmonicanalysisForinstancewepresentanewcharacterizationoftheelementsofthespaceT1withrespecttothesequenceofaspeciallinearcombinationoftheirdiagonals,namely:•LetA∈B(2)beanuppertriangularmatrix.Thenthefollowingassertionsareequivalent:a)A∈T1;n11b)sup||sjA||<∞;nanj+1j=0c)sup||PnA||<∞.nHere1n1n1PnA=sjA,wherean=(n=0,1,2,...)anj+1j+1j=0j=0jandsjA=k=0Ak.ItisimportanttomentionthatinChapter4andalsointhesequel,oftentheobtainedmatricealresultsareonlyanalogoustobutdonotextendtheknownresultsfromharmonicanalysis.ThishappensbecausemostoftheresultsfromthesechaptersrefertomatricesconnectedtoSchattenclassesofmatricesandconsequentlytheirproofscannotbeappliedtoToeplitzmatrices.AnothertopicwhichwastouchedsomeyearsagoisthestudyofmatrixversionofHankeloperators.SomeespeciallyinterestingresultsabouttheseHankeloperatorswereobtainedbyS.Powerin[77].ForinstanceheshowedtherethefollowingmatrixversionofNehari’stheorem:•LetΦbeaninfinitematrixsuchthatΦA∈C2forallfiniteband-typematricesA.Thenthefollowingstatementsareequivalent:(a)HΦisaboundedlinearoperatoronT2.(b)ThereisΨ∈B(2)suchthatΨk=Φkforallk<0.(c)P−Φ∈BMOF(2).

11November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013PrefacexiSeeChapter5forallunexplainednotionsandnotationsinthistheorem.InChapter5wegiveadifferentandinouropinionmorenaturalproofofthisresultandinvestigatethistopicfurther.Forexample,wederiveasufficientconditioninorderthatamatrixversionofaHankeloperatortobenuclear.Foraparticularclassofitssymbolthisconditionisevennecessary.AclassofBanachspacesofanalyticfunctions,whichhasreceivedgreatattentioninthelasttwodecadesistheclassofBergmanspaces.Seee.g.[94],[28]andthereferencesgiventhere.WeinvestigatesomepropertiesoftheclassofBergman-SchattenspacesinChapter6.ForinstancewepresentanddiscusssomeinequalitiesvalidinBergman-Schattenspaces(see[76])e.g.thefollowing:•(Hausdorff-YoungTheorem)For1≤p≤∞,letqbethecon-jugateindex,i.e.1+1=1(forp=1wehaveq=∞).pqP1/q∞q(i)If1≤p≤2,thenA∈Tpimpliesthatn=0||An||Tp≤||A||Tp.(ii)If2≤p≤∞,then{||An||Tp}∈`qimpliesthat||A||Tp≤P1/q∞qn=0||An||Tp.Anotherimportantresult,whichappearsinChapter6isamatricealanalogueofaresultobtainedbyMateljevicandPavlovic[61]in1984,(see[62]):•LetAbeanuppertriangularmatrix.ThenA∈L1(D,`)ifanda2onlyifX∞||σn(A)||C1<∞.(n+1)2n=1Moreover,amatrixversionoftheusualBlochspaceisintroducedinChapter7.See[72].Theinterestofthisspaceconsistsmainlyinthefactthatitsatisfiestheequality
B(D,`)=H1(`),BMOA(`),222whereB(D,`)meansthematricealBlochspace,H1(`),BMOA(`)stand222forthematrixversionoftheHardyspaceH1,respectively,forthematrixversionofthespaceBMOA,and(X,Y)isthespaceofallSchurmultipliersbetweenspacesofinfinitematricesXandY.

12November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013xiiMatrixspacesandSchurmultipliers:MatricealharmonicanalysisThisequationisthematrixanalogueofaresultvalidforFouriermul-tipliersofperiodicalfunctionsprovedin1990byMateljevicandPavlovic[60].Moreover,in1995O.Blasco[12]provedthatavectorvaluedversionoftheequalityofMateljevicandPavlovicisvalid,generallyspeaking,onlyforfunctionswithvaluesinHilbertspaces.Consequently,thisversionoftheaboveequalitydeservessomespecialattention.TheequalityisprovedinChapter8.Thischapterisdedicatedtoaveryimportanttoolinthetheory,namelySchurmultipliers,whichrepresentsthematrixversionofclassicalFouriermultipliers.Wealsopresent,proveanddiscusssomeotherresultsconcerningSchurmultipliersbetweenBanachspacesofinfiniteuppertriangularmatrices.Wementionjustthefollowingmatrixversionofawell-knownresultofPaley:P1P∞2•IfA=An∈H(`2),thenP(A):=k=1A2k∈H(`2).HereH1(`)isamatrixversionofHardyspaceintroducedinChapter24andH2(`)isthespaceofalluppertriangularHilbert-Schmidtmatrices.2AknowledgementThesecondauthorwaspartiallysupportedbytheCNCSISgrantID-PCE1905/2008.Moreover,wearebothgratefultoLule˚aUniversityofTechnologyforfinancialsupportforresearchvisitstobeabletofinalizethisbook.WearealsoverygratefultoDr.NiklasGripforhelpingandsupportingusinbothprofessionalandpracticalways.SpecialthanksareduetoourcolleaguesDr.A.N.MarcociandDr.L.G.Marcociwhoreadcarefullyapreliminaryversionofthisbookandmadevaluablesuggestionsandremarks.Finally,weemphasizethatwritingthisvolumehadnotbeenpossiblewithouttheexistenceofanatmospherefavorabletoscientificactivity,theatmosphereexistingintheInstituteofMathematics”SimionStoilow”oftheRomanianAcademyandattheDepartmentofMathematicsatLule˚aUniversityofTechnology.ThecoverimagesofthemidnightsunweretakenbyElinPerssonfromthenorthharbourofLule˚a,Sweden,atmidnightof16thJune2013.Theseimagesreflecttheempoweringsensestheauthorsfeltwhenthisspecialbookwasfinalizedundertheskiesofthisfamouslight.Lule˚a,June2013,undertheinfluenceofthemagicmidnightsunatmo-sphereclosetothePolarCircle.Lars-ErikPerssonandNicolaePopa

13November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013ContentsPrefacevii1.Introduction11.1Preliminarynotionsandnotations.............11.1.1Infinitematrices...................11.1.2Analyticfunctionsondisk.............41.1.3Miscellaneous....................51.1.4TheBergmanmetric................72.Integraloperatorsininfinitematrixtheory92.1Periodicalintegraloperators.................92.2Nonperiodicalintegraloperators..............172.3Someapplicationsofintegraloperatorsintheclassicalthe-oryofinfinitematrices....................182.3.1ThecharacterizationofToeplitzmatrices.....182.3.2ThecharacterizationofHankelmatrices......242.3.3Themaintriangleprojection............272.3.4B(`2)isaBanachalgebraundertheSchurproduct303.Matrixversionsofspacesofperiodicalfunctions333.1Preliminaries.........................343.2SomepropertiesofthespaceC(`2).............343.3AnothercharacterizationofthespaceC(`2)andrelatedresults.........................363.4Amatrixversionforfunctionsofboundedvariation...41xiii

14November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013xivMatrixspacesandSchurmultipliers:Matricealharmonicanalysis3.5ApproximationofinfinitematricesbymatricealHaarpoly-nomials............................443.5.1Introduction.....................453.5.2Aboutthespacems.................503.5.3ExtensionofHaar’stheorem............563.6Lipschitzspacesofmatrices;acharacterization......614.MatrixversionsofHardyspaces654.1FirstpropertiesofmatricealHardyspace.........654.2Hardy-Schattenspaces....................694.3AnanalogueoftheHardyinequalityinT1.........754.4TheHardyinequalityformatrix-valuedanalyticfunctions79p4.4.1Vector-valuedHardyspacesH..........79X4.4.2(Hp−`)-multipliersandinducedoperatorsforqvector-valuedfunctions...............804.5AcharacterizationofthespaceT1.............974.6AnextensionofShields’sinequality.............1015.ThematrixversionofBMOA1095.1FirstpropertiesofBMOA(`2)space............1095.2AnothermatrixversionofBMOandmatricealHankelop-erators.............................1115.3NuclearHankeloperatorsandthespaceM1,2.......1196.MatrixversionofBergmanspaces1216.1SchattenclassversionofBergmanspaces.........1216.2SomeinequalitiesinBergman-Schattenclasses......1326.3AcharacterizationoftheBergman-Schattenspace....1366.4UsualmultipliersinBergman-Schattenspaces.......1417.AmatrixversionofBlochspaces1497.1ElementarypropertiesofBlochmatrices..........1497.2MatrixversionoflittleBlochspace.............1618.Schurmultipliersonanalyticmatrixspaces175Bibliography185Index191

15October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Chapter1Introduction1.1PreliminarynotionsandnotationsInthissectionwecollectsomenotionsandfactsofthetheoryofinfinitematrices,thetheoryofanalyticfunctionsonthediskandthecircle,ofvector-valuedintegrationtheoryandofgeometryofthedisketc.1.1.1InfinitematricesForaninfinitematrixA=(aij),andanintegerkwedenotebyAkthematrixwhoseentriesaaregivenbyi,jai,jifj−i=k,ai,j=.0otherwiseThenAkwillbecalledthekth-diagonalmatrixassociatedtoA.Sometimesweusealsothenotationa(i,j)fortheentriesofthematrixA.AnimportantnotioninthetheoryofmatricesistheSchurproduct.LetA=(aij)i,jandB=(bij)i,jbetwoinfinitematrices.ThentheSchurproductC=(cij)i,jofAandB,denotedbyA∗B,hastheentriescij=aijbijforalli,j∈N.AninfinitematrixAsuchthatA∗B∈YforallB∈X,whereX,YareBanachspacesofinfinitematrices,iscalledaSchurmultiplierfromXintoY,andthespaceofallSchurmultipliersfromXintoY,endowedwiththenaturalnorm||A||(X,Y)=sup||A∗B||Y||B||X≤1isdenotedby(X,Y).1

16October20,20138:49WorldScientificBook-9inx6invers*11*oct*20132MatrixspacesandSchurmultipliers:MatricealharmonicanalysisInthecaseX=Y=B(2),whereB(2)isthespaceofalllinearandboundedoperatorson2,thespace(X,Y)isdenotedbyM(2)(anexplanationofthisnotationisgivenlaterinthissection)andamatrixA∈M(2)issimplycalledaSchurmultiplier.Weconsiderontheinterval[0,1)theLebesguemeasurableinfinitema-trixvaluedfunctionsA(r).Thesefunctionsmayberegardedasinfinitematrix-valuedfunctionsdefinedontheunitdiskDusingthecorrespon-∞iktdenceA(r)→fA(r,t)=k=−∞Ak(r)e,whereAk(r)isthekth-diagonalofthematrixA(r),theprecedingsumisaformaloneandtbelongstothetorusT.Wemayconsiderf(r,t),orf(z),withz=reit,asamatrixvaluedAAfunction,ordistribution,orjustaformalseries.SuchamatrixA(r)iscalledananalyticmatrixifthereexistsanuppertriangularinfinitematrixAsuchthat,forallr∈[0,1),wehaveAk(r)=Ark,forallk∈Z.kInwhatfollowsweidentifytheanalyticmatricesA(r)withtheircor-respondinguppertriangularmatricesAandcallthelatteralsoasanalyticmatrices.Aspecialclassofinfinitematricesisconsideredofteninthisbook,namelytheclassofToeplitzmatrices.LetA=(aij)i,j≥1beaninfinitematrix.Ifthereisasequenceofcomplex+∞numbers(ak)k=−∞,suchthataij=aj−iforalli,j∈N,thenAiscalledaToeplitzmatrix.+∞ForsimplicitywecanidentifyaToeplitzmatrixwithA=(ak)k=−∞,andtheclassofallToeplitzmatricesisdenotedbyT.G.Bennettprovedin1977thefollowinginterestingresult(seeTheorem8.1-[11])aboutSchurmultipliers:Bennett’sTheoremTheToeplitzmatrixM=(cj−k)j,k,where(cn)n∈Zisasequenceofcomplexnumbers,isaSchurmultiplierif,andonlyif,thereexistsa(bounded,complex,Borel)measureμon(thecirclegroup)Twithμ(n)=cnforn=0,±1,±2,....Moreover,wethenhave||M||M(2)=||μ||.Bennett’stheoremjustifiesthenotationM(2)sinceforaToeplitzma-trixthenotionsofSchurmultiplierandBorelmeasureonthetoruscoincide.

17October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Introduction3InthesequelwegivesomeresultsaboutcompactoperatorsontheHilbertspace2.(Seeforinstance[94].)Forexamplethefollowingdecompositionformulaisknown.SchmidtTheoremIfTisaself-adjointcompactoperatoronaHilbertspaceH,thenthereexistsasequenceofrealnumbers{λn}tendingto0andtherealsoexistsanorthonormalset{en}inHsuchthat∞Tx=λn(x,en)enn=1forallx∈H.IftheoperatorTiscompact,butnotnecessarilyself-adjoint,thenwefirstconsiderthepolardecompositionT=V|T|,where|T|=(T∗T)1/2ispositive(andhenceself-adjoint)andcompact.Bytheabovetheorem,thereisanorthonormalset{en}inHsuchthat|T|x=λn(x,en)en,x∈H,nwhere{λn}isanonincreasingsequenceofnonnegativenumberstendingto0.Letσn=Venforeachn;then{σn}isstillanorthonormalsetandwehavethatTx=λn(x,en)σn,x∈H.nThisiscalledthecanonicaldecompositionofacompactoperatorT.Thenon-increasinglyarrangedsequence{λn}iscalledthesequenceofsingularvaluesofT.ThenumberλniscalledthenthsingularvalueofT.NowweintroducetheSchattenclassoperators.Given0

18October20,20138:49WorldScientificBook-9inx6invers*11*oct*20134MatrixspacesandSchurmultipliers:Matricealharmonicanalysis1.1.2AnalyticfunctionsondiskInthissubsectionweintroducethedefinitionsofsomeimportantspacesofanalyticfunctionsonthedisk.FirstofallweconsidertheclassicalHardyspaceoffunctionsonthedisk.Let00isaconstantindependentofthechoiceoff.AnotheranalyticfunctionspacestudiedinconnectionwithHardyspaceisthespaceofallanalyticfunctionsofboundedmeanoscillationdenotedbyBMOA.Oneoftheequivalentdefinitionsofthisspaceisasfollows(see[33]):BMOAcoincideswiththespaceofallanalyticfunctionsfonthedisksuchthatthefollowingnormisfinite:1/2||f||:=sup|f(z)|2(1−|z|2)(1−|λ|2)|1−λz|−2dxdy.∗λ∈DDNowwerecallthedefinitionoftheclassicalBlochspaceforfunctionsonthedisk.TheBlochspaceBisthespaceofallanalyticfunctionsf:D→Csuchthat||A||:=sup(1−|z|2)|f(z)|+|f(0)|<∞.Bz∈DAnotherinterestingspaceofanalyticfunctionsonthediskistheso-calledlittleBlochspace.ThelittleBlochspaceofD,denotedbyB0,istheclosedsubspaceofBconsistingoffunctionsfwith(1−|z|2)f(z)→0(|z|→1−).

19October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Introduction51.1.3MiscellaneousWealsoneedsomenotionsofvector-valuedintegrationtheory.Wesaythatafunctionf:D→B(),isw∗-measurableifA◦fis2aLebesguemeasurablefunctiononDforeveryA∈C1,whereC1istheSchattenclassofalloperatorswithtrace,andAisconsideredasafunctionalonB(2).Thefunctionf:D→B(2)isstronglymeasurableifitisanormlimitofasequenceofsimplefunctions.(See[30]formoredetailsaboutvector-valuedmeasurability.)ThenwehavethefollowingparticularcaseofProposition8.15.3-[30]:Proposition1.1.Letf:D→B()beaw∗-mesurablefunction.Then2thefunction||f||:z→||f(z)||B(2)ismeasurable.Moreover,westatewithoutproofthefollowingtheorem,whichisaparticularcaseofTheorem8.18.2[30]forE=C1:Theorem1.2.ThetopologicaldualofL1(D,)maybeidentifiedwith2L∞(D,)bythedualitybilinearmap:21:=tr(A(s)[B(s)]∗)2sds,0whereA(·)∈L∞(D,),B(·)∈L1(D,).22HereL1(D,)andL∞(D,)aredefinedinSection6.1asparticular22casesofBanachspacesofvector-valuedfunctions.Wealsoneedthefollowingwell-knownlemma(seeforinstance[94]):Lemma1.3.Letz∈D,c∈R,t>−1,and(1−|w|2)tIc,t(z)=dA(w).D|1−zw|2+t+cThenwehavethat(1)ifc<0,thenIc,t(z)isboundedinz;(2)ifc>0,then1−Ic,t(z)∼(|z|→1);(1−|z|2)c(3)ifc=0,then1−I0,t(z)∼log(|z|→1).1−|z|2

20October20,20138:49WorldScientificBook-9inx6invers*11*oct*20136MatrixspacesandSchurmultipliers:MatricealharmonicanalysisProof.Sincet>−1,theintegralIc,tisdefinedforallz∈D.Letλ=1(2+t+c).Ifλiszerooranegativeinteger,thenclearlyc<0andI(z)2c,tisbounded.Ifλisnotzerooranegativeinteger,then∞1Γ(n+λ)nn=zw(1−zw)λn!Γ(λ)n=0andtherotationinvarianceof(1−|w|2)tdA(w)showsthat(1−|w|2)t∞Γ(n+λ)2dA(w)=|z|2n(1−|w|2)t|w|2ndA(w)D|1−zw|2λ(n!)2Γ(λ)2Dn=0∞Γ(n+λ)21=|z|2n(1−r)trndr(n!)2Γ(λ)20n=0∞Γ(n+λ)2Γ(t+1)Γ(n+1)=|z|2n(n!)2Γ(λ)2Γ(n+t+2)n=0Γ(t+1)∞Γ(n+λ)2=|z|2n.Γ(λ)2n!Γ(n+t+2)n=0ByStirling’sformula,Γ(n+λ)2∼nc−1(n→∞).n!Γ(n+t+2)∞c−12nClearlyn|z|isboundedinzforc<0.Ifc=0,thenn=1∞|z|2n1=log.n1−|z|2n=1Ifc>0,thenc−12n1n|z|∼(1−|z|2)csince∞1Γ(n+c)2n=|z|(1−|z|2)cn!Γ(c)n=0andΓ(n+c)/n!∼nc−1byStirling’sformula.Thiscompletestheproofofthelemma.

21October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Introduction71.1.4TheBergmanmetricInthissectionwerecallsomegeometricfactsaboutthedisk.Morespecif-icallyweconsidertheBergmanmetriconD,whichwillbeusefulinthestudyofmatricealBlochspace.Thepseudo-hyperbolicmetric.Recallthatforanyz∈D,ϕzistheMoebiustransformationofD,whichinterchangestheoriginandz,namelyz−wφz(w)=,w∈D.1−zwThepseudo-hyperbolicdistanceonDisdefinedbyz−wρ(z,w)=|ϕz(w)|=,z,w∈D.1−zwAnimportantpropertyofthepseudo-hyperbolicdistanceisthatitisMoebiusinvariant,thatis,ρ(ϕ(z),ϕ(w))=ρ(z,w)forallϕ∈Aut(D),theMoebiusgroupofD,andallz,w∈D.TheBergmanmetric.TheBergmanmetriconDisgivenby11+ρ(z,w)β(z,w)=log,z,w∈D.21−ρ(z,w)TheBergmanmetricisalsoMoebiusinvariant:β(ϕ(z),ϕ(w))=β(z,w)forallϕ∈Aut(D)andallz,w∈D.NotesMostoftheinformationinthischapterisclassicalbutnotsoeasytofindcollectedinthisformelsewhere.TheSchur(orHadamard)productiswellknowntospecialistsbuttherearefewmonographswhichtreatthismatter.TheauthorsknowonlythebookofG.Pisier[81].Bennett’sTheoremisalsoconsideredaspartoffolklore(seeforexample[8]).Foranaccessibleproofsee[11]and[8].

22November15,20139:58WorldScientificBook-9inx6invers*11*oct*20138MatrixspacesandSchurmultipliers:MatricealharmonicanalysisInthenextchapterwepresentanotherproofofthisimportantresult(seeTheorem2.14).MoreaboutSchmidtTheoremandSchattenclassoperatorscanbefoundinmanyexcellentbooks,forinstance[34].Analyticfunctiononthediskaretreatedalsoinmanybooksas[68],[33]and[23].WepayspecialattentiontodifferentspacesofanalyticfunctionsonthediskasHardyspaceH1,whichisintensivelystudiedin[23]and[33].MoreaboutthespaceofanalyticfunctionsofboundedmeanoscillationBMOAmaybefoundin[33].AveryinterestingspaceisalsotheBlochspaceofanalyticfunctionsB.Aclassicalreferenceis[3].Proposition1.1,Theorem1.2andrelatedfactsaboutvector-valuedin-tegrationtheoryaremainlytakenfrom[30].Lemma1.3maybefoundin[94].Moreover,thesubsectiondedicatedtotheBergmannmetricistakenfrom[94].

23October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Chapter2IntegraloperatorsininfinitematrixtheoryThecontentofthepresentchapteristakenfromthemasterdissertationofV.LieatUniversityofBucharestunderthesupervisionofthesecondauthor(see[50]).Westartdefininganddiscussingsomeimportantdevicesweneedinwhatfollowsinthestudyofinfinitematrices.Themainideaistoconsideraninfinitematrixasasequenceoffunctions(resp.distributions).Thisnewpointofviewhastheadvantagetousethemorerefinedresultsfromfunctiontheoryinthetheoryofinfinitematrices.Forinstanceinthefirstsectionwedefinetheimportantnotionsofsquarefunctionandmatricealoperatorassociatedtoamatrix.Usingthesenotionsweprovethefirstmainresult,namelyTheorem2.8.Inthesecondsectionthecentralresultisthenon-periodicalanalogueofTheorem2.8(seeProposition2.2).2.1PeriodicalintegraloperatorsLetthematrix⎛⎞b11b12b13...⎜⎜b21b22b23...⎟⎟⎜...⎟B=⎜⎜.........⎟⎟∈B(2).⎜⎝bn1bn2bn3...⎟⎠............SinceB∈B(2)itfollowsthat,forallk∈N,thesequence(bkj)j≥1∈9

24October20,20138:49WorldScientificBook-9inx6invers*11*oct*201310MatrixspacesandSchurmultipliers:Matricealharmonicanalysis2(N).Thereforewecandefinethefunctions:∞L(B)(t)=be2πijt∈H2([0,1]),kkj0j=1and∞L∗(B)(t)=be2πijt·e−2πikt=L(B)(t)e−2πikt.kkjkj=1Consequently,toeachrowkinthematrixBitcorrespondsauniquefunctionfromtheHardyspace,H2([0,1]),ofallanalyticfunctionsh(t)=0∞xe2πikt,∀t∈[0,1].ThisfunctionisdenotedbyL(B),andk=1kkLk(B)(j)=bkj,forallk,j≥1.ForbrevitywedenoteinwhatfollowsL(B)simplebyL,andL∗(B)kkkbyL∗.kThusthematrixBcanbewrittenasfollows:⎛1−2πit1−4πit⎞0L1(t)edt0L1(t)edt...⎜1L(t)e−2πitdt1L(t)e−4πitdt...⎟⎜0202⎟⎜⎟⎜....⎟B=⎜.....⎟,⎜⎜1L(t)e−2πitdt1L(t)e−4πitdt...⎟⎟⎝0n0n⎠.........orB=⎛1∗1∗−2πit1∗−4πit⎞0L1(t)dt0L1(t)edt0L1(t)edt...⎜1L∗(t)e2πitdt1L∗(t)dt1L∗(t)e−2πitdt...⎟⎜⎜020202⎟⎟⎜....⎟⎜........⎟.⎜⎜1L∗(t)e2πi(n−1)tdt1L∗(t)e2πi(n−2)tdt1L∗(t)e2πi(n−3)tdt...⎟⎟⎝0n0n0n⎠............WeusethebothexpressionsofBinwhatfollows.Ifx=(xj)j≥1∈2,then12||Bx||2=L(t)xe−2πit+xe−4πit+...dt+...21120

25October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory1112+L(t)xe−2πit+xe−4πit+...dt+...n120∞2πijt2Denotebyh(t)thesumj=1xje∈H0([0,1]).Then1212||Bx||2=L(t)h(−t)dt+···+L(t)h(−t)dt+...21n00∞12=Lk(t)h(−t)dt,0k=1and,sinceL∈H2([0,1])forallk≥1,wehavethatk011Lk(t)h(−t)dt=Lk(t)g(−t)dt,00forallg∈L2([0,1])suchthat(h−g)(n)=0foralln≥1.Thus,∞121/2||B||B(2)=sup||Bx||2=supLk(t)h(−t)dt||x||2≤1||h||≤10H2k=1∞121/2=supLk(t)g(−t)dt.||g||≤10L2k=1Consequently,thespaceB(2)maybeconsideredasasubspaceofthe∞2spacen=1Hn,whereHn=H0([0,1])foralln≥1.Moreover,ifwedenoteby∞L:=(L1,L2,...)∈Hn,n=1and∞121/2||L||H2(∞)=supLk(t)h(−t)dt,||h||≤10H2k=1thenitfollows∞2H0(∞):={L∈Hn|||L||H2(∞)<∞},0n=12andH0(∞),||·||H2(∞)isaBanachspace.0

26October20,20138:49WorldScientificBook-9inx6invers*11*oct*201312MatrixspacesandSchurmultipliers:MatricealharmonicanalysisMoreover,thelinearoperatordefinedby2L:B(2),||||B(2)→H0(∞),||||H2(∞)0L(B)=LB,whereLB=(L1(B),L2(B),...)isanisometrybetweenB()andH2(∞).20Inthesequelweintroducetheperiodicalsquarefunctionassociatedtoamatrix.∞2LetL=(L1,L2,...)∈n=1Hnandh∈H0([0,1]).WedefineVL,h:[0,1]→R+by:∞1/22VL,h(x)=|(Lk∗h)(x)|.k=1∞Proposition2.1.LetL=(L1,L2,...)beafixedelementinn=1Hn.WehavethatsupVL,h(0)=sup||VL,h(·)||∞.||h||≤1||h||≤1H2H2Proof.First,itisclearthatVL,h(0)≤||VL,h||∞.Fortheconverse,observethatVL,h0(x)≤supVL,h(x),||h||2≤1whereh∈H2([0,1])isfixedwith||h||≤1.0002WedefinetheisometricoperatorSx(h)=hx,wherehx(t)=h(x+t).ThensupVL,h(x)=supVL,hx(0)=supVL,hx(0)=supVL,h(0),||h||2≤1||h||2≤1||hx||2≤1||h||2≤1and,hence,VL,h0(x)≤supVL,h(0),||h||2≤1forallh0suchthat||h0||2≤1,andforallx∈[0,1].Inotherwordssup||VL,h||∞=sup||VL,h0||∞≤supVL,h(0).||h||2≤1||h0||2≤1||h||2≤1

27November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory13Remark2.2.a)Bythepreviousdiscussionitfollowsthat||L||H2(∞)=sup||VL,h||∞,0||h||≤1H2and,consequently,Y∞2H0(∞)={L∈Hn|sup||VL,h||∞<∞}.||h||≤1n=1H2b)ItisclearfromthedefinitionthatthevalueofVL,hineachpointx∈[0,1]hasadefinitemeaning.Q∞Proposition2.3.LetL=(L1,L2,...)∈n=1Hn,andBthematrixcanonicallyassociatedtothiselementinthesensedefinedpreviously.Thenthefollowingassertionsareequivalent:i)B∈B(`2).ii)L=L∈H2(∞).B0iii)Forallh∈H2([0,1])wehaveV∈C([0,1]),whereC([0,1])0L,hss:={f:[0,1]→C|∃(fn)n≥1suchthatfn:[0,1]→Carecontinuousfunctions,(fn)n≥1isaboundedsequenceinthesup-normandfn(x)→f(x)asn→∞,forallx∈[0,1]}.Proof.Theimplicationi)⇒ii)followsfromthedefinitionoftheoper-atorL.i)⇒iii)WeshowthatVL,h(x)≤M||h||2,forh∈H2([0,1]),x∈[0,1],andM:=||B||.0B(`2)Ofcourse,itisenoughtoprovethatVL,h(x)≤M,for||h||2≤1.Ifnot,thenthereexistx∈[0,1],andh∈H2([0,1]),with||h||≤100002suchthatVL,h0(x0)>M.P∞2πiktBut,forh0(t)=k=1ake,wehavethatVL,h0(x0)=(VL,h0)x0(0)=||By0||2,wherey=(y1,y2,...),andyk=ae2πikx0.0000kSince||y0||2=||h0||H2itfollowsthatthereexistsy0,with||y0||2≤1suchthat||By0||2>||B||B(`2)=M,whichisacontradiction.

28October20,20138:49WorldScientificBook-9inx6invers*11*oct*201314MatrixspacesandSchurmultipliers:MatricealharmonicanalysisTherefore,foreachh∈H2([0,1]),andforeachx∈[0,1],wehave0VL,h(x)≤M||h||2.FixinghitfollowsthatthereexistsC>0suchthat,forallx∈[0,1],wehavethatVL,h(x)0with|a|≤C∀i,j≥1},andletL:=LBbeitssequenceofijBkk≥1distributions.WecallthematricealdistributionassociatedtothematrixBtheexpressionL∈D([0,1]×[0,1])Bgivenbytheformula∞L(t,x)=LB(t)e2πikx.Bkk=1B∞22Remark2.5.i)IfLB=Lkk≥1∈n=1H0([0,1]),andh∈H0([0,1]),thenweobservethat1121/2VLB,h(u)=LB(t,x)h(u−t)dtdx.00Hence,1121/2B∈B(2)⇔supLB(t,x)h(t)dtdx<∞.||h||≤100H2

29October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory15ii)ThenotionofsquarefunctionassociatedtoamatrixmaybeextendedfromtheclassB(2)totheclassPM(2)asfollows:LetP([0,1])bethelinearspaceofallanalyticpolynomials(i.e.trigono-metricalpolynomialshavingFouriercoefficientsofnonpositivesindicesequaltozero).Thenwedefine,∀h∈P([0,1]),11VLB,h:=supLB(t,x)h(u−t)r(x)dtdx.r∈P([0,1]);||r||≤100H2Definition2.6.LetB∈PM(2)andletLBdenotethematricealdistri-butionassociatedtothematrixB.Then,theoperatorTB:P([0,1])×P([0,1])→C([0,1]×[0,1]),definedby11TB(r⊗h)(u,v)=LB(t,x)r(u−x)h(v−t)dtdx,00iscalledthematricealoperatorassociatedtoB.Proposition2.7.LetB∈PM(2).Thefollowingassertionsareequiva-lent:i)B∈B(2).ii)ThereexistsacontinuousoperatorT:H2([0,1])⊗H2([0,1])→B00Cs([0,1]×[0,1]),suchthatTBP([0,1])⊗P([0,1])=TB.Proof.i)⇒ii)LetB∈B().Thenitfollowsthat,forallh∈H2([0,1]),20andforallx∈[0,1],wehavethat(a)VLB,h(x)≤M||h||2,withM=||B||B(2).Letr,h∈P([0,1])befixedpolynomials.Then:∞11T(r⊗h)(u,v)=LB(t)e2πikxr(u−x)h(v−t)dxdtBk00k=1∞11=LB(t)h(v−t)dtr(u−x)e2πikxdxk00k=1∞1=LB∗h(v)r(x)e−2πikxdxe2πiku.k0k=1

30October20,20138:49WorldScientificBook-9inx6invers*11*oct*201316MatrixspacesandSchurmultipliers:MatricealharmonicanalysisHence,|TB(r⊗h)(u,v)|≤VLB,h(v)||r||2≤(by(a))≤M||h||2||r||2,and||TB(r⊗h)||∞≤||B||B(2)||r⊗h||.SinceP([0,1])isdenseinthenorm||||2,itfollowsthatthereexistsauniquecontinuousextensionTBlikeinthestatementoftheproposition.ii)⇒i)LetTBbeasinthehypothesis.ThenthereexistsM>0suchthat,forallr,h∈P([0,1]),wehavethat||TB(r⊗h)||∞≤M||r⊗h||=M||r||2||h||2.Wefixr,h∈P([0,1]).Thenitfollowsthat∞1B−2πikx||TB(r⊗h)||∞≥|TB(r⊗h)(0,0)|=Lk∗h(0)r(x)edx,0k=1and,therefore,∞1B−2πikxLk∗h(0)r(x)edx≤M||r||2||h||2.0k=1Nextwetakethesupremumoverallr∈P([0,1])with||r||2≤1andwegetthatVLB,h(0)≤M||h||2∀h∈P([0,1]).Hence,||B||B(2)=supVLB,h(0)≤M<∞,||h||2≤1i.e.B∈B(2).Consequently,wehave:Theorem2.8.LetB∈PM(2).Thefollowingassertionsareequivalent:i)B∈B(2).ii)L∈H2(∞).B0iii)Forallh∈H2([0,1]),wehavethatV∈C([0,1]).0LB,hsiv)sup||h||2≤1||VLB,h||∞<∞,and||B||B(2)=sup||h||2≤1||VLB,h||∞.v)ThereexistsacontinuousoperatorT:H2([0,1])⊗H2([0,1])→C([0,1]×[0,1]),B00ssuchthatTBP([0,1]×[0,1])=TB.

31October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory17Remark2.9.a)Theequivalencei)-iv)aboveholdsalsointhemoregeneralcaseB∈PM(2).b)Bythediscussionabovewehavethat:∞121/2||B||=supLB(t)h(−t)dt.B(2)k||h||2≤10k=1Finally,ifB∈M(2),A∈B(2),x=(xn)n≥1∈2(N),andh∈H2([0,1]),withhˆ(n)=x,foralln≥1,thenwehavethat0n∞12||(B∗A)x||2=LB∗LA(t)h(−t)dt.2kk0k=1Hence,wefindthat∞121/2||B||=supsupLB∗LA(t)h(−t)dt.M(2)kk||LA||H2(∞)≤1||h||2≤1k=1002.2NonperiodicalintegraloperatorsInwhatfollowswepresentanothermethodtousethefunctionsintheframeworkofmatrixtheory.LetB=(b)∈B()andx∈(N∗).Theniji,j≥122∞22||Bx||2=|bk1x1+bk2x2+...|.k=1WedefineP(0,∞):={f:(0,∞)→Cameasurablefunctionf(k,k+1]=cta.e.∀k∈N},andL2(0,∞):=P(0,∞)∩L2(0,∞).Thenwehaveaone-to-onecorrespondencebetweentheclassofallse-quencesfrom(N∗)andthespaceL2(0,∞),givenby2x=(x)∈(N∗)↔h∈L2(0,∞),kk≥12withh(k,k+1]=xk+1∀k∈N.

32October20,20138:49WorldScientificBook-9inx6invers*11*oct*201318MatrixspacesandSchurmultipliers:MatricealharmonicanalysisDefinition2.10.LetB∈PM(2).ToBitcorrespondsauniquefunctionCB(·,·)givenbyi)CB:(0,∞)×(0,∞)→C,ii)CB(·,t)∈P(0,∞),andCB(y,·)∈P(0,∞)∀t,y>0,iii)CB(y,t)=bify∈(k−1,k]andt∈(j−1,j],forallk,j≥1,kjwhereB=(bkj)k≥1;j≥1.Then,wehavethat∞∞2∞∞2||Bx||2=CB(k,t)h(t)dt=CB(y,t)h(t)dtdy.2000k=1Proposition2.11.LetB∈PM()andCB(·,·)beitsassociatedfunc-2tion.Thefollowingassertionsareequivalent:i)B∈B(2).ii)TheoperatorT:L2(0,∞)→L2(0,∞),Bgivenby∞T(h)(y)=CB(y,t)h(t)dtB0isacontinuousoperator.2.3SomeapplicationsofintegraloperatorsintheclassicaltheoryofinfinitematricesInthissectionweusethepreviousresultstogivedifferentproofsforsomeclassicaltheoremsofinfinitematrixtheory.2.3.1ThecharacterizationofToeplitzmatricesWerecallthedefinitionofToeplitzmatrices.Definition2.12.LetA∈PM(2).ThematrixAisaToeplitzmatrixifthereexistsasequenceofcomplexnumbers(an)n∈Zsuchthat⎛⎞a0a1a2...an...⎜⎜a−1a0a1...an−1...⎟⎟⎜.....⎟A=⎜⎜.............⎟⎟.(2.1)⎜⎝a−na−n+1a−n+2...a0...⎟⎠..................

33October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory19TosuchamatrixitispossibletoassociateauniquepseudomeasurefAgivenby+∞f(t)=ae2πikt,Akk=−∞equalitybeingtakeninthedistribution’ssense,andt∈[0,1].InwhatfollowswefindthenecessaryandsufficientconditionsinorderthataToeplitzmatrixAbelongtoB(2),resp.toM(2).Morespecifically,wehave:Theorem2.13.LetAbeaToeplitzmatrixlikein(2.1).ThenA∈B()⇔f∈L∞([0,1]).2AProof.ByProposition2.7wehavethat,ifTAisthematricealoperatorassociatedtoA,givenbyTA:P([0,1])⊗P([0,1])→C([0,1]⊗[0,1]),11TA(r⊗h)(u,v)=LA(t,x)r(u−x)h(v−t)dxdt,00thenthefollowingassertionsareequivalent:1)A∈B(2).2)||TA(r⊗h)||∞≤C||r||2||h||2,whereC>0isanabsoluteconstantandr,h∈H2([0,1])arearbitraryfunctions.0Consequently,itisenoughtoprovethat2)⇔f∈L∞([0,1]).ATherefore,letusconsiderthepseudomeasures∞LA(t)=ae2πijt=L∗A(t)e2πikt∀k≥1.k−k+jkj=1ThematricealdistributionassociatedtoAis∞L(t,x)=LA(t)·e2πikx.Akk=1Forr,h∈P([0,1])wehavethat∞11T(r⊗h)(u,v)=L∗A(t)e2πik(t+x)r(u−x)h(v−t)dxdtAk00k=1

34October20,20138:49WorldScientificBook-9inx6invers*11*oct*201320MatrixspacesandSchurmultipliers:Matricealharmonicanalysis∞11=r(u−x)e2πikxdxL∗A(t)e2πikth(v−t)dtk00k=1∞11=r(x)e−2πikxdxL∗A(t)e2πikth(v−t)dte2πiku.k00k=111L∗A(t)e2πikth(v−t)dt=f(t)e2πikth(v−t)dtforallk≥1.kA00Hence,TA(r⊗h)(u,v)=∞11f(t)h(v−t)r(x)e−2πikxdxe2πik(u+t)dtA00k=11=fA(t)h(v−t)r(u+t)dt.0Inthiswaywehavetoprovetheequivalenceofthefollowingtwocon-ditions:i)f∈L∞([0,1]).Aii)ThereexistsaconstantC>0suchthat,forallh,r∈P([0,1]),wehavethat1||fA(t)h(·−t)r(·+t)dt||∞≤C||h||2||r||2.0i)⇒ii)BySchwarzinequalityandatrivialestimatewehavethat11fA(t)h(v−t)r(u+t)dt≤||fA||∞|h(v−t)r(u+t)|dt≤00||fA||∞||h||2||r||2,forfixedu,v∈[0,1].ii)⇒i)Since111/2supf(t)h(−t)r(u+t)dt=|f(t)r(u+t)|2dt,AA||h||2≤100byusingii)wehavethat:11|f(t)r(t)|2dt≤C|r(t)|2dt∀r∈P([0,1]).A00Since1sup|f(t)r(t)|2dt=||f||2,AA∞||r||2≤10itfollowsthat||f||2≤C,A∞i.e.f∈L∞([0,1]).A

35October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory21WegiveadifferentproofofBennett’sTheorem(see[11]).Theorem2.14.LetAbeaToeplitzmatrix.ThenA∈M(2)⇔fA∈M([0,1]),wherethislastspaceisthespaceofallboundedcomplexmeasureson[0,1].Proof.LetB∈B(2)befixed.Ityieldsthat11TA∗B(r⊗h)(u,v)=LA∗B(t,x)r(u−x)h(v−t)dxdt.00∞∞AB2πikxAB∗2πik(x+t)LA∗B(t,x)=Lk∗Lk(t)e=Lk∗Lk(t)e,k=1k=1forA↔(LA),andB↔(LB).kk≥1kk≥1Since∗LA∗LB(t)=L∗A∗L∗B(t),kkkkwehavethatTA∗B(r⊗h)(u,v)∞11=L∗A∗L∗B(t)e2πik(x+t)r(u−x)h(v−t)dxdtkk00k=1∞11=f∗L∗B(t)e2πik(x+t)r(u−x)h(v−t)dxdtAk00k=1⎛⎞1⎜⎜11∞⎟⎟=f(s)⎜LB(t−s)e2πik(x+s)r(u−x)h(v−t)dxdt⎟dsA⎜k⎟0⎝00k=1⎠!LB(t−s,x+s)1=TB(r⊗h)(u+s,v−s)ds.0Thus,weconcludethat1TA∗B(r⊗h)(u,v)=TB(r⊗h)(u+s,v−s)ds,0foraToeplitzmatrixAandarbitraryB∈B(2).

36October20,20138:49WorldScientificBook-9inx6invers*11*oct*201322MatrixspacesandSchurmultipliers:MatricealharmonicanalysisMoreover,byusingTheorem2.13,wefindthat1TB(r⊗h)(u,v)=fBh(v−t)r(u+t)dt,0foraToeplitzmatrixB∈B(2).⇐SupposethatfA∈M([0,1]).ByusingProposition2.7itisenoughtoshowthat:∀B∈B(2),∃CB>0suchthat∀r,h∈P([0,1]),itfollowsthat||TA∗B(r⊗h)||∞≤CB||r||2||h||2.(2.2)But,forB∈B(2),TB:P([0,1])⊗P([0,1])→C([0,1]×[0,1])canbeextendedcontinuouslytoanoperatorTB.Hence,wehavethat1TA∗B(r⊗h)(u,v)=fA(s)TB(r⊗h)(u+s,v−s)ds,0and,sincefA∈M([0,1]),itfollowsthat|TA∗B(r⊗h)(u,v)|≤||fA||M([0,1])||TB(r⊗h)||∞.But||T(r⊗h)||≤C1||r||||h||.B∞B22Consequently,||TA∗B(r⊗h)||∞≤CB||r||2||h||2,whereC=C1||f||.BBAM⇒LetA,BbeToeplitzmatriceswithA∈M(2),andB∈B(2),||B||B(2)≤1.Then,wehavethat1TB(r⊗h)(u,v)=fB(t)r(u+t)h(v−t)dt,0and,byusingTheorem2.13andtherelation(2.2),itfollowsthatthereexistsC>0suchthat,foreachf∈L∞([0,1]),with||f||≤1,weBB∞have,∀r,h∈P([0,1]),that11fA(s)fB(t−s)h(−t)r(t)dtds≤C||h||2||r||2.(2.3)00

37October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory23NextwerecalltwonotionsfromclassicalFourieranalysis:theCesarokerneln"#2|j|2πijt1sin(n+1)πtKn(t)=1−e=≥0,n+1n+1sinπtj=−nandtheDirichletkerneln2πijtsinπ(2n+1)tDn(t)=e=.sinπtj=−nWeintroduceinrelation(2.3)12πi(n+1)th(t)=√Dn(t)e=r(t).2n+1Thenr,h∈P([0,1]),with||h||2=||r||2=1,and(2.3)becomes1fA(s)σ2n+1(fB)(−s)ds≤C∀n≥1,0whereσn(fB)(s)=(Kn∗fB)(s)aretheCesarosumsoftheordernassociatedtofB.Then,foralln≥1,andforallf∈L∞([0,1]),with||f||≤1,weBB∞havethat1fB(−s)σ2n+1(fA)(s)ds≤C,(2.4)0and,hence,foralln≥1,1||σ2n+1(fA)||1=supfB(−s)σ2n+1(fA)(s)ds≤C.||fB||∞≤10Moreover,wedefinethesequenceoffunctionalsSn:C([0,1])→C,by1Sn(g)=σ2n+1(fA)(s)g(s)ds.0ThenSnarelinearoperatorswith||Sn||≤Cforalln≥1,and,byAlaoglu’sTheorem,thereexistsalinearboundedoperatorS:C([0,1])→CsuchthatSn→Sweakly.ApplyingnowRieszTheorem,thereexistsμ∈M([0,1])suchthat1S(g)=gdμ.0Itisclearthatμ=fA∈M([0,1]).

38October20,20138:49WorldScientificBook-9inx6invers*11*oct*201324MatrixspacesandSchurmultipliers:Matricealharmonicanalysis2.3.2ThecharacterizationofHankelmatricesDefinition2.15.LetA∈PM(2).ThenAiscalledaHankelmatrixifthereexiststhecomplexsequence(an)n∈N∗suchthat⎛⎞a1a2a3an⎜⎟⎜a2a3an⎟⎜⎟⎜a3an⎟A=⎜⎟.⎜an⎟⎜⎟⎝an⎠TosuchamatrixweassociateauniquepseudomeasuregAgivenby∞g(t)=ae2πiktwheret∈[0,1].Akk=1LikeinthecaseofToeplitzmatriceswestudywheneverthematrixAbelongstothespacesB(2),orM(2).Morespecifically,wehavethatTheorem2.16.LetAbeaHankelmatrix.ThenA∈B(2)⇔gA∈BMO.Proof.LetTAbethematricealoperatorassociatedtoA11TA(r⊗h)(u,v)=LA(t,x)r(u−x)h(v−t)dxdt00∞11=LA(t)h(v−t)dtr(u−x)e2πikxdx.k00k=1Since11LA(t)h(v−t)dt=g(t)e−2πi(k−1)th(v−t)dt,kA00wehavethatTA(r⊗h)(u,v)=1∞1g(t)e2πith(v−t)r(x)e−2πikxdxe2πik(u−t)dt.A00k=1

39October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory25Thus,1T(r⊗h)(u,v)=g(t)h(v−t)e2πitr(u−t)dt,(2.5)AA0or,denotingbyhv(t)=h(v+t),andbyru(t)=r(u+t),1T(r⊗h)(u,v)=g(t)e2πit(hr)(−t)dt.AAvu0Nextweprovetheimplication⇒.IfA∈B(2),wehavethat,r,h∈P([0,1]),1|T(r⊗h)(0,0)|=g(t)e2πit(hr)(−t)dt≤C||r||||h||.AA220InviewofthetheoryofHardyspaces,thislastfactisequivalenttothefollowinginequality:1supg(t)e2πitf(−t)dt≤C,Af∈H1,||f||1≤100i.e.,tothat||gA||BMO≤C.FortheconverseimplicationletgA∈BMO.Wehavethat12πit|TA(r⊗h)(u,v)|=gA(t)e(hvru)(−t)dt≤||gA||BMO||hvru||H10≤||gA||BMO||h||2||r||2,i.e.,||TA||≤||gA||BMO.Theorem2.17.LetAbeaHankelmatrix.Wehave:i)A∈M()impliesthatg∈M(H1,H1),wherethislastspaceisthe2A00spaceofallFouriermultipliersofH1.0ii)gA∈M([0,1])impliesthatA∈M(2).Proof.LetB∈B(2).Then∞11T(r⊗h)(u,v)=LA∗LB(t)h(v−t)r(u−x)e2πikxdtdxA∗Bkk00k=1111=g(s)e2πisLB(t−s)e2πik(x−s)h(v−t)r(u−x)dtdxds.Ak000

40October20,20138:49WorldScientificBook-9inx6invers*11*oct*201326MatrixspacesandSchurmultipliers:MatricealharmonicanalysisThus,1T(r⊗h)(u,v)=g(s)e2πisT(r⊗h)(u−s,v−s)ds.(2.6)A∗BAB0i)WeassumethatA∈M(2).ThenthereexistsC>0suchthat,foreachHankelmatrixB,B∈B(2),with||B||B(2)≤1,wehavethat,forr,h∈P([0,1]),||TA∗B(r⊗h)||∞≤C||r||2||h||2.(2.7)Therefore,byusingtherelations(2.5),(2.6),and(2.7),wehavethat11g(s)e2πisg(t)e2πit(rh)(−t)dtds≤C||r||||h||.ABu−sv−s2200Wetakeu=v=0,ands+t=yintherelationabove,obtainingthat11g(s)g(y−s)e2πiy(rh)(−y)dyds≤C||r||||h||,AB2200or,equivalently,11supsupgB(s)gA(y−s)f(−y)dydsgB∈BMO;||gB||BMO≤1f∈H1;||f||1≤1000≤C.Thus,theoperatorS:H1→H1,A00givenby1SA(f)(s)=gA(s−y)f(y)dy,0isaboundedoperatorifandonlyifg∈M(H1,H1).A00ii)Accordingtorelation(2.7),foreveryB∈B(2),andforgA∈M([0,1]),wehavethat||TA∗B(r⊗h)||∞≤||gA||M||TB(r⊗h)||∞.SinceB∈B(2),itfollowsthat||TB(r⊗h)||∞≤CB||r||2||h||2,therefore,||TA∗B(r⊗h)||∞≤||gA||MCB||r||2||h||2.Hence,A∗B∈B(2)∀B∈B(2),thatisA∈M(2).Remark2.18.AnequivalentconditiontothestatementthataHankelmatrixA∈M(2)wasgivenbyG.Pisierin[81]:A∈M()⇔g∈M(H1(S1),H1(S1)),2AwhereH1(S1)istheoperatortraceclass-valuedHardyspace.

41October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory272.3.3ThemaintriangleprojectionInwhatfollowswepresentanewproofofaresultin[49],namely:Whatisthegrowthratewithrespecttonoftheexpressionsup||Pn(A)||B(2),||A||B(2)≤1whereaijifi+j≤n+1Pn(A)=(bij)i,j≥1,forbij=0otherwise,withA=(aij)i,j≥1.Theorem2.19.LetPnthetriangleprojectionoftheordern.Thenwehavethatsup||Pn(A)||B(2)=O(logn)forn→∞.||A||B(2)≤1Moreover,thereexistsC>0suchthatsup||P(A)||≥Clogn∀n∈N∗.nB(2)||A||B(2)≤1Proof.LetA∈B(2)andx∈2(N).Then,byusingthecorrespondencesA∞2πijtA↔(Lk)k≥1,andx=(xj)j≥1↔h(t)=j=1xje,asinSection2.1,wehavethefollowingequality:n12||P(A)x||2=LA∗D(t)h(−t)dt,n2kn+1−k0k=1wherenD(t)=e2πiktnk=−nistheDirichlet’skernel.Therefore,n112||P(A)x||2=D(s)LA(t−s)h(−t)dtdsn2n+1−kk00k=1n1A2≤||Dn+1−k||1|Dn+1−k(s)|Lk∗h(−s)ds0k=1n12≤sup||D||sup|D(s)|LA∗h(−s)dsn+1−k1n+1−kk1≤k≤n01≤k≤nk=1

42October20,20138:49WorldScientificBook-9inx6invers*11*oct*201328MatrixspacesandSchurmultipliers:Matricealharmonicanalysis$$$$≤sup||D||$sup|D(s)|$V2.n+1−k1$n+1−k$LA,h∞1≤k≤n1≤k≤n1Since||Dn||1=O(logn),forn→∞,itremainstoprovethat11|sinks|sup|Dk(s)|ds=O(logn)⇔supds=O(logn).01≤k≤n01≤k≤nsInordertoprovethiswetakegs:[1,u]→R+,givenby|sinxs|gs(x)=,swheres∈(0,1].Therearetwodistinctcases:1)s>1,whichimpliesthat2n1gs(x)<∀x∈[1,n],sand2)s≤1whichimplies00,suchthatsup||Pn(A)||B(2)≥Blogn,||A||B(2)≤1

43October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory29weobservethatsup||Pn(A)||B(2)=sup||Tn(A)||B(2),||A||B(2)≤1||A||B(2)≤1whereA=(aij)i,j≥1,and⎛⎞nn+1a1100...00...⎜⎟⎜a21a220...00...⎟⎜⎟⎜⎜............⎟⎟.........Tn(A)=⎜⎟.n⎜⎜an1an2an3...ann0...⎟⎟n+1⎜000000...⎟⎝⎠.....................WeconsidernextHilbert’smatrixgivenby⎛11⎞01...23⎜−1011...⎟⎜2⎟H=⎜−1−101...⎟,⎝2⎠...............12nandxn=(1,1,...,1,0,...).Then||H||B(2)<∞,and2222111||Tn(H)xn||2=1+1++···+1++···+22n−1n22222≥Clog2+log3+···+logn=Clogxdx∼Cnlogn.2Therefore||T(H)x||2nlog2n||T(H)||2≥nn2≥C,nB(2)||x||2nn2thatis−1sup||Tn(A)||B(2)≥||H||B(2)||Tn(H)||B(2)≥Clogn.||A||B(2)≤1

44October20,20138:49WorldScientificBook-9inx6invers*11*oct*201330MatrixspacesandSchurmultipliers:Matricealharmonicanalysis2.3.4B(2)isaBanachalgebraundertheSchurproductNextwegiveadifferentproofofanoldresultofI.Schur(seealso[11]).Theorem2.20.B(2,∗)isaBanachalgebra.Proof.LetA,B∈B(),and(LA)(LB),bethesequencesof2kk≥1,kk≥1,functionsassociatedtoA,andtoB,respectively.Wehavethat∞12||(B∗A)x||2=LA∗LB(t)h(−t)dt2kk0k=1∞112=LA(s)LB(t−s)h(−t)dtdskk00k=1∞11LA2B2≤k(s)dsLk∗h(−s)ds,00k=1wherex=(xj)j≥1∈2(N),and∞h(t)=xe2πijt∈H2([0,1]).j0j=1Therefore,1/21/2121||(B∗A)||≤supLA(s)dsV2(s)ds2kLB,hk≥100≤||A||B(2)||VLB,h||2.Finally,||B∗A||B(2)≤||A||B(2)sup||VLB,h||2≤||A||B(2)||B||B(2).||h||2≤1NotesThemainideaofthischapteristheinterpretationofaninfinitematrixasasequenceoffunctions(or,moregenerally,distributions).Wetakeinthiswaytheadvantageofarichclassofnotionsandtechniques,whichareusualinfunctiontheory.Forinstance,inSection2.1wedefinethenotionsofa

45October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Integraloperatorsininfinitematrixtheory31squarefunction,respectivelyofamatricealoperatorassociatedtoagivenmatrixA.ThemainresultofSection2.1isofcourseTheorem2.8.InSection2.2thecentralresultisProposition2.11,whichisthenon-periodicalanalogueofTheorem2.8.Section2.3isdedicatedtoapplicationsofresultsfromSection2.1.Inthiswaywegivedifferentproofsofsomeclassicalresultsfrominfinitematrixtheory:Theorem2.13isofcoursewell-known(see[94]).Theorem2.14isknownasBennett’sTheorem(see[11]).Theorem2.16,knownasNehari’sTheorem(see[65]),isherepresentedwithanewproof.Finally,wementionTheorem2.19,firstprovedin[49].Theorem2.20wasdiscoveredapparentlybyI.Schur(see[11]foramoregeneralresult).

46May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

47October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Chapter3MatrixversionsofspacesofperiodicalfunctionsAninterestingproblemconcerninginfinitematricesisthefollowing:LetA∈B(2).WhenisthematrixAapproximablebymatricesoffinitelybandtypeintheoperatornorm||·||B(2)?InwhatfollowswedealwiththisprobleminsomespacesofinfinitematriceswhichcanberegardedasextensionsofclassicalBanachspacesoffunctionsC(T)andL1(T).TheseBanachspacestogetherwiththespacesB(2)andM(2)areofinterestinordertodevelopsomeresultsextendingknowntheoremsofclassicalharmonicanalysisintheframeworkofmatrices.OnemainaimofthepresentchapteristoextendintheframeworkofmatricesFejer’stheoryforFourierseries.(Seeforinstance[96].)AsitwasstatedintheIntroductionwehaveasimilaritybetweentheexpansionintheFourierseriesf=aeikxofaperiodicalfunctionfkkonthetorusTandthedecompositionA=k∈ZAk,whereAkisthekthdiagonalofAfork∈Z.Moreover,thereisasimilaritybetweentheconvolutionproductf∗goftwoperiodicalfunctionsandtheSchurproductoftwomatricesAandB,C=A∗B.FirstwementionthefollowingresultsobtainedbyFejer,whichhavebeenguidingforourinvestigation:(A)Afunctionf(θ)=aeikθiscontinuousonT(thatisf∈k∈ZkC(T))ifandonlyiftheCesarosumsk|k|σ(f)=a1−eikθnkn+1k=−nconvergeuniformlyonTtof.(B)Afunctionf(θ)=meikθ∈L1(T)ifandonlyifk∈Zk||σn(f)−f||L1(T)→0asn→∞.33

48October20,20138:49WorldScientificBook-9inx6invers*11*oct*201334MatrixspacesandSchurmultipliers:Matricealharmonicanalysis3.1PreliminariesInviewofFejer’sresult(A)itisnaturaltogivethefollowingdefinition:Definition3.1.LetA∈B(2).Wedenotebyσn(A)theCesarosumassoci-nn|k|atedtoSn(A):=k=−nAk,thatisσn(A)=k=−nAk1−n+1.ThenwesaythatAisacontinuousmatrixiflim||σn(A)−A||B(2)=0.n→∞LetusdenotebyC(2)thevectorspaceofallcontinuousmatricesandconsideronittheusualoperatornorm.NowrecallthatthespaceofallSchurmultipliersM(2)isacommutativeunitalBanachalgebrawithrespecttoSchurproduct.MoreoverwehaveBennett’stheorem(seeforinstanceTheorem2.14):TheToeplitzmatrixM=(cj−k)j,k,where(cn)n∈Zisasequenceofcom-plexnumbers,isaSchurmultiplierif,andonlyif,thereexistsa(bounded,complex,Borel)measureμon(thecirclegroup)Twithμ(n)=cnforn=0,±1,±2,....Moreover,wethenhave||M||M(2)=||μ||.Wealsomentionthefollowingwell-knownfact(seeTheorem2.13):TheToeplitzmatrixMrepresentsalinearandboundedoperatoron2ifandonlyifthereexistsafunctionf∈L∞(T)withFouriercoefficientsf(n)=mnforalln∈Z.Moreover,wehave||M||B(2)=||f||L∞(T).3.2SomepropertiesofthespaceC(2)Firstofallletusobservethefollowingfact:Remark3.2.ByFejer’stheorem(A)wehavethataToeplitzmatrixT=def(t)∈C()ifandonlyiff(θ)=teikθ∈C(T),andinthiskk∈Z2Tk∈Zkwaywecanseethatthenotionofacontinuousmatrixmayberegardedasananalogueofthatofacontinuousfunction.

49October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions35NowletC∞denotethespaceofallmatricesdefiningcompactoperators.Proposition3.3.C(2)isaproperclosedidealofB(2)withrespecttoSchurmultiplicationwhich,initsturn,containsC∞properly.Proof.Wehave:$$n$$$|k|$||σn(A)||B(2)=$Ak1−$≤||Mn||M(2)||A||B(2),$n+1$k=−nB(2)whereMnisthen-bandtypeToeplitzmatrixwiththeentries⎧⎨|j−i|1−if|j−i|≤n,mij=n+1⎩0otherwise.HenceC(2)isaclosedsubspaceofB(2).NowweobservethatforA,B∈C(2),σn(A∗B)=σn(A)∗Bandthenwehavethat,forA∈C(2),B∈B(2)||A∗B−σn(A∗B)||B(2)=||[A−σn(A)]∗B||B(2)≤||A−σn(A)||B(2)·||B||M(2)≤||B||B(2)·||A−σn(A)||B(2).Herewehaveusedthesimplefactthat||B||M(2)=||B∗E||M(2)≤||B||B(2)·||E||M(2)=||B||B(2),E=(Eij)whereEij=1foralli,j∈N,and||E||M(2)=1.Hence,C(2)isaclosedidealofB(2)withrespecttoSchurmultipli-cation.NextwenotethatC(2)isaproperidealofB(2).Denotingbyeijthematrixwhosesinglenon-zeroentryis1ontheithrowandonthejthcolumn,weconsiderthematrixA=k∈NAk,whereA=e,k≥0,whichbelongstoB(),since(AA∗)1/2=I(Iiskk+1,2k+12theidentitymatrix).Moreover,$$n$$$1$k||σn(A)−A||B(2)=$Ak+kAk$=max∨1=1$n+1$k≤nn+1k>nk=0B(2)forallnand,thus,A/∈C(2).NowletA∈C∞.Denotingbyaiji,j≤nPn(A)(i,j)=0otherwise,

50October20,20138:49WorldScientificBook-9inx6invers*11*oct*201336MatrixspacesandSchurmultipliers:Matricealharmonicanalysiswehavethat||Pn(A)−A||B(2)→0,asn→∞.But,byBennett’stheorem,wehavethatfork>n:$$$n||$$$||Pn(A)−σk(Pn(A))||B(2)=$(Pn(A))$$k+1$=−nB(2)$$$n||$$iθ$≤$e$·||Pn(A)||B(2)→0ask→∞.$k+1$=−nM(T)Hence,Pn(A)∈C(2)foralln∈Nand,consequently,C∞⊂C(2).Sinceitiseasytoseeandwell-knownthataToeplitzmatrixdoesnotrepresentacompactoperator,bythepreviousremarkitfollowsthatC∞isapropersubspaceofC(2).Theproofiscomplete.3.3AnothercharacterizationofthespaceC(2)andrelatedresultsWewillgiveanothercharacterizationofthespaceC(2)byusingcontinuousvector-valuedfunctionsbutfirstwenotethefollowingsimplefact:Remark3.4.ConsiderthefunctionfA:T→B(2)givenbyfA(t)=A∗ei(j−k)t.Thenj,k≥0||fA(θ)||B(2)=||A||B(2)forallθ∈T.Indeed,byBennett’smultipliertheorem,wehavethat||fA(θ)||B(2)≤||A||B(2)||δ−θ||=||A||B(2),whereδθ∈M(T)denotetheDiracpointmassatθ∈T.Similarly,sinceA=f(θ)∗ei(j−k)t,wehavethat||A||≤||f(θ)||.Aj,k≥0B(2)AB(2)AneasyconsequenceofthisremarkisthatfAiscontinuousonTifandonlyifitiscontinuousatonesinglepoint.NowweaskourselveshowthematrixAshouldbeinorderthatthefunctionfAshallbecontinous.Theanswertothisquestionisasfollows:Theorem3.5.LetAbeaninfinitematrix.ThenfAisaB(2)-valuedcon-tinuousfunctionifandonlyifA∈C(2),withequalityofthecorrespondingnorms.

51October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions37Proof.ByRemark3.4itfollowsthat||σn(fA)−fA||C(T,B(2))=||σn(A)−A||B(2).NowreasoningasintheproofofFejer’sresult(A)(seeforinstance[41])wegetthatforacontinuousfunctionfA:T→B(2)itfollowsthat||σn(A)−A||B(2)→0,asn→∞.ThusA∈C(2).TheconverseimplicationfollowseasilyfromRemark3.4.Nowweshallstudythefollowingquestion:WhatcanwesayaboutsubspacesofM(2)inconnectionwiththemultiplierproperty?ThefollowingtheoremgivesajustificationofintroducingC(2)andalsoapartialanswertotheabovequestion.ItisthematricealanalogueofTheorem11.10,Chap.IV-[96].Theorem3.6.TheToeplitzmatrixM=(mk)k∈ZisaSchurmultiplierfromB(2)intoC(2)iffmeikθ∈L1(T).kk∈ZProof.Ifweidentifyf∈L∞(T)withitscorrespondingToeplitzoper-atorTf=f(j−k)inB(2),thenitisstraightforwardtoseethatj,k≥0aSchurmultiplierM=(mj−k)j,k≥0mappingB(2)intoC(2)inducesaFouriermultipliersequencem={m}∞mappingL∞(T)intoC(T),nn=−∞whichisknowntocorrespond,inthemannerindicatedinthestatementabove,toafunctionfromL1(T)(see[96]).Theconversefollowsalsobythesamelines.Guidedby[11]weproposeformatricesasimilarnotiontothatofLebes-gueintegrablefunctions.Definition3.7.WesaythataninfinitematrixAisanintegrablematrixifσn(A)→Aasn→∞inthenormofM(2).Thespaceofallsuchmatrices,endowedwiththenorminducedbyM(),isdenotedbyL1().22OfcourseL1()isaBanachspace.2Remark3.8.IfA∈L1(),thenitfollowsthatA∗B∈C()forall22B∈B(2).

52October20,20138:49WorldScientificBook-9inx6invers*11*oct*201338MatrixspacesandSchurmultipliers:MatricealharmonicanalysisIndeed,forB∈B(2),inviewofthefactthat||σn(A∗B)−A∗B||B(2)≤||σn(A)−A||M(2)·||B||B(2)→0,asn→∞,wefindthatA∗B∈C(2).NowitisclearthatL1()isaclosedidealofM()withrespecttothe22Schurproduct.WehavethefollowinganalogueoftheRiemann-LebesgueLemma:Lemma3.9.LetM∈L1().Then2lim||Mk||L1(2)=0.|k|→∞Proof.ItisclearfromDefinition3.7that,forany>0,thereisanumbern()suchthat,for|k|≥n(),itfollowsthat||Mk||L1(2)≤andtheproofiscomplete.Remark3.10.ItiseasytoseethatforadiagonalmatrixAk,k∈Z,wehavethat||Ak||B(2)=||Ak||M(2).Thus,inLemma3.9wecantake||Mk||B(2)insteadof||Mk||L1(2).InviewofTheorem3.5andRemark3.8itisnaturaltoask:IfAisaSchurmultiplierwhichmapsB(2)intoC(2)doesitfollowthatA∈L1()?2Theanswertotheabovequestionisnegative.Infact,wehavethatExample3.11.LetAbethefollowingmatrix:⎛⎞111...A=⎜000...⎟⎝⎠.............ThematrixAisaSchurmultiplierwhichmapsB(2)intoC(2)butitdoesnotbelongtoL1().2Infact,AisaSchurmultiplierwiththepropertythatA∗B∈C(2)forallB∈B(2)sincethematrixA∗Bhasrank1andthereforerepresentsacompactoperatorandconsequentlyitbelongstoC(2).Moreover,AdoesnotbelongtoL1()byLemma3.9.2

53October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions39ThereforetheBanachspace(B(2),C(2))ofallmultipliersfromB(2)intoC()isdifferentfrombothM()andL1().Thus,itseemsthatthis222spacedeservestobestudiedinmoredetail.Ontheotherhandthespace(C(2),C(2))ofallinfinitematricesAsuchthatA∗B∈C(2)forallB∈C(2)canbedescribedeasily.Morepreciselywehave:Theorem3.12.(C(2),C(2))isexactlythespaceM(2)ofallSchurmul-tipliers.Proof.Sinceσn(A∗M)=σn(A)∗MitfollowseasilythatM∈(C(2),C(2))ifM∈M(2)andA∈C(2).Conversely,assumingthatM∈(C(2),C(2)),wehaveforA∈B(2)that||M∗σn(A)||B(2)≤C||σn(A)||B(2).Moreover,σn(A)→AintheweaktopologyofoperatorsinB(2),thatis<σn(A)x,y>→forallx,y∈2,where<·,·>isthescalarproductin2(use2-sequenceswithafinitenumberofnonzerocomponentsandastandardapproximationargument).Thisyieldsthat||M∗A||B(2)≤C||A||B(2),thatisMisaSchurmultiplier.Theproofiscomplete.NextwegiveacharacterizationofanintegrablematrixinthespiritofTheorem3.5:Theorem3.13.LetA∈M()andf(θ)=A∗(ei(j−k))forθ∈T.2Aj,k≥0ThenfAisapointwisewell-definedfunctionfA:T→M(2)suchthat||fA(θ)||M(2)=||A||M(2)forallθ∈T.Furthermore,fA∈C(T,M(2)),thatisfiscontinuous,ifandonlyifA∈L1().A2Proof.Forμ∈M(T)letusintroducethenotationTμfortheToeplitzmatrixwithsymbolμ,thatisT=(μ(j−k)),whereμ(n)=e−intdμ(t)μj,k≥0isthenthFouriercoefficientofμ.NotethatfA(θ)=A∗Tδ0,whereδ0∈M(T)denotestheunitpointmassatθ∈T.ByBennett’smultipliertheorem(see[11])weobtainthat||fA(θ)||M(2)≤||A||M(2).Similarly,sinceA=fA(θ)∗Tδ0,wehavethat||A||M(2)≤||fA(θ)||M(2).AssumenextthatA∈L1().Wethenhavethat2||σN(fA)(θ)−fA(θ)||M(2)=||σn(A)−A||M(2)→0.

54October20,20138:49WorldScientificBook-9inx6invers*11*oct*201340MatrixspacesandSchurmultipliers:MatricealharmonicanalysisThus,σn(fA)→fAuniformlyandweobtainthatfA∈C(T,M(2)).AssumenowthatfA∈C(T,M(2)).WeconsiderthentheM(2)-valuedintegral1fN(θ)=fA(θ−t)KN(t)dt,θ∈T,2πTwhereKNistheNthFejerkernel.ItisstraightforwardtoseethatfN−fAconvergesinC(T,M(2))asN→∞.AneasycomputationyieldsthatN|n|f(θ)=1−f((n)einθ,NAN+1n=−Nwheref(A(n)isthenthFouriercoefficientoffA.TocomputetheFouriercoefficientf(A(n)weneedonlytoobservethattheoperationM→mjkoftakingthe(j,k)thentryisaboundedlinearfunctionalonM(2).Bythisweclearlyhavethatf(A(n)=An.Summingup,wehaveshownthatN|n|lim1−Aeinθ=f(θ)nAN→∞N+1n=−NinC(T,M(2)).Forθ=0thisyieldsσN(A)→AinM(2).Theproofiscomplete.WealsoremarkthatfiscontinuousonTifandonlyifitiscontinuousatonesinglepoint.Thisisclearbythefirstassertionoftheabovetheorem.ThenextremarkisaneasyconsequenceofFejer’stheory.Remark3.14.LetAbeaToeplitzmatrix.ThenA∈L1()ifandonlyif2itrepresentsafunctionfromL1(T).Nowwerecallthefollowingwell-knownresult(see[41],Chapter2).AfunctionFonTbelongstoL∞(T)ifandonlyifsup||σn(f)||L∞(T)<∞.nWehavethefollowingmatrix-versionofthepreviousresult:Proposition3.15.LetAbeaninfinitematrix.ThenAbelongstoB(2)ifandonlyifsup||σn(A)||B(2)<∞.n

55October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions41Proof.Assumethatsupn||σn(A)||B(2)<∞.Then,byreasoningasintheproofofTheorem3.12,wegeteasilythatA∈B(2).TheconverseimplicationcanbeprovedbyusingthesameargumentsasintheproofofProposition3.3.Proposition3.16.A∈M(2)ifandonlyifsup||σn(A)||M(2)<∞.nProof.Assumethatsup||σn(A)||M(2)<∞nandfixanarbitraryB∈B(2).Wethenhavethat||σn(B∗A)||B(2)≤||B||B(2)sup||σn(A)||M(2)<∞.nItfollowsthatσn(B∗A)→B∗AintheweaktopologyofoperatorsinB(2).(SeetheproofofTheorem3.12.)Inparticular,B∗A∈B(2).SinceBisarbitrarythismeansthatAisaSchurmultiplier.Conversely,letA∈M(2).Thenσn(A)=A∗σn(M),whereM=(mi)i∈Zwithmi=1.Thus,byusingBennett’stheorem,wefindthat||σn(A)||M(2)≤||A||M(2)·||σn(M)||M(2)≤||A||M(2).Theproofiscomplete.3.4AmatrixversionforfunctionsofboundedvariationFollowing[75]weintroducenowamatrixversionoffunctionswithboundedvariationandprovetheanalogueofclassicalJordan’stheoremontrigono-metricseries.WedenotebyAthematrixkA.k∈ZkDefinition3.17.AsintheFourierseriesframeworkwesaythatamatrixAisamatrixofboundedvariationifA∈M().2ThespaceofallmatricesAofboundedvariationisdenotedbyBV(2)anditisaBanachspaceendowedwiththenorm||A||=||A||+||A||.BV(2)0B(2)M(2)

56October20,20138:49WorldScientificBook-9inx6invers*11*oct*201342MatrixspacesandSchurmultipliers:MatricealharmonicanalysisFinally,wesaythatamatrixAisabsolutelycontinuousifA∈L1()2andA0∈B(2).LetusremarkthatbyLemma3.9itfollowsthat||A||=o1forkB(2)kanabsolutelycontinuousmatrixA.Moreover,itiseasytoseethatforamatrixM∈M(2)wehavethat||Mk||M(2)≤||M||M(2),forallk∈Z.Ontheotherhand,itispossiblethat||Mk||M(2)→0,as|k|→∞.ForinstanceifMcoincideswithE,thematrixhavingonly1asentries,thenobviouslylim||Mk||M(2)=1.|k|→∞NextweintroduceanotherinterestingsubspaceofB(2)byU(2)={A∈C(2);suchthat||Sn(A)−A||B(2)→0asn→∞}endowedwiththenorm||A||U(2)=sup||Sn(A)||B(2).nObviously,U(2)⊂C(2).Usingthewell-knownexampleofDuBoisRaymond(see[96])thereexistsaToeplitzmatrixfromC(2)whichdoesnotbelongtoU(2),sothatU(2)isinfactapropersubspaceofC(2).Buttherearemoresophisticatedsuchexamplesofmatrices.Forin-stancewecanadaptanexamplefoundbyFejertotheframeworkofinfinitematriceswhicharenotToeplitzmatrices.Wesketchinwhatfollowsthisexample:Foranyintegersn,μandforallx∈Rweput(see[96]-page168)nsinkxQ(x,μ,n)=−cos(μ+n)x.kk=1Sincethepartialsumsoftheseriessinx+1sin2x+...arelessthana2constantCinabsolutevalue,wehavethat|Q|≤C,foreveryx,μ,n.IfwedenotebyQ(μ,n)theinfiniteToeplitzmatrixassociatedtotheperiodicalfunctionQ(x,μ,n),thenwegetthat||Q(μ,n)||B(2)≤C.Now,foreveryμ,n,wechoosethedecreasingsequencesaμ,n={al},suchthatthereexist0

57October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions43∞ll+1Sincel=1|aμ,n−aμ,n|<∞,forallμ,n,wegetthat[aμ,n]∈M(2),(see[18]),and||[aμ,n]||M(2)≤b−a,forallμ,n.Let{nk},{μk}besetsofintegerswhichweshalldefineinamoment,andletαk>0,α1+α2+···<∞.Wedefinethematrices[aμk,nk]asdescribedabove.Thentheseries∞αk[aμk,nk]∗Q(x,μk,nk)k=1convergesuniformlytoacontinuousmatrix,whichwedenotebyG∈C(2).Ifμk+2nk<μk+1(k=1,2,...),thenQ(x,μk,nk)andQ(x,μl,nl)donotoverlapforn=l.3Ifα=k−2,μ=n=2k,thenthecontinuousmatrixGdefinedkkkabovehasanexpansionG=k∈ZGkwhichdoesnotconvergeintheoperatornorm.Indeed,thesequencea={(al)−1}isincreasingμk,nkμk,nkl≥1andalsobounded,and,therefore,[aμk,nk]∈M(2),anditsnorminM(2)islessthana−1−b−1.Thus,√)*α(logn)/2

58October20,20138:49WorldScientificBook-9inx6invers*11*oct*201344MatrixspacesandSchurmultipliers:Matricealharmonicanalysisnnnn=||1+σn+k−1(A)−1+A−σn−1(A)+A||B(2)kkkknn≤1+||σn+k−1(A)−A||B(2)+||σn−1(A)−A||B(2)→0.kkThus,ifA∈C(2),thatisσν(A)→A,asν→∞,andA∈BV(2),thenitfollowsthat|k|||Ak||B(2)≤Cforallk∈Zand,bythesecondrelationabove,wegetthatn+k−1n+k−11k−1||σn,k(A)−Sn(A)||B(2)≤||Aν||B(2)≤C≤C.νn|ν|+1n+1Now,if>0andk=[n]+1,thenCk−1≤C.Sincen

59October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions453.5.1IntroductionTheclassicalformofHaar’stheorem.LetTbetheone-dimensionaltorusidentifiedwiththeinterval[0,2π).NowweconsidertheHaarL2(T)-normalizedfunctionshgivenbyh(t)=1k0fort∈Tand,forn=2k+m,k≥0andm∈{0,...,2k−1},by⎧⎪⎪2k/2,t∈Δ(k+1),⎪⎪2m⎪⎨h(t)=−2k/2,t∈Δ(k+1),n⎪⎪2m+1⎪⎪⎪⎩(k)0,t∈T\Δm,whereΔ(k)m·2π,m+1·2π).m=[2k2kWecannowstatethefollowingwell-knowntheoremofapproximationofcontinuousfunctionsonT(i.e.periodicalcontinuousfunctionson[0,2π])bymeansofHaarfunctions(periodicallyextendedonR)duetoHaar(see[37]).TheoremA.IffisacontinuousfunctiononT(i.e.iff∈C(T))andif>0,thenthereexistsaHaarpolynomialofdegreen()∈Nn−1Sn(f)=αkhk,αk∈C,k=0suchthat||f−Sn(f)||L∞(T)<.TranslationofTheoremAtoamatricealframework.Thefollowingresultaswellasthereafterremarkconstitutethestartingpointofthewholetheorypresentedhere(see[18]).+∞Theorem0.AToeplitzmatrixA=(ak)k=−∞belongstoB(2)ifandonlyifthereexistsauniquefunctionf∈L∞(T)whoseFouriercoefficientsAf((n)=12πf(t)e−intdtareequaltoa,forn∈Zand,moreover,A2π0n||A||B(2)=||fA||L∞(T).

60November15,20139:58WorldScientificBook-9inx6invers*11*oct*201346MatrixspacesandSchurmultipliers:MatricealharmonicanalysisInordertodevelopthetheoryweconsiderinthepreviousresulttwodifferent”geometric”directionstobefollowed.Model1:Diagonalmatrix.Aswenotedalreadyintheintroductorychapter,foraninfinitematrixA=(aij),andanintegerk,possiblynegative,wedenotebyAkthematrixwhoseentriesa0aregivenbyi,j0ai,jifj−i=k,ai,j=0otherwise.ThenAkwillbecalledthekth-diagonalmatrixassociatedtoA.Intheprecedingtheoremweremarkthatthereisaone-to-onecorre-spondencebetweenAandfb(k)forA∈B(`)andf∈L∞(T).kA2Consequently,wemayimagine(Ak)k∈Z,asthe”matricealFouriercoef-ficients”associatedtothematrixA.Model2:Cornermatrix.Inthesequelweuseanothernotation,whichismoreappropriateforouraims.FortheentriesofthematrixA,weputlal,l+k,k≥0,l=1,2,3...,ak=al−k,l,k<0,l=1,2,3,...,anddenoteAsometimesasA=(al).kl≥1,k∈ZLetA(l)=(bm),wherel∈N∗,bethematrixgivenbykk∈Z,m≥1alifm=l,bm=kk0ifm6=l.WecallthematrixA(l),thelth-cornermatrixassociatedtoA.Now,ifforanycorner-matrixA(l)=(bm)weassociateakk∈Z,m≥1distributiononT,denotedbyfsuchthatbl=fb(k),weget,incaselklA∈T∩B(`),thatf=f∈L∞(T),foralll∈N∗.2lUsingthemodels.a)Model1.InthiscasewerecallthatAkplaystheroleofthe“kthFouriercoefficientofthematrixA.”Theorem0allowsustowritetheformula[T∩B(`)]∗=L∞(T),2whereby[H]∗wedenotetheimageofthespaceHofmatricesbythecorrespondenceA→fA.

61October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions47Remark3.19.Forbrevitywewriteinwhatfollowsequationslikethepre-viousoneinthefollowingmanner:T∩B()=L∞(T),2C(2)∩T=C(T).b)Model2.(l)WecanidentifythematrixA=(A)l∈N∗withitssequenceofassoci-ateddistributionsf=(fl)l∈N∗,writingthisfactasA=Af.ByTheorem0wehavethefollowingcorrespondences:f∈L∞(T)ifandonlyifA∈T∩B()f=(f,f,f,...)f2g∈L∞(T)ifandonlyifA∈T∩B()g=(g,g,g,...).g2Then,ofcourse,itfollowsthatfg∈L∞(T)ifandonlyifA∈T∩B()wherefg=(fg,fg,fg,...).fg2WerecallthatthematrixA=(aij)issaidtobeofn-bandtypeifaij=0for|i−j|>n.Havingthesenotionsinmindweintroduceacommutativeproductofinfinitematrices:Definition3.20.LetA=AfandB=Agbetwoinfinitematricesoffinitebandtype.WeintroducenowthecommutativeproductgivenbyAB:=Afg.Remark3.21.(1)WementionthatinthepreviousdefinitionsinceA=Af,andB=Agareinfinitematricesoffinitebandtypeityieldsthatfandgaretrigonometricpolynomials,andwemayconsidertheproductfg.2)ThisproductcanbedefinedalsoforallmatricesA,B∈B(2),butABdoesthennotingeneralbelongtoB(2)ascanbeeasilyseen.3)Ofcourse,ifAf,Ag∈T∩B(2),thenitfollowsthatAfAg=Afg∈T∩B(2).

62October20,20138:49WorldScientificBook-9inx6invers*11*oct*201348MatrixspacesandSchurmultipliers:MatricealharmonicanalysisWeendthepresentationofthismodelbyconsideringanimportantparticularcase:Letα=(α1,α2,α3,...)beasequenceofcomplexnumbersandB=Af∈B(2),wheref=(f1,f2,...).ConsideringαasasequenceofconstantfunctionsonT,weget,byDefinition3.20,thatAαB=Aαf,whereαf=(α1f,α2f,...).12ForbrevitywedenoteAαBbyαB.InwhatfollowsitwillbeimportanttoknowmoreaboutthesequencesαsatisfyingtheconditionB∈B(2)⇒αB∈B(2).Actually,theentirenextsubsectionwillbedevotedtothisquestion,butforthemoment,forunderstandingitsimplicationswewillrewritetheoperationunderadifferentform.Weassociatetoanysequenceα=(α1,α2,...),thematrix[α]whoseentries[α]lareequaltoαl,forl≥1andk∈Z.kThenitisclearthatαB=[α]∗B.Definition3.22.Wesaythatthesequenceα∈msifithasthepropertythatαB∈B(2)∀B∈B(2),or,equivalently,[α]∈M(2).Onmsweconsiderthenorm||α||ms:=||[α]||M(2).Thenmsisauni-talcommutativeBanachalgebrawithrespecttousualmultiplicationofsequences.Remark3.23.Anyconstantcomplexsequenceα=(α,α,...)belongstoms.InordertogetanextensionofHaar’stheoremwehavetofindtheappropriateanaloguesinthematrixcontext.Theyaresummarizedbelow:ThefunctioncaseThematrixcase1.norm||·||L∞(T)norm||·||B(2)2.spaceC(T)spaceC(2)3.multiplicationofafunctionbyascalarmultiplication.

63October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions49Thecorrespondencegivenby3.becomesmoretransparentifweremarkthat,forα∈Candforf∈L∞(T),denotingbyαthesequence(α,α,...),andbyftheconstantsequence(f,f,...),wegetthatαAf=[α]∗Af=Aαf.DenotingbyHktheToeplitzmatrixassociatedlikeinTheorem0totheHaarfunctionhk,fork=0,1,2,...andbySn(f)theconstantsequencen−1(Sn(f),Sn(f),...),whereSn(f)=k=0αkhk,forf∈C(T),αk∈C,andk∈{0,n−1},wegetthefollowingtranslationofTheoremAintheToeplitzmatricessetting:TheoremB.LetA=Af∈C(2)beaToeplitzmatrixandlet>0.Thenthereisamatricealpolynomialgivenbyn−1n−1ASn(f)=αkHk=αkHkk=0k=0suchthat||A−ASn(f)||B(2)<,whereαk=(αk,αk,...).NowitisnaturaltoaskourselvesabouttheexistenceofaclassofmatriceslargerthanC(2)∩TsuchthatTheoremBstillholds.TheaimofournextTheoremistogiveananswertothisquestion.Moreprecisely,weprovethefollowingtheoremalsoformulatedinourpreface:TheoremC.LetA=(al)beamatrixbelongingtoC()suchkl≥1,k∈Z2thatallsequencesa=(al),k∈Zbelongtotheclassms.kkl≥1Then,forany>0thereisann∈N∗andsequencesα∈ms,kk∈{0,...,n−1}suchthatn−1||A−αkHk||B(2)<.k=0Itisalsoworthwhiletomentionthefollowingopenproblem:Openproblem.DoesTheoremCstillholdifthematrixAsatisfiesonlytheconditionA∈C(2)?Ifnot,whatisthebestversionofTheoremCinthiscase?

64October20,20138:49WorldScientificBook-9inx6invers*11*oct*201350MatrixspacesandSchurmultipliers:Matricealharmonicanalysis3.5.2AboutthespacemsAsweremarkedintheprevioussubsection(seealsothestatementofThe-oremC)thespacemsplaysanimportantroleforourtheoryand,conse-quently,itisdesirabletoknowmorefactsaboutit.Inthiscontext,weagainnotethatanyconstantsequencebelongstoms(seeRemark3.23).Ourprimarygoalhereistoprovethatthisalgebraisfarmorerichthanthat;thisrichnesswillquantifythelevelofextensionofthetheoremofHaarinthematrixcase,sinceinthefunctionscase,correspondingtoToeplitzmatrices,(seeTheoremB)thealgebramsisreducedtoexactlytheconstantsequences.Hereisanoutlookforthissubsection:Wegivesomesufficientconditionsforasequencetobelongtoms,fol-lowingtwocomplementaryways:Thefirstoneisbasedondefiningaparticularalgebrapmsandshowingthatpmsisintimatelyconnectedwithms.(SeeProposition3.25.)Asaconsequencewederivepropertiesformsdisplayingsomenecessaryandsomesufficientconditionsforasequenceinordertobelongtopms;(seeTheorem3.26)thesecondapproach(Theorem3.28)isinvolvedwiththestructureofmsratherthanofpms.ForaninfinitematrixA=(aij)i≥1,j≥1,wedefineitsuppertriangularprojectionPT(A)asfollows:ai,jifi≤j,PT(A):=0otherwise.Definition3.24.Asequenceb=(bn)n≥1belongstopmsifandonlyifB:={b}=PT([b])∈M(2).Thenpmsendowedwiththenorm||b||=||{b}||M(2)becomesaBanachalgebrawithrespecttousualproductofsequences.Proposition3.25.Letb=(bn)n≥1beasequenceofcomplexnumbers.Then1.b∈pms⇒b∈ms(sopms⊂ms.)2.ifwewrite(b1,b2,...,bn,...)=(b1,0,b3,0,...)+(0,b2,0,b4,...),or,equivalently,b=b10+b20,denotingbyb1=(b,b,...,b,...)and132n−1byb2=(b,b,...),wehavethatbi∈pms⇔bi0∈msfori∈{1,2}and24sobi∈pms,i∈{1,2}⇒b∈ms.Theproofisobvious,soweleaveoutthedetails.

65October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions51Wenowpasstoastudyofthealgebrapms.WeintroduceanewmethodforestimatingthenorminthespaceB(2).Weassociatetoeverysequencex=(xj)j≥1from2(N)thefunctionh(t)=∞xe2πijt∈H2([0,1]),whereH2([0,1])consistsofallfunctionsj=1j001h:[0,1]→CfromtheHardyspaceH2suchthath(t)dt=0.0IfA=(a)∈B(),wedefineL(t):=∞ae2πijt∈H2([0,1]).kj2kj=1kjItfollowsthat∞121A=sup(|Lk(t)h(s−t)dt|)2<∞foranys.(3.1)B(2)h≤102k=1SeeChapter2foraproofofthisformula.Theorem3.26.Letb=(bn)n≥1beasequenceofcomplexnumbers.1)If(in)n≥1isastrictlyincreasingsequenceofnaturalnumberswithi1=0,andzin:=maxin0suchthat||b||ms=||B||M(2)≤Rinf(||(zin)n≥1||2+||(zinln(in+1−in))n||∞).(in)ln2nn+p22)Ifb∈pms,thensupn≥1;p≥1nk=p|bk|<∞.3)If(|bk|)k≥1isadecreasingsequence,thenb∈pmsifandonlyif|b|=O1.klnkProof.1)LetA∈B(2)andx∈2(N).Byusingtherelation(3.1)forBAitfollowsthatthereexistsR1>0suchthat∞12(BA)x2≤R|b|2L(t)(h−S(h)(−t))dt,21kkk−10k=1whereSk(h)istheFourierpartialsumoforderk(i.e.ifDkistheDirichletkernel,thenSk(h)(t)=(hDk)(t)istheconvolutionofhandDk).Therefore,wehavethat∞1212||(BA)x||2≤2|b|2L(t)S(h)(−t)dt+L(t)h(−t)dt2kkk−1k00k=1∞12∞12≤2|b|2L(t)S(h)(−t)dt+2||b||2L(t)h(−t)dtkkk−1∞k00k=1k=1∞12≤2|b|2L(t)S(h)(−t)dt+2||b||2||A||2||h||2.kkk−1∞B(2)20k=1

66October20,20138:49WorldScientificBook-9inx6invers*11*oct*201352MatrixspacesandSchurmultipliers:MatricealharmonicanalysisLet(in)n≥1beastrictlyincreasingsequenceofnaturalnumberssuchthati1=0.Thenwehavethat||b||2≤||(z)||2(3.2)∞inn≥12and∞12|b|2L(t)S(h)(−t)dtkkk−10k=1∞in+112=|b|2L(t)S(h)(−t)dtkkk−1n=1k=in+10∞in+112≤2|b|2L(t)S(h)(−t)dtkkinn=1k=in+10∞in+112+2|b|2L(t)(S−S)(h)(−t)dtkkk−1inn=1k=in+10∞in+112≤2z2L(t)S(h)(−t)dt+inkinn=1k=in+10∞in+1122z2L(t)(S−S)(h)(−t)dt.inkkinn=1k=in+10Moreover,byusingtheformula(3.1),wegetthatin+112L(t)S(h)(−t)dt≤A2||h||2kinB(2)20k=in+1andin+112Lk(t)(Sk−1−Sin)(h)(−t)dt0k=in+1in+1−in12∼D(t){[LS−S(h)](−t)e2πinit}dtk−1k+inin+1in0k=1

67November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions53≤sup||Dk−1||L1(0,1)1≤k≤in+1−inin+1X−inZ1

2×|Dk−1(t)|Lk+in?Sin+1−Sin(h)(−s)ds0k=1Z!1≤Clog(in+1−in)sup|Dk−1(s)|ds01≤k≤in+1−inin+1X−in
2×|Lk+in?(Sin+1−Sin)(h)(·)|k=1L∞≤C0||A||2||(S−S)(h)||2|ln(i−i)|2.B(`2)in+1in2n+1nThus,using(3.2),wegetthatk(A?B)xkB(`2)≤RkAkB(`2)||h||2(||(zin)n≥1||2+||zinln(in+1−in)||∞).2)LetB∈M(`2).TakingA∈T∩B(`)suchthatal=1forallj∈Z\{0}andforall2jjl∈N∗andal=0foralll∈N,weobtainthatB?˜A˜∈B(`),where02111···b1b1b1b1···23b2b2b2b2···11B˜:=1··················andA˜:=22.bnbnbnbn···111··················32············n1ifk∈{p,...,n+p}Lettingxp=(xk)k≥1withxk=,wherep,n∈0otherwiseN∗arefixed,wegetthatnX+p2ln2(n+1)|b|2≤CB?˜A˜xn≤C(n+1).kp2k=pHence,2nX+pln(n+1)2sup|bk|<∞.n≥1;p≥1n+1k=p3)Let(|bk|)k≥1
beadecreasingsequence.Then,accordingto2),wegetthat|b|=O1.nlnn

68October20,20138:49WorldScientificBook-9inx6invers*11*oct*201354MatrixspacesandSchurmultipliers:MatricealharmonicanalysisConversely,definingr=(r)withr=1foralln≥1,wenn≥1nln(n+1)havethat||B||M(2)=||{b}||M(2)≤C||{r}||M(2).Bychoosingi=2nforalln≥2andi=0,itfollows,by1),thatn11zin=r2n+1∼n.Consequently,,$$$$$$$$-$1$$1n$||{r}||M(2)≤R$$+$ln2$<∞$n$$n$n≥1n≥12∞thatisB∈M(2).Theproofiscomplete.ObservethatresultslikeTheorem7.1orTheorem8.6[11]cannotbeappliedinoursituation.Remark3.27.Fromthepreviousresultswededucethat2(N)⊂ms⊂∞(N)andthat{(b)||b|=O1}⊂ms,withproperinclusions.nn≥1nlnnNextwechangetheviewandderiveanothersetofsufficientsconditionsinorderthatb∈ms.Theseresultsusetheestimateoftheabsolutevalueofdifferencesofsequence’stermsratherthantheabsolutevalueofthetermsthemselves.Theorem3.28.Letb=(bn)n≥1beasequenceofcomplexnumbers.(1)Ifsupn|b−b|2<∞,thenb∈ms,n≥1j=1jn∞(2)If||b||BV(N)=|b1|+n=1|bn+1−bn|<∞,thenb∈ms.Proof.1.Weusethefollowingresultfrom[11]AmatrixM∈M(2)iffthereexistsaP∈B(2,∞)andQ∈B(l1,2)suchthatM=PQand||M||M(2)≤||P||2,∞||Q|1,2.WerecallthatifQ=(qjk),j≥1,k≥1andP=(pjk),j≥1,k≥1,then⎛⎞1⎛⎞122||Q||=sup⎝|q|2⎠and||P||=sup⎝|p|2⎠.1,2jk2,∞jkk≥1j≥1j≥1k≥1

69October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions55Let[b]=Bb+Cb,where⎛⎞b1b2b3···⎜⎜b1b2b3···⎟⎟Bb=⎜⎝b1b2b3···⎟⎠,............and⎛⎞0b1−b2b1−b3···⎜⎜00b2−b3···⎟⎟Cb=⎜⎝000···⎟⎠.............1n22Bb∈M(2)∀b∈∞,and,since||Cb||1,2=supnj=1|bj−bn|<∞,byBennett’stheoremitfollowsthatCb∈M(2).2.IfA∈B(),(L),h∈H2([0,1]),x∈l2(N)andh=kDareas2kk≥1kkbefore(3.1)anddefiningf(t)=∞be2πijt(inthesenseofdistributions)j=1jwegetthat([b]A)x2=2∞112f(t)(Lh)(−t)dt+f(t)(L(h−h))(0)e−2πiktdt≤kkkk00k=1∞122g(s)(Lh)(−s)ds+2b2A2x2,kk∞B(2)20k=1wherekfˆ(k)−fˆ(j)2πijsk−1fˆ(j+1)−fˆ(j)gk(s)=j=1e=j=1Dj(s).But∞12gk(s)(Lkh)(−s)ds≤0k=1⎛⎞∞∞∞⎝fˆ(j+1)−fˆ(j)⎠fˆ(j+1)−fˆ(j)|(hL)(0)|2≤jkj=1j=1k=2⎛⎞2∞⎝fˆ(j+1)−fˆ(j)⎠h2A2,2B(2)j=1∞sothat([b]A)x2≤CAB(2)x2j=1|bj+1−bj|+b∞,thatis[b]∈M(2).

70October20,20138:49WorldScientificBook-9inx6invers*11*oct*201356MatrixspacesandSchurmultipliers:Matricealharmonicanalysis3.5.3ExtensionofHaar’stheoremAsweannouncedthissectionisdedicatedtoprovethegeneralizedHaartheorem-seeTheoremCfromsubsection3.5.1.WestartourexpositionbyintroducingausefulvectorspaceE(2).Afterthat,wedefinethenotionofgeneralizedscalarproductformatriceswhichallowsustogiveamoreusefulformforE(2)andalsotoseesomesimilaritieswithwhathappensinthefunctioncase.Proposition3.29isusefultoidentifytheconstraintsinthedefinitionofE(2)andalsotopointoutsomeofthedifficultiesinthistheory.Finally,wedefinethespaceCr(2)andprovethatthisspaceadmitsaSchaudertypedecomposition(seeProposition3.30,Theorem3.31).InthismoregeneralframeTheoremCwillfollowasaCorollary.WedefinethevectorspaceE(2)asfollows:nE(2):={A∈B(2);A=k=0αkHk,suchthatαkHk∈B(2)forall0≤k≤n,n∈N},whereαk∈∞,andαkHk=[αk]Hk.WeintroduceageneralizedscalarproductofmatricesforA=AfandB=Bg,wheref=(f1,f2,...)andg=(g1,g2,...),inthefollowingway:=(,,...).Wesaythatafamily(Φk)k∈Nisanorthonormalsystemifthefollowingorthogonalityrelationshold:Φk,Φl=0∈∞fork=landΦk,Φk=1∈forallk∈N∗.∞Bytheorthogonalityofthesystem(Hk)kwededucethatA∈E(2)nimpliesthatA=l=1Hl∈E(2).Therefore,nE(2)={A∈B(2);A=l=1A,HlHl,suchthatA,HH∈B()foralll≤n,n∈N∗}.ll2Proposition3.29.1)ThereisA∈B(2)suchthatA,H1∈∞andA,H1H1∈/B(2).2)If0]∈M(2),which,inturn,impliesthatHk∈B(2)foranyk∈N∗.Proof.1)LetA=Awitha2k−1=1,a2k=0fork∈N∗andal=0,if111kk=1andk∈N.ThenA,H=(x,0,x,0,...)∈,wherexissome111∞1constant.

71October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions572iHenceA,H1H1=x1B,whereπ⎛⎞010101...35⎜..⎟⎜⎜−100000.⎟⎟⎜⎜..⎟⎟⎜000101.⎟⎜3⎟⎜1..⎟B=⎜−0−1000.⎟.⎜3⎟⎜..⎟⎜000001.⎟⎜⎟⎜..⎟⎜−10−10−10.⎟⎝53⎠.....................But,||I−PT(B)||B(2)isinfiniteand||I−PT(B)||B(2)≤||B||B(2),whereIistheunitfortheusualnon-commutativemultiplicationofinfinitematrices.Thisresultisnotsurprising,since,byusingProposition3.25andThe-orem3.26,weobtainthat(x1,0,x1,0,...)∈ms⇔x1=0.2)Letp≤2.By[94],ityieldsthatanyA∈B(2)belongstoSpifandponlyif,foranyorthonormalbasis(ek)in2,wehavethatkAek<∞,pp∞∞22∞∞22hencek=1j=1|akj|<∞,j=1k=1|akj|<∞.Thus,byusingCauchy-Schwarzinequalityandtheaboveinequalities,wegetthatpppp≤CHkB(2)ASp<∞forsomeconstantC>0.ByRemark3.27itfollowsthat[]∈M(2).Thelastimplica-tionisnowobvious.Theproofiscomplete.ObservealsothatthereexistsanA∈B(2)suchthatA,HkHk∈B(2),forallk∈N,butforak0∈N,wefindthatA,Hk0∈/ms.IndeedA=A0=(an)n≥1∈∞\msgivestheanswertotheaboveproblemfork0=0.Therefore,inthedefinitionofE(2),weprefertheweakerconditionHk∈B(2)forallkratherthan∈msforallk.Onthems-moduleE(2)weconsiderthenormm|||A|||:=supHk.m≤nk=0B(2)

72October20,20138:49WorldScientificBook-9inx6invers*11*oct*201358MatrixspacesandSchurmultipliers:MatricealharmonicanalysisSinceE(2)∩TcanbeidentifiedwithEd([0,1]),thespaceofalldyadicstepfunctions,whosecompletionwithrespecttothesupremumnormisequaltothespaceofallcountablepiecewisecontinuousfunctionswithdiscontinuitiesatdyadicpointsof[0,1].ThisspaceisdenotedbyCr([0,1]),andwecallCr(2)thecompletionof(E(2),|||·|||).Inwhatfollowswegivesomeknownclassesofmatriceswhichareem-beddedinCr(2).Examples.1)ObviouslyallToeplitzmatrices,associatedtofunctionsfromCr([0,1])belongtoCr(2).2)TheHilbert-SchmidtmatricesA=al,j∈Z,l≥1,withA=jHS∞∞212√j=1l=1alj<∞belongtoCr(2)andACr(2)≤2AHS.∨Wedenotebyg(t)=g(−t)andP∞ae2πijt=∞ae2πijt.lj=−∞jj=ljThen,bytheFubinitheoremandtheCauchy-Schwarzinequality,wegetthat∞12P(S(A))2=supS(f)(Pg)(−t)dtTnB(2)nllg≤10H2([0,1])l=1∞12∨=supflSnPlg(−t)dtg≤10H2([0,1])l=1$$22$∨$2≤AHS·sup$Plg$=AHS.g≤1L2H2([0,1])√Hence,A≤2A.Cr(2)HS3)LetAbeadiagonalmatrixhavingasnon-zeroentriestheelementsofthesequenceα=(αi)i≥1∈ms.ThenA∈Cr(2)andACr(2)≤αms.TheproofisstraightforwardusingthetrivialobservationsthatmsisanalgebrawithrespecttousualmultiplicationandCr(2)isams-modulewith|||αX|||≤α·|||X|||.ms4)IfA=alissuchthat∞a<∞,wherea:=jj∈Z,l≥1j=−∞jmsjal,thenA≤∞aandA∈C().jl≥1Cr(2)j=−∞jmsr2Thisstatementfollowseasilyfrom3).5)IfAisthemaindiagonalmatrixhavingasnon-zeroentriestheel-ementsajwith(aj)j∈∞,thenA∈Cr(2)andAB(2)=ACr(2).(Notethat(aj)jmaynotbelongtoms.)Proposition3.30.Ifthesequenceofmatrices(An)isaCauchyse-n≥1quenceofE()withrespecttonorm|||·|||,thenAn,HHconvergesto2kk

73October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions59someαHinthisnorm.Moreover,αH∈B()andAn,H→αkkkk2kknin∞.Proof.StepI.WefirstprovethatA,Hk∞≤2AB(2)forallk∈NandA∈B(2).(3.3)IfA=Af,wheref=(f1,f2,...),andQlAisthematrixwithentriesalk=l,j∈Z[QA]k=j,lj0k=l.then,bytheCauchy-Schwarzinequality,itfollowsthat⎛⎞12∞⎝al2⎠||A,Hk||∞=(fl,hk)l∞≤supflL2=supjl∈N∗l∈N∗j=−∞√≤2supQlAB(2)≤2AB(2).l∈N∗StepII.Letnow(An)beaCauchysequenceinE().Then,foran≥12fixedk∈N,wehavethatAn,H→αin.kk∞nIndeed,using(3.3)andthefactthatA≤|||A|||,thestatementB(2)followsbyStepI.StepIII.(An,HH)isaCauchysequenceofE()forkkn≥12allkandforaCauchysequence(An)inE().HenceAn,Hn≥12kH→Bk∈C()inthenorm|||·|||.Thus,by(3.3),itfollowskr2n$./$thatlim$An,H−Bk,H$=0,andbyStepIIitfollowsthatkk.n/∞α=Bk,H.kk./StepIV.IfweshowthatBk=Bk,HH,thenProposition3.30kkisproved.ButbyStepIIIwehavethatAn,HH→BkinB().kk2nThentheentriesofthematricesAn,HHconvergewithrespecttonkktothecorrespondingentryofthematrixBk.Moreover,accordingtoStep././I,An,H→Bk,Hinand,hence,itfollowsthatBk,HHkk∞kkn=Bk.Theproofiscomplete.WeuseProposition3.30inordertoprovetheexistenceofsomekindofSchauderbasisinCr(2)givenbythesequence(Hk)k≥0.Morespecifically,wehavethefollowingresult:Theorem3.31.LetA∈Cr(2).Thenwehavethedecomposition∞A=A,HkHk,k=0inthenorm|||·|||.

74October20,20138:49WorldScientificBook-9inx6invers*11*oct*201360MatrixspacesandSchurmultipliers:MatricealharmonicanalysisProof.LetA∈C().ThenthereexistsaCauchysequenceAn∈r2E()suchthatA=limAn.2nByProposition3.30weobtainthatlim|||An,HH−αn→∞kkkHk|||=0forallk≥0.Letε>0.Therefore,thereexistsnε≥0suchthatforalln≥nε,andk>j,wegetthatkkk|||An,HH−αH|||≤limsup|||An,HH−(3.4)iiiiiim→∞i=ji=ji=jkkAm,HH|||+lim|||Am,HH−αH|||≤ε.iiiiiim→∞i=ji=jBytheorthogonalityrelationssatisfiedbythesequence(Hk)kandusingk(3.4),wefindthatthereexistsanumberlεsuchthat|||i=jαiHi|||<εforallk>j>lε.∞Thus,i=0αiHi=B∈Cr(2).But,bytakingj=0andk≥k(n)nnmax(k(n),lε),wherei=0A,HiHi=A,in(3.4),wefindthat,fornk(n)allε>0,andforalln≥nε,|||A−i=0αiHi|||<ε.∞ThusA=B=i=0αiHiand,usingtheorthogonalityrelationssatisfiedby(Hk)kandthefactthattheoperatorA→A,Hi:Cr(2)→∞∞iscontinuous,weconcludethatA=i=0HitheseriesbeingconvergentinCr(2).Theproofiscomplete.Inparticular,wegetthefollowingextensionofHaar’stheoremforma-trices:∞Corollary3.32.LetA∈Cr(2).ThenA=k=0A,HkHk,inthenormofB(2).OfcoursethereexistsA∈C(2)\Cr(2),forinstanceAisthediagonalmatrixA1givenbythesequence(an)n≥1,wherea2n−1=1anda2n=0foralln=1,2,3,...ProofofTheoremC.LetAbeaninfinitematrixasinTheoremCandlet>0.SinceA∈C(2)thereisk∈Nsuchthat||σk(A)−A||B(2)<.2Then,byhypothesisandbyExample4,itfollowsthatσk(A)∈Cr(2),andn−1consequently,byTheorem3.31,thereisaHaarpolynomiali=0αiHin−1suchthat||σk(A)−i=0αiHi||B(2)<.Theproofiscomplete.2

75October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions613.6Lipschitzspacesofmatrices;acharacterizationLet1≤p<∞andletA∈Cp.Wedefine1/p2πpωp(δ)=ωp(δ;A):=sup||A(x+h)−A(x)||Cpdx,0

76October20,20138:49WorldScientificBook-9inx6invers*11*oct*201362MatrixspacesandSchurmultipliers:MatricealharmonicanalysisAddingtheseinequalitieswefindthat4α||A||2≤Ch2α,kC24α−1|k|≥1hand,consequently,(3.5)implies(3.6).)*Assumenow(3.6).Putin(3.6)n=1+1.Then,inviewof(3.7),ithfollowsthat12ππhπhπ2h2||A(x+)−A(x−)||2dx≤4Ch2α+4||A||2k2,2π44C21kC21601|k|≤handwemustonlyestimateI=h2k2||A||2.hkC2|k|≤n−1Let∞γ={||A||2+||A||2}(k>0),kmC2−mC2m=ksuchthat1γk≤C1.k2αRepresentnowIhasn−1I=h2k2(γ−γ).hkk+1k=1Thus,I≤h2{γ+γ(22−1)+γ(32−22)+···+γ[(n−1)2−(n−2)2]}h123n−1n−1n−11≤2h2kγ≤2Ch2≤Ch2(n−1)2−2α≤Ch2α,k1k2α−1k=1k=1whereCdoesnotdependonh.Hence,0πhπ3ω≤hα8πC+C,2122andtheproofiscomplete.

77October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Matrixversionsofspacesofperiodicalfunctions63Wenotethat2πn−1min||A(x)−Beirx||2dx=2π||A||2.rC2kC2Br0r=−n+1|k|≥nThisformularepresentsthesquareofthebestapproximationinthe2-metricofthematrixAwithrespecttobandtypematricesoftheorder

78May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

79October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Chapter4MatrixversionsofHardyspaces4.1FirstpropertiesofmatricealHardyspaceTheresultsfromthissectionwerecommunicatedtousbyV.Lie.WeintroduceamatrixversionoftheHardyspace,whichwillcoincidewiththeclassicaloneontheclassofallToeplitzmatricesT.LetA=(ajk)j≥1;k≥1beaninfinitematrix.WeassociatewithAthematricealperiodicaldistribution(function)LA(x,t)on[0,1]×[0,1]definedby∞∞L(x,t):=ae2πijxe2πikt.Akjk=1j=1Theaboverelationmayberewrittenas∞∞L(x,t)=LA(x)e2πikt=CA(t)e2πijx,Akjk=1j=1whereLAisthedistribution(function)associatedwithrowkwhereasCAkjisthedistributionassociatedwithcolumnj.BecauseweworkonlywithuppertriangularmatricesitisconvenienttodefineLA(x):=LA(x)e−2πikx.kkUsingthesenotationswehavethat∞L(x,t)=LA(x)e2πik(x+t).Akk=1WeremarkthatifAisanuppertriangularToeplitzmatrix,thenLA:=kLAforallk,andL(x,t)=LA(x)∞e2πik(x+t).Ak=165

80October20,20138:49WorldScientificBook-9inx6invers*11*oct*201366MatrixspacesandSchurmultipliers:MatricealharmonicanalysisTheimportanceofmatricealdistributionsinwhatfollowsisstressedbythefollowingequalities:11LC∗D(t,s)=LC(t−μ,v)LD(μ,s−v)dμdv,001LCD(x,t)=LD(x,s)LC(−s,t)ds.0ForaninfinitematrixAwedenotebyAthematrixwhosematricealdistributionisgivenby∞LA2πik(x+t)LA(x,t):=k(x)e.k=1Weremarkthattheabovedefinitioncoincideswiththefollowingimpli-cation:ifA=A,thenA=kA,kkk∈Zk∈ZwhereAkisthekthdiagonalofA.Remark4.1.IfweconsiderthedistributionsLAasactingonthetorus,kthatis,ifLA(x)=LA(e2πix)kkandifweworkonlywithuppertriangularmatricesA,thenwemayregardLAasbeingalimit(inthespaceofdistributions)attheborderofananalytickdistribution,thatislimLA(re2πix)=LA(e2πix).kkr→1Inthiswaywearriveatthefollowingnotation:GivenanuppertriangularmatrixAwesaythatA(r)(0

81October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces67TheabovedefinitionreflectsthebehaviourofthematrixAwithrespecttoitsrows;ofcoursewemayalsodefinethespacescorrespondingtothepcolumnsofA,namelyL(2):={A|||A||Lp(<∞},wherecc2)⎛p/2⎞1/p1∞||A||p=sup⎝|CA(s)|2|x|2ds⎠.Lc(2)kk||x||2(N)≤10k=1Remark.LetA0bethemaindiagonalsubmatrixofamatrixA∈2k2L(2).Since||A0||A∈L2(2)=supk|a0|,itfollowsthatL(2)isnotisomor-phictoaHilbertspace.ThisremarkwillbeofspecialinterestinChapter6.ForsimplicityinwhatfollowswewriteLp()insteadofLA().2r2Definition4.3.WedefinethematricealHardyspaceHp()ofindexp,21≤p≤2,inthefollowingway:Hp():={A|Auppertriangular;A∈Lp()}.22Here⎛p/2⎞1/p1∞⎝A22⎠||A||Hp(2):=sup|Lk(s)||xk|ds.||x||2(N)≤10k=1AninterestingpropertyofthespaceH1()isthefollowingHardy-2Littlewoodtypeinequality:Proposition4.4.LetA∈H1().Thenwehavethat2⎧⎛⎞2⎫1/2⎪⎪1∞2A2πiθ21/2⎪⎪⎨1⎜r0j=1|xj||(Lj)(se)|dθ⎟⎬sup(1−r)⎝ds⎠dr||x||≤1⎪⎪⎩001−s⎪⎪⎭≤||A||H1(2).Proof.Letx∈(N)begiven,sothat||x||≤1,andletg∈L2([0,1])be22afixedpositivefunctionsuchthat||g||2≤1.UsingKhintchine’sinequalitywegetthat,foralls∈[0,1],1∞(|x|2|(LA)(se2πiθ)|2)1/2dθ∼jj0j=1

82October20,20138:49WorldScientificBook-9inx6invers*11*oct*201368MatrixspacesandSchurmultipliers:Matricealharmonicanalysis11∞|x(ω)(LA)(se2πiθ)|dθdω.jjj00j=1Herej(ω)standsforthejthRademacherfunction.Byduality,itisenoughtoprovethat1√r11∞A2πiθ00|j=1xjj(ω)(Lj)(se)|dθdω1−rg(r)dsdr001−s!I≤||A||H1(2).ApplyingFubini’stheoremwefindthatI:=11r1∞A2πiθ√0|j=1xjj(ω)(Lj)(se)|dθ1−rg(r)(ds)drdω≤0001−s⎛12⎞1/211r|∞x(ω)(LA)(se2πiθ)|dθ⎝0j=1jjj⎠(1−r)dsdrdω.0001−sWeget,byusingCauchy-Schwarzinequality,that1r12πiθ2|f(se)|dθ(1−r)0dsdr≤001−s1rr12ds2πiθ(1−r)|f(se)|dθdsdr≤00(1−s)2001r12|f(se2πiθ)|dθdsdr≤(byFubini’stheorem)≤000112(1−r)|f(re2πiθ)|dθdr≤(byinequality(HL)onpage4)00≤C2||f||2H1.2πiθ∞A2πiθHence,denotingbyf(se)=j=1xjj(ω)(Lj)(se),wegetfromtheinequalitiesabovethat11∞I≤C|x(ω)(LA)(e2πiθ)|dθdω∼(byFubini’stheoremandjjj00j=1⎛⎞1/21∞Khintchine’sinequality)∼⎝|x|2|(LA)(e2πiθ)|2⎠dθjj0j=1≤||A||H1(2)||x||2.Theproofiscomplete.

83October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces694.2Hardy-SchattenspacesIn1983A.ShieldsprovedaninterestinginequalitywhichholdsinSchattenclassC1(see[84]).Thisinequalityissimilartothefollowingwell-knowninequalityofHardyandLittlewood(seee.g.[96]):1/p∞p−2p(n+1)|an|≤C(p)||f||Hp,0k0otherwiseforsomefixedk∈N.Therefore,theanalogueoftheHardyspaceHp(T),0

84October20,20138:49WorldScientificBook-9inx6invers*11*oct*201370MatrixspacesandSchurmultipliers:MatricealharmonicanalysisForp=1theinequality(4.2)holdswithK(1)=π,thatis∞∞(n+1)−1|a|≤π||A||,A∈Ti,i+nT11n=0i=1whichwasprovedbyShieldsin1983(see[84]).Wediscussthisfactfurtherinthenextsection.Foranother,moregeneral,proofsee[15]-Thm.2.2-a)andalsoSection4.4.Nowwebrieflydescribethemaincontentofthesection.Firstwecon-siderthecase1

85October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces71and,similarly,that⎡⎛⎞p/2⎤1/p∞∞∞||(AA∗)1/2||=⎢⎣⎝|a|2⎠⎥⎦,kkCpijk=0i=1j=isothat⎧⎡⎪⎨p/2⎤1/p∞j⎣2⎦||A||Tp(2)+Tp(2)=inf|aij|(4.3)RCAk=A+A,k≥0⎪⎩kkj=1i=1⎡⎤⎫⎛⎞p/21/p⎪⎪∞∞⎬+⎢⎣⎝|a|2⎠⎥⎦.ij⎪⎪i=1j=i⎭Moreover,therelation(I.13)-[53]impliesthat⎛⎞1/2⎝2⎠||Ak||Cp≤||A||Tp(2)+Tp(2).(4.4)RCk≥0Wealsonotethat⎛⎞1/p⎝p⎠||An||Tp(2)+Tp(2)=||An||Cp=|ai,i+n|,1≤p≤2.(4.5)RCi≥1Nowwestateandprovethefollowingresult:Theorem4.5.Let1

86October20,20138:49WorldScientificBook-9inx6invers*11*oct*201372MatrixspacesandSchurmultipliers:MatricealharmonicanalysisNowdenote(n+1)AnbyA˜nandfixs>0.Then,A(t)=Φs(t)+Ψs(t),t∈T,whereA(t)if||A(t)||Tp>sΦs(t)=,0if||A(t)||Tp≤sandΨs(t)=A(t)−Φs(t),t∈T.Itisclearthat[A(t)]n=[Φs(t)]n+[Ψs(t)]n,n=0,1,2,...;and,there-fore,An∗[Et]n=[Φs]n∗[Et]n+[Ψs]n∗[Et]n,where[Φs]n=Anand[Ψs]n=0if||A||Tp=supt∈T||A(t)||Tp>s.ThesameholdsforΨs.Hence,(n+1)p−2||A||p=||A˜||pμ(n)(4.7)nCpnCpn≥0n≥0⎛⎞≤2p⎝||[Φ˜]||pμ(n)+||[Ψ˜]||pμ(n)⎠.snCpsnCpn≥0n≥0Putnowα(s)=μ{n;||[Φ˜s]n||Cp>s}andβ(s)=μ{n;||[Ψ˜s]n||Cp>s}:=μ(Es).Thens2β(s)≤||[Ψ]||2.(4.8)snCpn≥0Wedenotetheset{n≥0;||[Φ˜s]n||Cp>s},byFs.Since||An||Cp≤||A||Cp,n∈N,(see[34])wehavethatμ(Fs)=μ{n;(n+1)||[Φs]n||Cp>s}≤μ{n;(n+1)||Φs||Cp>s}=1n+114≤4dx≤,(n+1)2nx2n0+1{n;||Φ||>s}{n;||Φ||>s}sCpn+1sCpn+1wheren=min{n;||Φ||>s}.0sCpn+1Thus,4||Φs||Cpα(s)=μ(Fs)≤,s>0.(4.9)sAccordingto(4.9)wehavethat∞∞∞||[Φ˜]||pμ(n)=−spdα(s)=psp−1α(s)ds≤snCpn=000∞||Φ||||A||Cp4pp−1sCpp−2p4psds≤4p||A||Cpsds≤||A||Tp,0s0p−1

87October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces73forallt∈T.Moreover,by(4.8)and(4.4),itfollowsthat∞∞∞∞||[Ψ˜]||pμ(n)=psp−1β(s)ds≤psp−3||[Ψ]||2ds=snCpsnCpn=000n=0∞∞2p∞p||A||2sp−3ds≤||A||p−2||A||2≤nCp2−pCpnCp||A||Cpn=0n=0pp−222−p||A||Tp||A||Tp(2R)+Tp(2C).Byusing(4.7)wegetthat∞:;(n+1)p−2||A||p≤K(p)||A||p+||A||p−2||A||2.(4.10)nCpTpTpTp(2)+Tp(2)RCn=0Butin[38]itisprovedthat||A||Cp≤||A||Tp(2)+Tp(2)and,conse-RCquently,wegetthefollowinginequalityofHardy-Littlewoodtype:(n+1)p−2||A||p≤K(p)||A||p,1=tr(AB∗),kkk=0

88October20,20138:49WorldScientificBook-9inx6invers*11*oct*201374MatrixspacesandSchurmultipliers:MatricealharmonicanalysiswhereA=k≥0Ak,B=k≥0Bk,wehavethefollowing:Theorem4.7.Let2≤q<∞andA=k≥0Aksuchthatn≥0(n+1)q−2||A||q<∞.nCqThenA∈T(2)∩T(2)andqRqC⎛⎞∞∞∞j⎝(2q/21/q2q/21/q⎠||A||Tq(2)∩Tq(2):=max(|ai,j|)),((|ai,j|))RCi=1j=ij=1i=1⎛⎞1/q≤C(q)⎝(n+1)q−2||A||q⎠.nCqn≥0nProof.Letp=q/(q−1)andG=k=0Gkbeafinitetypebandmatrixwith||G||Tp(2)+Tp(2)≤1.RnCLetSn(A)=k=0Ak.Thennn∞||=|tr(GA∗)|≤|ga|≤(byH¨older’snkki,k+ii,k+ik=0k=0i=1nninequality)≤(|g|p)1/p(|a|q)1/q=||G||||A||i,k+ii,k+ikCpkCqk=0iik=0n≤(againbyH¨older’sinequality)≤(||G||p(k+1)p−2)1/pkCpk=0n(||A||q(k+1)q−2)1/q≤(byTheorem4.5)kCqk=01/qnqq−2≤C(p)||G||Tp(2)+Tp(2)||Ak||Cq(k+1)RCk=01/qn≤C(p)||A||q(k+1)q−2.kCqk=0Hence,||Sn(A)||Tq(2)∩Tq(2)=sup||RC||G||Tp(2)+Tp(2)≤1RC1/qn≤C(p)||A||q(k+1)q−2,kCqk=0forallnandtheproofiscomplete.

89October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces75NowwediscussaninequalityofHausdorff-Youngtype.Theorem4.8.(Hausdorff-Young’sinequality)For1≤p≤∞,letqbetheconjugateindex,with1+1=1.pq1/q∞q(i)If1≤p≤2,thenA∈Tpimpliesthatn=0||An||Tp≤||A||Tp.(ii)If2≤p≤∞,then{||An||Tp}∈qimpliesthat1/q∞q||A||Tp≤||An||Tp.n=0∞∞21/2Proof.Incase(ii),forp=q=2,ifn=0l=1|al,n+l|<∞,then∞∞21/2clearlyA=(aij)∈T2and||A||T2≤n=0l=1|al,n+l|,inotherwordsthemapT({An}n≥0)=n≥0Anhasnormlessthan1fromthespace2(2)intoT2.Here2(2)meansthespaceofallmatrices(aij)i,j∞∞2suchthatn=1i=1|ai,i+n|<∞.Ifp=∞itfollowsthatq=1andthemapThasthenormlessthan1∞from1(∞)intoT∞,since,clearly,||A||T∞≤n=0||An||T∞.Usingcomplexinterpolation(see[19]),wegettheconclusion.Case(i)followsbyduality,since()∗=(),and(T)∗=T.pqqppq4.3AnanalogueoftheHardyinequalityinT1InthissectionwepresentanimportantinequalityduetoA.Shields[84].Infact,thepaperofShieldscontainingthisinequalitywasthestartingpointofthematerialdescribedinthepresentbook.LetusstatetheanalogueoftheinequalityofHardy,LittlewoodandFej´er.Inthenextsectionwegivealsoamoregeneralusefulinequality,whichwillbeprovedusingdifferentmethods.Theorem4.9.LetM∈C1havetheuppertriangularformwithrespecttotheorthonormalbasis{en}(n=1,2,...)of2.Then∞k|M(j,k)|≤π||M||T1,1+k−jk=1j=1withequalityonlywhenM=0.

90October20,20138:49WorldScientificBook-9inx6invers*11*oct*201376MatrixspacesandSchurmultipliers:MatricealharmonicanalysisItiseasytoobservethatanotherformoftheaboveinequalityisasfollows:∞1||Mk||T1≤π||M||T1.k+1k=0ThisinequalityissimilarwiththeclassicalinequalityofHardy[23].InordertoproveTheorem4.9werequirethreelemmas.Throughoutourdiscussiontheorthonormalbasis{en},n=1,2,...willbefixed;uppertriangularitywillalwaysbewithrespecttothisbasis.Lemma4.10.LetRdenoteeitherthespaceB(2),withtheweakoperatortopology,oranyoftheBanachspacesCp(1

91October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces77Proof.LetusdenotebyEnthesubspacespan{e1,...,en},foralln∈N.WefirstprovethelemmaundertheadditionalassumptionthatPisone-to-oneoneachofthesubspacesE(n≥1).ThenP1/2isalsoone-to-noneoneachofthesespaces.LetF=P1/2E.ThenF⊂F⊂...,nn12anddimFnforalln.Hence,thereisanorthonormalset{fk}suchthatFn=span{f1,...,fn}.DefineanoperatorVby:Vfn=en(n≥1),andV=0ontheorthogonalcomplementofthespanof{fn}.ThenVisapartialisometry.LetB=VP1/2.ThenBE⊂Eand,thus,Bhasnntheuppertriangularform.Moreover,B∗B=P1/2V∗VP1/2=P,sinceV∗Vistheprojectionontothespanof{f},whichcontainstherangeofnP1/2.Finally,||B||≤||V||||P1/2||=1,and,therefore,1=||P||≤T2T2T1||B∗||||T||≤1,whichcompletestheproofinthiscase.T2T2NowsupposethatPisnotone-to-one.LetSbeafixedpositiveoperatorfromCwithtrivialkernel.LetPn=(P+n−1S)d,whered=||P+1nnn−1S||−1.ThenPnhasnormone,andPn→P,inC.InviewoftheT11resultprovedabove,thereisasequence{Bn}ofoperatorsinC,having2theuppertriangularform,withPn=(Bn)∗Bn,||Bn||=1,foralln.ByT2passingtoasubsequencewemayassumethat{Bn}isweaklyconvergentinC:Bn→BforsomeBintheunitballofC.ThelimitoperatorB22musthavetheuppertriangularformand,byLemma4.10,wehavethatB∗B=P.Fromthiswehavethat||B||≥1and,hence,thenormmustT2beequaltounity.Theproofiscomplete.Lemma4.12.LetM∈C1havetheuppertriangularformwith||M||T1=1.ThenthereexistuppertriangularoperatorsA,B∈C2withM=ABand||A||T2=||B||T2=1.Proof.WefirstprovethelemmawiththeadditionalassumptionthatMisone-to-oneoneachofthespacesEn;thisisequivalenttorequiringthatalldiagonalmatrixentriesaredifferentfrom0;=0forallj.LetM=UPbethepolardecompositionofM.ThenP=(M∗M)1/2isapositiveoperatorofnormoneinC1andUmapstherangeofPisometri-callyontotherangeofM.Since||Pf||=||Mf||forallfweseethatPhasthesamekernelasM.Therefore,Pisone-to-oneoneachofthespacesEn.ByLemma3.3,P=B∗B,whereBisanuppertriangularoperatorofnormoneinC2.WeseethatBmustbeone-to-oneoneachofthespacesEn.NowletA=UB∗.ThenAisintheunitballofC,andAB=M.From2thisweseethat||A||C2=1.ToshowthatAhastheuppertriangularformwemustshowthatitmapseachspaceEnintoitself.SinceBisone-to-one

92October20,20138:49WorldScientificBook-9inx6invers*11*oct*201378MatrixspacesandSchurmultipliers:MatricealharmonicanalysisonEnandEnisfinite-dimensionalwehavethatEn=BEn.Hence,AEn=ABEn=MEn=En.NowsupposethatMisnotone-to-oneoneachofthespacesEn,thatis,somediagonalmatrixentriesare0.LetSbeamatrixfromC1,withnon-zerodiagonalentriespreciselyinthoseplaceswhereMhasazero.LetMn=(M+n−1S)d,whered=||M+n−1S||−1.ThenMnsatisfiesallnnC1theconditionsofthelemmaand,inaddition,isone-to-oneoneachofthespacesEn.BywhatwasprovedabovethereareuppertriangularoperatorsAn,BnintheunitballofC,Ball(C),withMn=AnBn.Bypassingtoa22subsequencewemayassumethatthesequences{An}and{Bn}areweaklyconvergentinC:An→A,Bn→B,whereA,B∈Ball(C).ByLemma224.10wehavethatAnBn→AB,andsoM=AB.Thiscompletestheproofsinceweakconvergencepreservestheuppertriangularform.ProofofTheorem4.9.Withoutlossofgeneralitywemayassumethat||M||T1=1.ThenbyLemma4.12thereareuppertriangularoperatorsA,BofnormoneinC2suchthatM=AB.Letmij,aij,bijdenotethematrixentriesofM,A,B,respectively.Thefollowingsummationsarewrittenwitheachvariablegoingfrom1to∞.Becauseoftheuppertriangularity,however,thetermsareequaltozeroifj>k,orifj>r,orifr>k.Thus,wereallyhavethat1≤j≤r≤k<∞.WeusetheboundednessofthesecondHilbertmatrix,thatisthematrixwithentries(n−m)−1whenn=m,and0whenn=m(n,m=1,2,...);thenweusetheCauchy-Buniakovsky-Schwarzinequalityandobviousestimatestoobtainthat|mjk||ajrbrk||ajrbrk|≤=1+k−j1+k−j1+k−jj,kj,krrj,k⎛⎞1/21/2≤π⎝|a|2⎠|b|2jrrkrjk⎛⎞1/21/2≤π⎝|a|2⎠|b|2=π.jrrkrjrkWehavestrictinequalitybecausetheboundπforthesecondHilbertmatrixisnotattended.Theproofiscomplete.

93October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces794.4TheHardyinequalityformatrix-valuedanalyticfunc-tionsHerewepresentanotherproofofthepreviousinequalityofShields.Infactwepresentamoregeneralinequalityforvector-valuedanalyticfunctionsemphasizingthespecialcaseofmatrixBanachspaces.AlltheresultsofthissectionareduetoO.BlascoandA.Pelczynski[15].Werecallthatiff=aeijtisananalytictrigonometricpolyno-j≥0jmial,thenπ|a|(j+1)−1≤C|f(t)|dt,j1−πj≥0⎛⎞1/2π⎝|a2⎠≤C|f(t)|dt,2k|2−πk≥0whereC1andC2arenumericalconstantsindependentoff(cf.[23]).ThefirstfactiscalledtheHardyinequality;thesecondisaparticularcaseofatheoremofPaley,where(2k)isreplacedbyanysequence(n)ofkpositiveintegerswithinfknk+1/nk>1.Itisalsoknownthatbothoftheseinequalitiesarefalseifanalytictrigonometricpolynomialsarereplacedbyarbitrarytrigonometricpolynomials.Inwhatfollowsweareinterestedinfindingunderwhichadditionalcon-ditionsonaBanachspaceXtheinequalitiesremaintrueiftheFouriercoefficientsaj’sareelementsofXandabsolutevaluesareeverywherere-placedbynorms.WeremarkthatitisknownthatinthatgeneralsettingforarbitraryBanachspacestheinequalitiesarefalse(seeforinstance[15]).ItappearsthatthevalidityofX-valuedversionsoftheseinequalitiesdependsongeometricpropertiesofX.WearespeciallyinterestedinthecasewhenXissomematrixspace,forinstanceifX=C1,theBanachspaceofalltraceclassmatrices.pThemainideaoftheproofsistousevector-valuedHardyspacesHXandtoconsiderandusesomeoperatorsinducedbyboundedmultipliersfromH1into.1p4.4.1Vector-valuedHardyspacesHXAllBanachspacesareconsideredtobetakenoverthecomplexnumberfieldC.GivenaBanachspaceXandp∈[1,∞)(respectivelyp=∞)wedenote

94October20,20138:49WorldScientificBook-9inx6invers*11*oct*201380MatrixspacesandSchurmultipliers:MatricealharmonicanalysispbyLthespaceofallX-valued2π-periodicfunctionsonthereallineXR,whichareBochnerabsolutelyintegrableinthepthpower(respectivelyessentiallybounded)underthenorm"π#1/p||f||=(2π)−1||f(t)||pdtfor1≤p<∞p−π(respectively||f||∞=esssupt∈R||f(t)||).Givenf∈L1andanintegerj,thejthFouriercoefficientoffisdefinedXbyπf(j)=(2π)−1e−ijtf(t)dt.−πIfforsomenonnegativeintegern,f(j)=0for|j|>n,thenfiscalledanX-valuedtrigonometricpolynomialofdegree≤n;if,moreover,f(j)=0forj<0,thenfiscalledanX-valuedanalytictrigonometricpolynomial.pGivenp∈[1,∞)theHardyspaceHisdefinedtobetheclosureofallXX-valuedanalytictrigonometricpolynomialsunderthenorm||·||p;orinotherwordsppH={f∈L:f(j)=0forj<0}.XX4.4.2(Hp−)-multipliersandinducedoperatorsforqvector-valuedfunctionsLetm=(mj)j≥0beacomplexsequenceandletXbeaBanachspace.DefinetheoperatormXfromX-valuedanalytictrigonometricpolynomialsintotheeventuallyzeroX-valuedsequencesbymX(f)=(mjf(j))j≥0.WecallmXtheoperatorinducedbythemultiplierm.TheoperatormXissaidtobe(p,q)-boundedprovidedthatthereexistsaconstantK=K(m,X)suchthat,foreveryX-valuedanalytictrigonometricpolynomialf,⎛⎞1/q∞⎝||mf(j)||q⎠≤K||f||.jpj=0IfmXis(p,q)-bounded,thenitcanbeuniquelyextendedtoanoperatorp(alsodenotedbymX)fromHXintotheBanachspace(q)X,where1/q()={(x)⊂X:||(x)||=||x||q<∞}.qXjjqjWecallman(Hp−)-multiplierif,forX=C,mis(p,q)-bounded.qCDefinition4.13.ABanachspaceXisof(H1−)-Fouriertypeprovided1that,forevery(H1−)-multiplierm,theinducedmultipliermis(1,1)-1Xbounded.

95October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces81Recallthefollowingbeautifuldescriptionof(H1−)-multipliersgivenby1Ch.Feffermaninanunpublishedmanuscript:Theorem4.14.Ascalarsequencem=(m)isan(H1−)multiplierjj≥01ifandonlyif⎛⎛⎞2⎞1/2∞(k+1)sρ(m)=⎜⎝|m2+|m|2+sup⎝|m|⎠⎟⎠<∞.0|1js≥1k=1j=ks+1Followingthelinesofthepaper[88]wegiveaproofoftheabovetheorem.LetusdenotebyΛthelatticeofallintegersofRandbyQtheintervalα{x∈R:α−/2≤x<α+/2},whereα∈Λand>0.NowwestatethefollowingtheorembelongingtoSleddandStegenga[88]:Theorem4.15.LetμbeapositiveBorelmeasureonR\{0}.Thensup|f|dμ<∞(4.13)||f||≤1H1(R)ifandonlyif1/2supμ(Q)2<∞.(4.14)α>0Moreover,thecorrespondingsupremaareequivalent.Corollary4.16.Let{mα}α∈Λbenonnegativenumbersanddefineamea-sureonR\{0}byμ=α=0mαδα,whereδαisthepointmassatx=α.Thensup|f(α)|mα<∞(4.15)||f||≤1H1(T)α=0ifandonlyifμsatisfiescondition(4.14).ItiseasytoseethatCorollary4.16isnothingelsethanTheorem4.14.NowweproceedtotheproofofTheorem4.15.Proof.Werecallthatanatoma(x)correspondingtoanintervalQisameasurablefunctionsupportedonQwhichhaszeromeanandisboundedby|Q|−1(|·|meaningtheLebesguemeasure).ByafundamentalresultofR.Coifman[21]wemaytakeasadefinitionofafunctionfofH1(R)theequalityf=iλiai,wherei|λi|<∞and||f||H1(R)=inf{i|λi|},forall{λi}asbefore.

96October20,20138:49WorldScientificBook-9inx6invers*11*oct*201382MatrixspacesandSchurmultipliers:MatricealharmonicanalysisThus,thesufficiencyofcondition(4.14)followsifthereisac<∞with|a|dμ≤c(4.16)forallatomsa.Partoftheproofof(4.16)isstraightforward.Ifaisanatomcorre-spondingtoanintervaloflengthδ,thenitiseasytoseethat|a(y)|≤c|y|δfory∈Q,where=δ−1.Herecisaconstantnondependingofa.Nowit0isclearthat(4.14)implies−1|x|dμ(x)≤cQ0and,hence,(4.16)willfollowfrom|a|dμ≤c(4.17)R\Q0whereisrelatedtoaasabove.Thisresultisnoweasilyseentobeaconsequenceofcondition(4.14)andthefollowingtheoremTheorem4.17.Thereisaconstantc<∞suchthatifa(x)isanatomcorrespondingtoanintervalwiththelength2δand=δ−1,thensup|a|2≤c.QααProof.Itsufficestoassumethataissmoothandsupportedintheinterval[−δ/2,δ/2].FixanintervalIoflengthandassumethatfiscontinuouslydiffer-entiableonI.Itiselementarytoseethatsup|f−b|≤|f|,wherebisIItheaverage|I|−1f.Hence,I"#2122sup|f|≤2|f|+|f|.IIINormalizingtheFouriertransformsothat||f||2=||f||2,weobtainthat"∞∞#2122sup|a|≤2|a|+|a|Q−∞−∞ααδ/2δ/2122=2|a|+|2πixa|dx−δ/2−δ/2fromwhichTheorem4.17follows.

97October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces83ThefollowingLemmaisusefulforourpurposes.Lemma4.18.Letg∈L2(R)andassumethatg=0on|y|≤1.If1f=gχ[−1,1],thenf∈Hand||f||H1≤c||g||2.(Hereχ[−1,1]isthechar-acteristicfunctionfortheunitintervalcenteredattheorigin.)Proof.AssumethatgisaC∞-functionwithcompactsupportin|y|>1.Thenfistheconvolutiong∗χ[−1,1]and,hence,isarapidlydecreasingfunction,whichvanishesinaneighborhoodoftheorigin.ThusfisinH1.Ifu∈BMO(R)andbisitsaverageover[−1,1],then,bytheSchwarzinequality,21/2|u−b||fu|=|f(u−b)|≤c||g||22dx≤c||g||2||u||BMO.1+|x|Thefirstinequalityisawell-knownestimateforχ[−1,1]andthesecondaslightextensionofinequality(1.2)in[32].Nowusetheduality.TheproofofTheorem4.15iscompleteonceweestablishthenecessityofthecondition(4.14).However,if(4.13)holdswithsupremumA,thenfromLemma4.18wededucethat[μ(y+[−1,1])]2dy≤cA2.(4.18)Hence,thereisanM<∞,δ>0,forwhichμ(Qδ)≤cA2,wherec|α|≥Mαisanabsoluteconstant.Butthenadilationargumentgivesthisinequalityforallδ>0and(4.14)nowfollowsinanelementaryway.Thus,alsotheproofofTheorem4.15iscomplete.ProofofCorollary4.16.ThespaceH1(T)isthesubspaceofH1(T)0consistingoffunctionswithzeromean.Givenf∈H1(R),wedefinePf(x):=f(x+α).α∈ΛSincef∈L1(R)wehavethatPf∈L1(T)and,bythePoissonsumma-tionformula(see[90]),itfollowsthatf(α)=(Pf)(α)forα∈Λ.Theproofofthecorollaryisanimmediateconsequenceofthefollowingtheorem:Theorem4.19.ItyieldsthatP(H1(R))=H1(T).0Proof.Letφbeanonnegativerapidlydecreasingfunctionforwhichφhassupportcontainedintheopeninterval(−1,1)andφ(0)=1.Putϕ(x)=ϕ(x)for0<<1.ForapolynomialF(x)=ae2πiαxletf=F∗ϕ.α

98October20,20138:49WorldScientificBook-9inx6invers*11*oct*201384MatrixspacesandSchurmultipliers:MatricealharmonicanalysisClaim.lim→0||f||H1(R)≤||F||H1(T).Westartwiththeeasilyderivedfactthatlimg|ϕ|=gdx(4.19)→0RTforallcontinuousfunctionsgonT.Observethat||ϕ||1=1.LetS,RdenotetheRieszprojectionsonT,R.Then,by(4.19),weobtainthatlimsup[||f||1+||Rf||1]≤||F||H1(T)+limsup||Rf−(SF)(ϕ)||1→0→0sothatwemustshowthatthesecondtermontherighthandsideiszero.SinceFisapolynomialitsufficestofixα∈Λwithα=0,puth(y−α)=(y/|y|−α|α|)ϕ(y−α)andshowthatlim→0||h||1=0.Nowϕissupportedin[−,].Thus,wemayassumethath(y)=m(y)ϕ(y),wheremissmooth,allderivativesuptoorder2areboundedbyaconstant,and|m(y)|≤c|y|.Theconditionsonmimplythat||Dh||1≤c,whereD=d2/dy2.Hence,|h(x)|≤c|x|−2.Clearly,lim||h||=→010sothatlim→0||h||∞=0and,thus,theaboveestimateimpliesthatlim→0||h||1=0.Thisprovestheclaim.TocompletetheproofwefixFinH1(T)andnotethattherearepoly-01nomialsFn∈H0(T)with||Fn||H1<∞andF=Fn.Usingtheabove1wefindthatfn∈H(R)with||fn||H1(R)<∞andPfn=Fn.Thus,Pf=Fwheref=fisafunctioninH1(R).Theproofiscomplete.nBysummingupwenotethatalsotheproofofTheorem4.14iscomplete.InthesequelFMmeanstheBanachspaceofallscalarsequencessat-isfyingtheaboverelationequippedwiththenormρ(·).Nowwepresentadualdescriptionof(H1−)Fouriertypespaces.1Proposition4.20.ForeveryBanachspaceXthefollowingstatementsareequivalent:(i)Xisan(H1−)Fouriertypespace;1(ii)thereisC>0suchthat,foreverym∈FMandf∈H1,X||mjf(j)||≤Cρ(m)||f||1;j≥0(iii)thereisC>0suchthatforeveryeventuallyzerosequence(x∗)jj≥0ofelementsofX∗thereisanX∗-valuedtrigonometricpolynomialg∗suchthatg∗=x∗forj≥0;||g∗||≤Cρ((||x∗||)).j∞jj≥0

99October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces85Proof.(i)⇒(ii).Putρ(m)=sup{||mf(j)||;f∈H1;||f||=1}.XjX1j≥0UsingtheBairecategoryargumentwegetthatρX(·)isaboundednormonFM.(ii)⇒(iii).Letx∗=0forj≥N.WedefineonH1thelinearfunctionaljX∗∗N∗1∗φ0byφ0(f)=j=0xj(fj)forf∈HX.Itfollowsfrom(ii)that||φ0||≤Cρ((||x∗||)).Letφ∗beanorm-preservingextensionofφ∗ontoL1.Letjj≥00XVbetheNthdelaVall`ePoussinkernel,i.e.,V(j)=1for|j|≤N,V(j)=0for|j|≥2NandV(j)linearfor−2N≤j≤−NandforN≤j≤2N.Itiswellknownthat||V||≤2.Define,forj=0,±1,±2,...,y∗∈X∗by1jy∗(x)=V(j)φ∗(xe)forx∈X,jjwheree(t)=eijt.Putg∗(t)=y∗eijt.Then(denotingbya∗bthej|j|≤2Njconvolutionofthefunctionsaandb)=φ∗(V∗f)forf∈L1.XThus,usingtheinequality||V||1≤2,wegetthat||g∗||≤||φ∗||||V||≤2Cρ((||x∗||)).∞1jj≥0Ontheotherhand,takingintoaccountthatx·e∈H1forj≥0andjXx∈X,wegetthat,for0≤j≤N,g∗(j)(x)=y∗(x)=V(j)ϕ∗(xe)=V(j)ϕ∗(xe)=x∗(x).jj0jjThus,g∗(j)=x∗for0≤j≤N.j(iii)⇒(i).Letm∈FMandletfbeanX-valuedanalytictrigono-metricpolynomialofdegreeN.Forj=0,1,...,Npicky∗∈Xsothatj||y∗||=1andy∗(f(j))=||f(j)||.Putx∗=|m|y∗for0≤j≤Nandjjjjjx∗=0forj>N.Obviously,ρ((||x∗||))≤ρ(m).Wehavethatjjj≥0∞||mf(j)||=≤||g∗||||f||≤Cρ((||x∗||))||f||≤j∞1jj≥01j=0Cρ(m)||f||1.HencemXis(1,1)-bounded.Theproofiscomplete.

100October20,20138:49WorldScientificBook-9inx6invers*11*oct*201386MatrixspacesandSchurmultipliers:MatricealharmonicanalysisNowwerecallthataBanachspaceYiscrudelyfinitelyrepresentableinaBanachspaceXifthereisK≥1suchthatforeveryfinite-dimensionalsubspaceEofYthereisalinearoperatoru:E→Xsuchthat||e||≤||u(e)||≤K||e||fore∈E.Asausefulexampleconsiderfor1≤p<∞pthespaceH(D)ofallX-valuedanalyticfunctionsontheunitdiskD=Xp{z∈C;|z|<1}suchthatforeach00,suchthat,foreverym∈FM,∞||mjfk(j)||X≤Cρ(m)||fk||X(k=0,1,...).j=0Summingoverallk,wegetthat∞∞∞∞||mjf(j)||(1)X=||mjfk(j)||X≤Cρ(m)||fk||Xj=0j=0k=0k=0=Cρ(m)||f||(1)X.AnotherobviousbutusefulconsequenceofDefinition4.13isthefollow-ing:Corollary4.23.AssumethataBanachspaceXsatisfiesthefollowingcondition:

101October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces87(∗)thereisC>0suchthatforeveryf∈H1thereisacomplexvaluedXfunctionϕ∈H1suchthat||f(j)||≤|ϕ(j)|forj=0,1,...;||ϕ||1≤C||f||1.(4.20)ThenXisa(H1−)-Fouriertypespace.1Theorem4.24.ThespaceC1satisfies(∗)withC=1+,forall>0,and,hence,itisa(H1−)-Fouriertypespace.1TheproofofTheorem4.24isaconsequenceofthefollowingresult:Theorem4.25.(Thenoncommutativefactorizationtheorem[45].)Let>0.Foreveryf∈H1therearegandhinH2suchthatC1C2f=g·h,(1+)||f||C1,1≥||g||C2,2||h||C2,2≥||f||C1,1.Heref=g·hmeansthatf(t)=g(t)·h(t)fort∈R,i.e.ateachpointtthematrixf(t)istheproductofthematrixh(t)withthematrixg(t).Moreover,π1/r||f||:=(2π)−1||f(t)||rdtfor1≤p<∞and1≤r<∞.Cp,rCp−πRemark:InfactTheorem4.25holdsalsofor=0.ThisstrongversionisduetoSarason[85].Weneedonlytheweaker-versionofSarason’stheorem.Infact,weneedonlyamuchweakerversionofCorollary4.23(see[84]),namelythefollowing:Corollary4.26.AssumethattheanalyticmatrixBanachspaceXsatisfiesthefollowingcondition:(∗)thereisC>0suchthatforeveryA∈Xthereisacomplexvaluedfunctionϕ∈H1suchthat||Aj||≤|ϕ(j)|forj=0,1,...;||ϕ||1≤C||A||X.(4.21)ThenXhasthefollowingproperty:IfthereisaconstantK>0suchthat,forasequenceofcomplexnumbers(m)andf∈H1wehavethatj∞fj|∞j=0|mj≤K||f||H1,thenj=0||mjAj||X≤K||A||X.NextwepresentaproofofTheorem4.24byusingSarason’sfactorizationtheoremasintheHaagerupandPisier’sproof(see[45]):

102October20,20138:49WorldScientificBook-9inx6invers*11*oct*201388MatrixspacesandSchurmultipliers:MatricealharmonicanalysisProofofTheorem4.24.Letf∈H1andg,h∈H2satisfytheC1C2relationofTheorem4.25.Put∞jϕ(t)=||g(k)||||h(j−k)||eijt.C2C2j=0k=0Byusingtheinequality||A·B||C1≤||A||C2||B||C2andthatf=g·h,wegetthatjj|ϕ(j)|≥||g(k)·h(j−k)||C1≥||g(k)·h(j−k)||C1=||f(j)||C1.k=0k=0Ontheotherhand,notethatϕ=G·H,where∞∞G=||g(k)||eijt,H=||h(j)||eijt.C2C2j=0j=0Nextweobservethat||G||2=||g||C2,2and||H||2=||h||C2,2.NowusingtheSchwarzinequalityandTheorem4.25wegetthat||ϕ||1≤||G||2||H||2=||g||C2,2||h||C2,2≤(1+)||f||C1,1.Theproofiscomplete.Corollary4.27.ThedualofB()has(H1−)-Fouriertype.21Proof.WehaveC∗=B().Butitiswell-knownbytheLocalReflex-12ivityPrinciple[54]thattheseconddualofanyBanachspaceisfinitelyrepresentableinthespace.Hence,theproofoftheCorollaryfollows.NowwegivetheproofofTheorem4.25asgivenin[45].ProofofTheorem4.25.Firstwerecallthewell-knownfact(see[35])thattheprojectivetensorproduct2⊗2maybeisometricallyidentifiedwithC1.Here2⊗2isthecompletionofalgebraictensorproduct2⊗2underthenorm∞||u||=inf{||xi||·||yi||;u=xi⊗yi}.ii=1Then,intheframeworkoftensorproductsthestatementofTheorem4.25canbereformulatedasfollows:Let>0.Foranyf∈H1,therearesequences(g)and(h)inH22⊗2kk2suchthat∞∀z∈Df(z)=gk(z)⊗hk(z)(4.22)k=1

103October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces89and∞||f||H1≤||gk||H2||hk||H2.(4.23)2⊗222k=1Indeed,let(en)denotethecanonicalbasisof2.Letusdenoteby(gij(z))and(hij(z))thecoefficientsofthematricesg(z)andh(z)relativetothebasis(ei⊗ej),andsimilarlyforf.Wehave,bythefirstrelationinTheorem4.25,thatfij(z)=gik(z)hkj(z)kand,hence,f(z)=fij(z)ei⊗ej=gk(z)⊗hk(z),kwheregk(z)=gikeiandhk(z)=hkjej.ijThisprovesthat(4.22)and(4.23)followfromtherelationsinTheorem4.25.(Theconversedirectionisalsoeasy.)Letusprovetherelations(4.22)and(4.23).WedenotebyPthelinearsubspaceofH1formedbyallthepolyno-C1mialswithcoefficientsin2⊗2.Moreover,wedenoteby||||thenorminH1,andby||||thenorm1C12inH2.2Clearly,foreveryfinPtherearepolynomialswithcoefficientsin2gi,hisuchthatn∀z∈Df(z)=gi(z)⊗hi(z).i=1WeintroduceanormonPbyn||f||=inf{||gi||2||hi||2},iwheretheinfimumrunsoverallpossiblerepresentations.Notethatweobviouslyhavethat,||f||1≤||f||and||||isindeedanormonP.ThemainpointoftheproofofTheorem4.25istocheckthatactuallythis”new”norm||f||coincideswith||f||1.Usingduality,wewillshowthat

104October20,20138:49WorldScientificBook-9inx6invers*11*oct*201390MatrixspacesandSchurmultipliers:MatricealharmonicanalysisthisfollowsratherdirectlyfromknownresultsinthetheoryofvectorialHankeloperatorsduetoS.Parott[70].Toexplainthismoreprecisely,weneedtoidentifythedualspacestoPequippedwiththenorms||||1and||||.LetusdenotebyΛthespaceofallsequencesa=(an)n≥0withan∈B(2)suchthattheHankelmatrixHawithentries(Ha)ij=ai+j(i≥0,j≥0)definesaboundedoperatoron2(2)=2⊗2.Bydefinition,weset||a||=||Ha||.(ForadefinitionofaHankelmatrixandforsomeitspropertiessee[65]andalsoChapter5.)LetusdenotebyX(resp.X1)thenormedspaceobtainedbyequippingPwiththenorm||||(resp.||||1).WemayintroduceadualitybetweenPandΛasfollows.Let(fn)denotetheTaylorcoefficientsofanelementfinP.Then,forallainΛ,wedefine∞:=.n=0(Notethatthissumisfinite.)Withthisduality,wehavethat||a||X∗=sup=sup(aijgj,hi)=||Ha||,ijwhereeachoftheabovesupremarunsoverallg,hinPsuchthat||g||2≤1and||h||≤1.(Ofcoursewehavethat||g||=(||g||2)1/2.)22jThisshowsthatΛcanbenaturallyidentifiedisometricallywiththedualofX.Similarly,letusdenotebyΛthespaceofallsequencesα=(αn)n∈Zwithα∈X∗⊂B()suchthatthematrixTdefinedbyn2α(Tα)ij=αi+j∀i,j∈Z(4.24)definesaboundedoperatoron2(Z,2).Bydefinition,weset||α||Λ:=||Tα||.HereagainitissimpletocheckthatL(T,C)∗=Λisometrically.11Equivalently,thismeansthatthenaturalmappingfromL2(2)⊗L2(2)intoL1(C1)isametricsurjection.ThiscanbeviewedasaconsequenceoftheidentityL1(C1)=L1⊗C1andthefactthateveryscalarfunctionwithL1-norm1istheproductoftwofunctionswithL2-norm1.LetusnowreturntoouroriginalproblemtoshowthatXcoincideswithX1,orsimplythat||f||≤||f||1forallfinP.ToprovethatitsufficestoshowthateveryaintheunitballofX∗definesanelementintheunitballofX∗∗.1

105October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces91Equivalently,itisenoughtoshowthatforanya=(an)n≥0intheunitballofΛ=X∗,thereisanα=(α)intheunitballofL(C)∗=Λ,whichnn∈Z11issuchthat=<α,f>forallfinP.Clearlythismeansthatαn=anforalln≥0.WehavethusreducedourproblemtothefactthateveryHankelmatrixwithcoefficientsinB(2)canbecompletedtoamatrixwithcoefficientsinB(2)oftheform(4.24)andofthesamenorm.ThisispreciselywhatParrottshowsin[70].Infact,hegivesanexplicitinductiveconstructionofthecoefficientsα−1,α−2,etc.whichcanbeaddedtothesequencea=(an)n≥0inordertoformanextendedsequencewiththedesiredproperty||Tα||=||Ha||.ThisallowsustoconcludethatXandX1areidentical.Sincetheircompletionsmustbealsoidentical,weobtaintheproofofthetheorem.NowwewishtoproveParrott’sresult,whichweusedpreviously.Inordertodothiswestateandprovesometechnicalresultsconcerningthecompletingmatrixcontractions.TheseresultsbelongalsotoParrott,butwefollowthepresentationfromPeller’smonograph[65].LetH,KbeHilbertspaces,AaboundedlinearoperatoronH,BaboundedlinearoperatorfromKtoH,andCaboundedlinearoperatorfromHtoK.TheproblemistofindoutunderwhichconditionsthereexistsaboundedlinearoperatorZonKsuchthattheoperatorABQZ=(4.25)CZonH⊕Kisacontraction,thatis,||QZ||≤1.Itiseasytoseethatiftheproblemissolvable,thentheoperatorsAandAB(4.26)CfromHtoH⊕KandfromH⊕KtoH,respectively,arecontractions.Itturnsoutthattheconverseisalsotrue.Theorem4.28.LetH,KbeHilbertspacesandlet,A:H→H,B:K→H,andC:H→Kboundedlinearoperators.ThenthereisanoperatorZ:K→KforwhichtheoperatorQZdefinedby(4.25)isacontractiononH⊕KifandonlyifA||||≤1and||AB||≤1.(4.27)C

106October20,20138:49WorldScientificBook-9inx6invers*11*oct*201392MatrixspacesandSchurmultipliers:MatricealharmonicanalysisNextwedescribealloperatorsZonKforwhichQZisacontraction.Inordertobeabletostatethisdescriptionweneedsomepreliminaries.Lemma4.29.LetH,H1,andH2beHilbertspaces,andletT:H1→HandR:H→Hbeboundedlinearoperators.ThenTT∗≤RR∗ifand2onlyifthereexistsacontractionQ:H1→H2suchthatT=RQ.Proof.SupposethatT=RQand||Q||≤1.Wehavethat(TT∗x,x)=(RQQ∗R∗x,x)=(Q∗R∗x,Q∗R∗x)=||Q∗R∗x||2≤||R∗x||2=(RR∗x,x).Conversely,assumethatTT∗≤RR∗.WedefinetheoperatorLonRangeR∗asfollows:LR∗x=T∗x,x∈H.2TheinequalityTT∗≤RR∗impliesthatLiswell-definedonRangeR∗and||L||≤1onRangeR∗.WecanextendLbycontinuitytotheclosureclosRangeR∗andputL|KerR=L|(RangeR∗)⊥=0.SetQ=L∗.ClearlyT=RQ.ForacontractionA:H1→H2thedefectoperatorDAisdefinedonH1byD=(I−A∗A)1/2.AItisalsoconvenientbesidesDAtoconsiderotheroperatorsDA:H1→HsuchthatD∗D=I−A∗A,whereHisaHilbertspace.InthiscaseAADA=VDAforsomeisometryVdefinedonclosRangeDA.Lemma4.30.LetH,K,HbeHilbertspaces,A:H→H,B:K→HHbeanoperatorsuchlinearoperatorssuchthat||A||≤1.LetDA∗:H→∗∗thatDADA∗=I−AA.Then||AB||≤1(4.28)ifandonlyifB=D∗H.A∗KforacontractionK:K→Proof.Itiseasytoseethat(4.28)isequivalenttothefactthatA∗AB∗≤IH,B∗∗∗∗whichmeansthatAA+BB≤Ior,whichisthesame,BB≤DADA∗.ByLemma4.29,thisisequivalenttothefactthatB=D∗A∗KforsomecontractionK:K→H.Theproofiscomplete.

107October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces93Remark.TheconclusionofLemma4.30isvalidifDA∗=DA∗=(I−AA∗)1/2.ItiseasytoseethatwecanchooseacontractionK:K→H∗suchthatRangeK⊂closRange(I−AA)andB=DA∗K.Clearly,suchacontractionKisunique.Lemma4.31.LetH,K,HbeHilbertspacesandletA:H→H,C:H→Kbelinearoperatorssuchthat||A||≤1.LetDA:H→HbeanoperatorsuchthatD∗D=I−A∗A.ThenAA$$$A$$$≤1(4.29)$C$ifandonlyifC=LDAforsomecontractionL:H→K.Proof.TheresultfollowsfromLemma4.30since(4.29)isequivalenttotheinequality||A∗C∗||≤1.Remark.AsinLemma4.30wecantakeD=D=(I−A∗A)1/2.AAClearly,onecanfindacontractionL:H→KsuchthatL|(Range(I−A∗A))⊥=0andC=LDA.AsinLemma4.30itiseasytoseethatsuchacontractionLisunique.NowweareinapositiontostatethedescriptionofthoseoperatorsZ:K→KforwhichtheoperatorQZdefinedby(4.25)isacontrac-tion.Aswehavealreadyobserved,theoperatorsin(4.26)arecontractions.Therefore(seetheRemarksafterLemmas4.30and4.31)thereexistuniquecontractionsK:K→HandL:H→Ksuchthat∗RangeK⊂closRange(I−AA),B=DA∗K,(4.30)L|(Range(I−A∗A))⊥=0,C=LD.(4.31)ATheorem4.32.LetH,KbeHilbertspaces,A:H→H,B:K→H,andC:H→Kboundedlinearoperatorssatisfying(4.27).LetK:K→HandL:H→Kbetheoperatorssatisfying(4.30)and(4.31).IfZ:K→Kisaboundedlinearoperator,thentheoperatorQZ,definedby(4.25),isacontractionifandonlyifZadmitsarepresentation∗Z=−LAK+DL∗MDK,(4.32)whereMisacontractiononK.

108October20,20138:49WorldScientificBook-9inx6invers*11*oct*201394MatrixspacesandSchurmultipliers:MatricealharmonicanalysisNotethatwemayalwaysassumethat⊥M|(RangeDK)=0andRangeM⊂closRangeDL∗.(4.33)Ifthesetwoconditionsaresatisfied,thenZdeterminesMuniquely,andsothecontractionsMsatisfying(4.33)parametrizethesolutionsZ.ItiseasytoseethatTheorem4.28followsfromTheorem4.32.Indeed,wecanalwaystakeM=0.ToproveTheorem4.32,weneedonemorelemma.Lemma4.33.LetA,BbeasaboveandletK:K→Hbeanoperatorsatisfying(4.30).ThentheoperatorD−A∗KAD(AB)=(4.34)0DKsatisfiesA∗D∗D=I−.(AB)(AB)H⊕K∗ABBProof.Wehavethat∗D−A∗KD−A∗KA∗AA+AB0D0DB∗KKD0D−A∗KA∗AA=+AB−K∗AD0DB∗KKD2−DA∗KA∗AA∗B=AA+−K∗ADK∗AA∗K+D2B∗AB∗BAKI−A∗A−DA∗KHA=∗∗2∗−KADAKK+DK−KDA∗DA∗KA∗AA∗B+.B∗AB∗B∗∗∗∗∗LetusshowthatDAA=ADA∗.Wehavethat(I−AA)A=A(I−AA∗).Itfollowsthatφ(I−A∗A)A∗=A∗φ(I−AA∗)foranypolynomialφ,sothesameequalityholdsforanycontinuousfunctionφ.Ifwetakeφ(t)=1/2∗∗t,t≥0,weobtainthatDAA=ADA∗.Similarly,DA∗A=ADA.Consequently,A∗D∗D+(AB)(AB)∗ABBI−A∗A−A∗BA∗AA∗BI0HH=+=.−B∗AI−B∗BB∗AB∗B0IH

109October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces95ProofofTheorem4.32.Supposethat||QZ||≤1.ByLemma4.29CZ=XYD(AB),whereXYisacontractionfromH⊕KtoKandD(AB)isdefinedby(4.34).ThenC=XDA.LetPbetheorthogonalprojectionfromHontoclosRange(I−A∗A).PutX=XP.Clearly,C=XDand,bytheRemarkAafterLemma4.31,wehavethatX=L.ItiseasytoseethatCZ=LYD(AB).(4.35)Clearly,P0LY=XY,0IHwhichprovesthatLYisacontraction.Then,byLemma4.30,theoperatorYadmitsarepresentationY=DL∗MforacontractionMonK.Formula(4.32)followsnowimmediatelyfrom(4.35).SupposethatZsatisfies(4.32),whereMisacontractiononK.Thenitiseasytoseethat⎛⎞⎛⎞ADL∗0I0ABI00=⎝D−A∗0⎠⎝0K⎠.(4.36)ACZ0LDL∗00M0DKTheresultfollowsfromthefactthatallfactorsontheright-handsideof(4.36)arecontractions,whichisaconsequenceofthefollowinglemma:Lemma4.34.LetTbeacontractiononaHilbertspace.ThentheoperatorTDT∗D−T∗Tisunitary.Proof.Wehave∗∗2∗∗TDT∗TDT∗TT+DTTDT∗−DTT=.∗∗2∗DT−TDT−TDT∗T−TDTDT∗+TT∗∗IthasbeenshownintheproofofLemma4.33thatTDT∗=DTTandTDT=DT∗T.Thus,∗TDT∗TDT∗I0=.D−T∗D−T∗0ITTSimilarly,∗TDT∗TDT∗I0=.D−T∗D−T∗0ITTTheproofiscomplete.

110October20,20138:49WorldScientificBook-9inx6invers*11*oct*201396MatrixspacesandSchurmultipliers:MatricealharmonicanalysisRemark.ThesameresultsarevalidifAisanoperatorfromH1toH2,BisanoperatorfromK1toK2,CisanoperatorfromH1toK2,andZisanoperatorfromK1toK2,whereH1,H2,K1,K2areHilbertspaces.Theproofgivenaboveworksalsointhismoregeneralsituation.Remark.ItisclearthatifwereplaceinTheorem4.28HbyKandconverselywegetthatthematrixwithoperatorsasitsentriesZCBACisacontractiononK⊕HifandonlyifBAandarecontractions.ANowletuspresenttheParrott’sargumenttocompletetheproofofTheorem4.25(see[70]):ConsidertheblockHankelmatrix(thatisamatrixhavingoperatorsasitsentries)ΓΩ={Ωj+k}j,k≥0,whereΩ={Ωj}j≥0isasequenceofboundedlinearoperatorsfromHtoK,fortwoseparableHilbertspacesH,K,andthematrixΓΩisoftheform:⎛⎞Ω0Ω1Ω2Ω3...⎜⎜Ω1Ω2Ω3Ω4...⎟⎟⎜.⎟⎜⎜Ω2Ω3Ω4.....⎟⎟.⎜.⎟⎜.⎟⎝Ω3Ω4.⎠......NowwewishtoconstructablockHankelmatrix⎛⎞ΓΩoftheform......⎜......⎟⎜⎟⎜Ω−2Ω−1Ω0...⎟⎜⎟⎜Ω−1Ω0Ω1...⎟⎜⎟⎜Ω0Ω1Ω2...⎟⎜⎟⎜..⎟⎜Ω1Ω2Ω3.⎟⎝⎠.........suchthat||ΓΩ||=||ΓΩ||.WeconstructΓΩinductively,firstconstructingaHankelmatrix⎛⎞ZΩ0Ω1Ω2...⎜⎜Ω0Ω1Ω2Ω3...⎟⎟⎜⎜Ω1Ω2Ω3Ω4...⎟⎟Γ−1=⎜.⎟⎜.⎟⎝Ω2Ω3Ω4Ω5.⎠...............

111October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces97suchthat||Γ−1||=||ΓΩ||.ThisofcourseinvolvesthechoiceofΩ−1=Z.Giventhatthisconstructionisalwayspossible,successiveiterationswillproduceΩ−2,Ω−3,...,andΓΩ.TochooseΩ−1wewrite⎛⎞⎛⎞Ω1Ω2Ω3...Ω0⎜⎜Ω2Ω3Ω4...⎟⎟⎜⎜Ω1⎟⎟A=⎜⎝Ω3Ω4Ω5...⎟⎠,B=⎜⎝Ω2⎟⎠,...............C=Ω0Ω1Ω2....ThematrixΓ−1canbeidentifiedwiththematrixZC.BAClearly,$$$C$||BA||=||ΓΩ||,$$$$=||ΓΩ||.AByTheorem4.28andthelasttworemarks,thereexistsanoperatorZsuchthat$$$ZC$$$$$=||ΓΩ||.BANowputΩ−1=Z.Wehave||Γ−1||=||ΓΩ||.TheproofofTheorem4.25iscomplete.4.5AcharacterizationofthespaceT1InthebookofM.Pavlovi´c([68]page96)thereisthefollowingbeautifulcharacterizationoffunctionsbelongingtotheHardyspaceH1:Pavlovi´cTheoremForafunctionf,whichisanalyticinD,thefol-lowingassertionsareequivalent:a)f∈H1;n11b)sup||sjf||<∞;nanj+1j=0

112October20,20138:49WorldScientificBook-9inx6invers*11*oct*201398MatrixspacesandSchurmultipliers:Matricealharmonicanalysisc)sup||Pnf||<∞.nHere,forafunctionfanalyticinDlet1n1n1Pnf=sjf,wherean=(n=0,1,2,...)anj+1j+1j=0j=0andsjfarethepartialsumsoftheTaylorseriesoff.Ananalogueofthisresultusingthevector-valuedHardyinequalitygiveninSection4.4isalsotrueandispresentedbelow.Moreprecisely,wehavethefollowingresult:Theorem4.35.LetA∈B(2)beanuppertriangularmatrix.Thefollow-ingassertionsareequivalent:a)A∈T1;n11b)sup||sjA||<∞;nanj+1j=0c)sup||PnA||<∞.nHere1n1n1PnA=sjA,wherean=(n=0,1,2,...)anj+1j+1j=0j=0jandsjA=k=0Ak.Proof.Obviouslyb)⇒c).a)⇒b).LetA∈T,andforfixedn≥2,w∈D,andr=1−1<1,1ndefinethematrix-valuedfunctiong(z)=(1−rz)−1[A∗C(rwz)](|z|≤1),whereC(z)istheToeplitzmatrixcorrespondingtothefunction1for1−zeachz∈D.Thenwehavethat∞∞∞g(z)=Arkwkzkrlzl=Awkrk+lzk+l=kkk=0l=0k,l=0

113October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces99∞m∞Awkrmzm=s(A∗C(w))rmzm.kmm=0k=0m=0Henceg(m)=s(A∗C(w))rm,m=0,1,2,...mItiswell-known(andeasytosee)that||smA||T1≤Cln(m+1)||A||T1∀A∈T1andm∈N,(4.37)whereC>0isanabsoluteconstant.Moreover,g∈H1since,by(4.37),wehavethatC11Cln(m+1)||sm(A∗C(w))||T1≤||smA||T1≤∀m∈Nand|w|<1.1−|w|1−|w|Therefore∞∞mmCm=0rln(m+1)||sm(A∗C(w))||T1r≤<∞.1−|w|m=0Weconcludethat∞1∞1||s(A∗C(w))||rj=||g(j)||jT1T1j+1j+1j=0j=0≤(byCorollary4.14forX=T1)||A∗C(rweit)||T1≤C||g||H1=itforallt∈[0,2π).T1|1−re|Sincerj=(1−1)j≥c∀0≤j≤n,wherec>0isanabsoluteconstant,nwehave:n2π1itdt||sj(A∗C(w))||T1≤C||g(re)||T1j+102πj=02πit||A∗C(rwe)||T1dt=C.0|1−reit|2πIntegratingthisinequalityoverthecircle|w|=1andsincesj(A∗C(w))=sj(A)∗C(w),wefind,usinglimw→eiθ||sj(A)∗C(w)||T1=||sA∗C(eiθ)||∀j,thatjT1n12πdθπ2π2π||A∗C(rei(θ+t))||dtdθiθT1||sjA∗C(e)||T1≤itj+102πc00|1−re|2π2πj=0

114October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013100MatrixspacesandSchurmultipliers:Matricealharmonicanalysis2π2ππdθdt=||A∗Pr(t+θ)||T1itc002π2π|1−re|2ππ1dt≤||A||T1it≤C||A||T1lnn,c2π0|1−re|wherePr(t+θ)istheusualPoissonkernelontheunitcircleandC>0isanabsoluteconstant.However,denotingbyEθtheToeplitzmatrixcorrespondingtoδθ(theDiracmeasureconcentratedinθ)wehavethat2π2π1iθ1||sjA∗C(e)||T1dθ=||sjA∗Eθ||T1dθ.2π02π0Since||B||T1=||B∗Eθ∗E2π−θ||T1≤||B∗Eθ||T1||E2π−θ||M(2)=||B∗Eθ||T1≤||B||T1||Eθ||M(2)=||B||T1∀θ∈[0,2π]itfollowsthatn1n12πdθ||sA||≤||sA∗C(eiθ)||≤C||A||lnn,jT1jT1T1j+1j+102πj=0j=1thatisn11||sjA||T1≤C1||A||T1anj+1j=0andb)holds.c)⇒a)ItisclearthatifAisafinitematrix,then||A||C1≤sup||PnA||C1.nNowassumethatAisanymatrixsuchthatsupn||PnA||C1<∞.LetEmbethecanonicalprojectionwhichprojectsamatrixtoitssubmatrixofordermattheleftuppercorner.SincePnandEmcommute,wefindthatsupsup||PnEmA||C1<∞.mnBytheprecedingremark,wehavethatsup||EmA||C1≤supsup||PnEmA||C1<∞;mmnwhenceA∈C1and||A||C1≤sup||PnA||C1.nThisinequalityholdswithouttheassumptionthatAisuppertriangular.Theproofiscomplete.

115October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces101AsimpleconsequenceofTheorem4.35isthefollowing:Corollary4.36.IfA∈T1,thenn11lim||A−sjA||=0(4.38)nanj+1j=0and,consequently,n11lim||sjA||=||A||.(4.39)nanj+1j=0Proof.Obviously(4.38)holdsifAisafinitematrix.SincefinitematricesaredenseinT1theproofof(4.38)follows.Thesecondassertionfollowsimmediatelyfrom(4.38).WeremarkthatB.Smith[89]provedin1983therelation(4.39)forf∈H1insteadofA∈T,andthisinfact,motivatedPavlovi´ctogivehis1theorem.AsaconsequenceofthistheoremwehavethatCorollary4.37.IfA∈T1,thenliminfn→∞||A−snA||T1=0.OfcoursethelastthreeresultsarematrixversionsofsometheoremsconcerningthestrongconvergenceinH1.(See[68].)4.6AnextensionofShields’sinequalityInthissectionwepresentanextensionofthematrixversionofShields’sinequality.TheresultsconcerningthefunctionsandanalyticmeasuresonTwereobtainedbyC.McGehee,L.PignoandB.Smithin1981(see[63]).Inparticular,theabovementionedauthorsprovedtheLittlewoodcon-jecture[40]from1948.InwhatfollowswedenotebyM(T)theusualconvolutionalgebraofBorelmeasuresonT.FirstwepresentthefollowinggeneralizationoftheclassicalHardyin-equality:Theorem4.38.ThereisarealnumberC>0suchthat,givenanysetS={n1

116October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013102MatrixspacesandSchurmultipliers:Matricealharmonicanalysisinequalityholds:∞|μ(nk)|≤C||μ||.kk=1AdirectconsequenceofTheorem4.38isthefollowingresult:NinkθCorollary4.39.Ifp(θ)=k=1ckewhere{n1

117October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces103|f(n)|=4−jifn∈S;(4.42)jjfjμ≥0.(4.43)∞inθPut|fj|=−∞cneanddefine−11inθhj(θ):={c0+2cne}.4−∞Moreover,observethat,viatheconditions(4.40),(4.41),(4.42),wehavethat||f||=2−j,(4.44)j2sothat√√22−j−j−3||hj||2≤||fj||2=2<3·2.(4.45)44LetF0=(1/5)f0anddefineinductively,−hj+11Fj+1=Fje+fj+1(j=0,1,2,...).5Noticethatsinceh=(1/4)|f|,wehavethate−hj∈L∞(T)becausejj|f|≤1fromconditions(4.40)through(4.42).ThusF∈L∞(T)andwejjfind,accordingto(4.40)andtheinequalitye−x/4+x/5≤1for0≤x≤1that||Fj||∞≤1(j=0,1,2,...).Moreover,sincesupph⊂Z−,wehavethatjsupp{ehj}⊂Z−(4.46)viapart(a)ofLemma4.40.Wenowclaimthat,foranym,11|Fm(n)−fj(n)|≤|fj(n)|ifn∈Sjandj≤m.(4.47)510Inordertoprovetheaboveassertionwefirstobservethat1−mh1−mhF=fe1k+fe2k+...m01551−hm1+fm−1e+fm;55asaconsequenceof(4.41)and(4.46).Hence,ifn∈Sjandj

118October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013104MatrixspacesandSchurmultipliers:Matricealharmonicanalysis:;fj+1−mh+ej+2k−1(n)5fm−1)−hm*+···+e−1(n).5Itisnowobviousfrompart(b)ofLemma4.40andtheCauchy-Schwarzinequalitythat1mm|Fm(n)−fj(n)|≤{||fj||2||hk||2+||fj+1||2||hk||2+...5j+1j+2+||fm−1||2||hm||2}.Thus,inviewoftheinequalities(4.44)and(4.45),weobtainthat|F(n)−1f(n)|≤3(4−j−1+4−j−2+...)=14−j=1|f(n)|.mjj5101010Letj>0andsupposethatn∈S.Then3k>4jand,hence,kj−j1|fj(nk)|=4>.(4.48)3kNoticethat1(Fmμ)(nk)≥fj(nk)μ(nk)(4.49)10becauseoftheinequalities(4.43)and(4.47).Moreover,asaconsequenceoftheinequalities(4.43),(4.48)and(4.49),weseethatifnk∈Sj,j≤m,then1(Fmμ)(nk)≥|μ(nk)|.(4.50)30PutBm=S0∪S1∪···∪SmandassumeforthemomentthatμisatrigonometricpolynomialonT.Then,ontheonehand|(F∗μ)(0)|≤||μ||1because||Fm||∞≤1,whileontheotherhand|(Fm∗μ)(0)|=|Fm(n)μ(n)|.n∈BmHence,theinequality(4.50)permitsustoconcludethat,foralltrigono-metricpolynomialsμwithsuppμ⊂S,∞|μ(nk)|≤30||μ||1.(4.51)k1

119November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces105Astandardapproximateidentityargumentimpliestheinequality(4.51)forallµ∈M(T)withsuppµb⊂S.Theproofiscomplete.PNinkθProofofCorollary4.39.Givenp(θ)=k=1cke(n1

120November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013106MatrixspacesandSchurmultipliers:MatricealharmonicanalysisCorollary4.42.(GeneralizedShields’sinequality)Thereisacon-stantC>1suchthatgivenanysetn11suchthat,givenanyset{n1

121October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionsofHardyspaces107AlsotheproofofSarason’snoncommutativefactorizationtheoremwiththeproofofU.HaagerupandG.Pisierisgiven.WementiontheinterestingandusefulParott’sresultaboutthecompletionofaHankelmatrixwithB(2)-coefficients.Inanexcellentmonograph[68]dedicatedtoanalyticfunctionsonthediskM.PavloviccharacterizedtheanalyticfunctionsfromH1bythecon-ditionn1sup||sjf||<∞.nanj=0Inspiredbythisresultweprovedin[73]acharacterizationofuppertriangularmatricesfromC1.ThischaracterizationisstatedandprovedinSection4.5.Finally,inSection4.6wepresenttheproofofanextensionofShields’sinequality.

122May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

123October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Chapter5ThematrixversionofBMOA5.1FirstpropertiesofBMOA(2)spaceWeintroducedifferentdefinitionsforthematrixversionofBMOA,whichmaybedifferentfromeachotherbutontheclassofToeplitzmatrices,theyallcoincidewiththeclassicalBMOA.OneofthesedefinitionsforthematrixversionofthespaceBMOA,denotedforshortbyBMOA(2),isasfollows:Forλ∈Dweputk(e2πiθ):=1anddenotebyKtheToeplitzλ1−λe2πiθλmatrixassociatedwiththefunctionkλ(·).Definition5.1.ThespaceBMOA(2)isdefinedbyBMOA(2):={A|Auppertriangular||A||BMOA(2)<∞},where1||A||2:=sup{||A(r)K||2(1−r2)(1−|λ|2)rdr}.BMOA(2)λrL2(2)λ∈D0HereL2()wasintroducedinSection4.1.2WeremarkthatBMOA(2)isaBanachspaceendowedwiththenorm||A0||B(2)+||A||BMOA(2).OurBMOA()isapropersubspaceofBMOA(L2())introduced2C2byBlascoin[13].LetVMOA(2)bethesubspaceofBMOA(2)consistingofthosema-tricesAsuchthat1lim{||A(r)K||2(1−r2)(1−|λ|2)rdr}=0.λrL2(2)λ→10109

124October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013110MatrixspacesandSchurmultipliers:MatricealharmonicanalysisWeneedtheabovedefinitionofBMOA(2)andthefollowingresultsinordertoextendanicetheoremofMateljevicandPavlovic(see[60]).Proposition5.2.IfA∈L2(),andB∈B(),thenwehavethatr22||AB||L2(2)≤||A||L2(2)||B||B(2).rrProof.Wenotethat1LAB(x,t)=LB(x,s)LA(−s,t)ds,0whichimpliesthat∞∞∞L(x,t)=LB(x)CA(t)=LB(x)ae2πijt,ABkkkjkk=1j=1k=1and,hence,∞LAB(x)=LB(x)a,jkjkk=1forallj≥1.Ofcourse,hereA=(ajk)j≥1;k≥1,andB=(bjk)j≥1;k≥1.Wededucethat1/21AB2||AB||L2(2)=sup|Lj(x)|dx=rj≥101∞supsup|LB(x)ah(x)dx|≤kjkj≥1||h||≤10L2k=1∞1/2∞1/21supsup|a|2|LB(x)h(x)dx|2jkkj≥1||h||≤10L2k=1k=1=||A||L2(2)||B||B(2)randtheproofiscomplete.Remark5.3.ByreasoningasintheproofofProposition5.2wecanderivethat||AB||L2r(2)≤||A||B(2)||B||L2c(2).(5.1)IfthematrixBisaToeplitzmatrix,then||B||L2(2)=||B||L2(2)rcand,consequently,(5.1)isequivalenttothat||AB||L2r(2)≤||A||B(2)||B||L2r(2)forBaToeplitzmatrix,andwemaydeletethesubscriptrintheaboveformulas.

125October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013ThematrixversionofBMOA111WegivenowasufficientconditiontoguaranteethatanuppertriangularmatrixbelongstoBMOA(2).nCorollary5.4.IfAisamatrixoffinitebandtypeA=k=0Ak,thenwehavethat1/21||A||≤||A(r)||2(1−r)dr.BMOA(2)B(2)0Proof.||A||BMOA(2)≤(byProposition5.2andRemark5.3)≤1/21Csup||A(r)||2||K||2(1−r)(1−|λ|2)rdr.B(2)λrL2(2)λ∈(0,1)0Wenotethat12dθ1||Kλr||L2(2)=2πiθ2=1−|λr|2,0|1−λre|2(1−|λ|)randsince1−|λr|2≤1wearedone.SeealsoProposition1.2-[13]wherearatherstrongresultwasprovedfortheBlascospaceBMOAC(X)withanarbitraryBanachspaceX.InspiredbyProposition1.3-[13]wegiveanexampleofamatrixbelong-ingtoBMOA(2).Letusdenotefirstbyeij,forfixedi≥1,j≥1,thematrixhaving1ontheintersectionoftheithrowwiththejthcolumnand0otherwise.∞1Example5.5.LetA=k=1nln(n+1)en,2n.Thenthecomputationsdonein[13]-Proposition1.3giveusthat1dr||A||BMOA(2)≤C12<∞.0(1−r)(ln(1−r))5.2AnothermatrixversionofBMOandmatricealHankeloperatorsInthelastthirtyyearsanumberofimportantresearchpapersaredevotedtoHankeloperators.Seeforinstancetheveryimpressivemonograph[65].InthatbookthereisachapterdedicatedtovectorialHankeloperators,whichareimportantforapplicationsandnowwerecallsomenotionsanddefinitionsfromthischapter.

126October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013112MatrixspacesandSchurmultipliers:MatricealharmonicanalysisLetusdenotebyH2(H)theHardyclassoffunctionswithvaluesinaseparableHilbertspaceHandbyB(H,K)thespaceofboundedlin-earoperatorsfromHtoKandbyL∞(B(H,K))thespaceofboundedweaklymeasurableB(H,K)-valuedfunctions.WerecallthattheHardyclassH2(H)isdefinedasfollows:H2(H)={F∈L2(H):Fˆ(n)=0,n<0},whereL2(H)isthespaceofweaklymeasurableH-valuedfunctionsFforwhich||F||2=||F(ζ)||2dm(ζ)<∞.L2(H)HTPutH2(H)=L2(H)H2(H).LetKbeanotherseparableHilbert−spaceandletΦbeafunctioninL2(B(H,K))(see[65],chapter2),i.e.s||Φ(ζ)x||2dm(ζ)<∞foranyx∈H.(5.2)KTRecallthatforfunctionssatisfying(5.2)theFouriercoefficientsΦ(ˆn)∈B(H,K)aredefinedbyΦ(ˆn)x=nζΦ(ζ)xdm(ζ),n∈Z,x∈H.TNowwecandefineforsuchfunctionsΦtheHankeloperatorHΦ:H2(H)→H2(K)onthesetofpolynomialsinH2(H)by−HF=PΦF,F∈H2(H),(5.3)Φ−wherewedenotebyPandPtheorthogonalprojectionsontoH2(H)and+−H2(K),respectively.−WedenotebyC2thesetofallHilbert-Schmidtmatrices,equippedwiththeusualHilbert-Schmidtnorm||·||C2.LetT2bethesetofalluppertriangularHilbert-Schmidtmatricesen-dowedwiththenorminducedby||·||C2.OfcourseT2isaHilbertspace.IfA∈T2andk=0,1,2,3,...wedenotebyAkthekth-diagonalofA.Then,formally,A=k∈ZAkandwedenotebyP+thetriangularprojectionP+A=k≥0Ak.Itiswell-knownthatP+:T2→T2isboundedwithnormlessthan1.NowputP−=I−P+,whereIistheidentityonT2.Ofcourse||P−||≤1.PutH∞(B(T,T)):={Φ∈L∞(B(T,T)):Φ(ˆn)=0,n<0}.2222

127October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013ThematrixversionofBMOA113Thenthefollowingstatementisknown(see[65]):VectorialNehariTheorem.LetΦbeaB(T2,T2)-valuedfunctionthatsatifies(5.2).ThentheoperatorHΦdefinedby(5.3)extendstoaboundedoperatoronH2(H)ifandonlyifthereexistsafunctionΨinL∞(B(T,T))suchthat22Ψ(ˆn)=Φ(ˆn),n<0,and∞||HΦ||=distL∞(Ψ,H(B(T2,T2)).nLetΦbeaninfinitematrixsuchthatforallmatricesA=k=0Ak,n∈N,wehavethatP−(ΦA)∈C2.Examplesofsuchmatricesareeitherallmatricesrepresentinglinearboundedoperatorson2,ormatricesΦsuchthatP−Φ=0.WedefinethematrixversionofHankeloperatorHΦasfollows:HΦ:T2→(T2)−:=C2T2,onthedensesubspaceinT2ofallmatricesA=nk=0Ak,n∈N,suchthatΦA∈C2,byHΦ(A)=P−(ΦA).ThematrixΦiscalledthesymboloftheHankeloperatorHΦ.IfΦisaninfinitematrixandifwedenotebyΦalsotheoperatorfunc-tiongivenbyΦ(ζ)=eikζΦ,thenitiseasytoverifythatthematrixk∈ZkversionoftheHankeloperatorHΦdefinedonT2coincideswiththerestric-tionofPeller’svectorialHankeloperatorHtothesubspaceofH2(T)Φ2consistingofallmatrixfunctions∞eikζA,forsomeA∈T.k=0k2In1985S.Power[77]gavethematrixversionsofNehari’sandHartman’stheorems.InwhatfollowswepresenttheseresultsaboutmatrixversionsofHankeloperatorsHΦwithdifferentproofs.Wedenoteasin[84],byA˜thematrixwhoseentries˜aklaregivenby⎧⎨−iaklifkl,⎩0ifk=l,whereaklaretheentriesofA.WerecallthatB(2)meansthespaceofrepresentingmatricesofalllinearboundedoperatorson2,equippedwiththeusualoperatornorm.

128October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013114MatrixspacesandSchurmultipliers:MatricealharmonicanalysisMoreover,wedenotebyBMOF(2)thespaceofallmatricesAsuchthatthereexistmatricesΦandΨfromB(2)withA=Φ+Ψ˜,equippedwiththenorm||A||BMOF(2):=inf{||Φ||B(2)+||Ψ||B(2);A=Φ+Ψ˜}.OfcourseBMOF(2)isaBanachspace.WealsodefineBMOAF:=BMOF(2)∩H∞(),whereH∞()isthesetofalluppertriangularmatricesA∈B().222Thenthefollowingholds[69](foramoregeneralresultsee[77],[59]):Theorem5.6.(MatrixversionoftheNeharitheorem)LetΦbeaninfinitematrixsuchthatΦA∈C2forallfiniteband-typematricesA.Thenthefollowingconditionsareequivalent:(a)HΦisaboundedlinearoperatoronT2.(b)ThereisΨ∈B(2)suchthatΨk=Φkforallk<0.(c)P−Φ∈BMOF(2).Proof.(b)⇒(a).BysimplecomputationswegetthatHΦ(A)=HΨ(A)nforallA=k=0Ak,n∈N.Hence,||HΦ(A)||C2=||HΨ(A)||C2≤||ΨA||C2≤||Ψ||B(2)||A||T2,thatis||HΦ||≤||Ψ||B(2)<∞.(a)⇒(b).WenotethatHΦ(A)=P−(ΦA)=P−[P−(Φ)A]forallA=nk=0Ak,n∈N.AssumethatHΦ:T2→(T2)−isaboundedlinearoperator.Then||H(A)||=sup||=sup|trH(A)B∗|=Φ(T2)−ΦΦ||B||(T2)−≤1||B||(T2)−≤1sup|trP[(PΦ)A]B∗|.(5.4)−−||B||(T2)−≤1But,obviouslytrP[(PΦ)A]B∗=tr(PΦ)AB∗forB∈(T)and−−−2−A∈T2.Hence,(5.4)canbewrittenas||H(A)||=sup|tr(PΦ)(AB∗)|.Φ(T2)−−||B||(T2)−≤1SinceA,B∗∈TandeveryA∈TmaybewrittenasA=BB,where21121/2B1,B2∈T2and||B1||T2=||B2||T2=||A||T1(see[84]),wehavethat∞>M=||H||=sup|tr(PΦ)AB∗|=Φ−||A||(T2)≤1,||B||(T2)−≤1

129October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013ThematrixversionofBMOA115sup|tr(P−Φ)A|=||P−Φ||(T1,0)∗,||A||(T1)≤1;A0=0whereT1,0={A∈T1;A=j≥1Aj}.UsingtheHahn-Banachtheoremwegetthat∞>||HΦ||=inf||Ψ||B(2).P−Φ−Ψ∈(T1,0)⊥But(T)⊥={Ψ∈B()|tr(ΨB)=0forallB∈T}=1,021,0{Ψ∈B(2)|P−Ψ=0}and,hence,thereisΨ∈B(2)suchthatP−Φ=P−Ψ.Thus(b)holds.(b)⇒(c).P−Φ=P−ΨforΨ∈B(2).ButP−Ψ=−iΨ+˜P+Ψ−Ψ0=−iΨ+Ψ˜−P−Ψ−Ψ0and,thus,PΨ=−1iΨ+˜1Ψ−1Ψ,thatisPΨ∈BMO().−2220−F2(c)⇒(b).LetP−Φ∈BMOF(2).ThenthereexistA,B∈B(2)withP−Φ=A+B.˜ButP−Φ=P−(A)+P−(B˜)=P−(A)+iP−B=P−(A+iB)=P−Ψ,Ψ∈B(2),andtheimplicationfollows.Theproofiscomplete.Corollary5.7.Ityieldsthat||H||=inf{||Φ−F||;F∈H∞()}=dist(Φ,H∞()).ΦB(2)2B(2)2Proof.Itfollowsdirectlyfromtheequality||H||=inf{||Ψ||;Ψ−PΦ∈(T)⊥}.ΦB(2)−1,0WeintendtogiveamatrixversionofHartman’sTheorem(see[65])aboutcompacityofHankeloperators.Weneedthefollowinglemma:Lemma5.8.LetΦ∈C∞,thespaceofallcompactoperatorson2.Thendist(Φ,H∞())=dist(Φ,T),whereT={A∈C|AupperB(2)2B(2)∞∞∞triangular}.

130October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013116MatrixspacesandSchurmultipliers:MatricealharmonicanalysisProof.Theinequalitydist(Φ,H∞())≤dist(Φ,T)istrivial.B(2)2B(2)∞Conversely,letΦ∈C,H∈H∞()andPbetheprojectiononB(),∞2n2givenbyPA=(a),whereniji,j≥1aijif1≤1,j≤naij=,n=1,2,3,....0otherwiseThen,sinceΦisacompactoperator,foreach>0thereexistsn0=n0()suchthat||Φ−Pn0Φ||<.Therefore,||Φ−H||B(2)≥||Pn0Φ−Pn0H||B(2)≥||Φ−Pn0H||B(2)−≥distB(2)(Φ,T∞)−and,sinceisarbitrary,wehavethatdist(Φ,H∞())≥dist(Φ,T).B(2)2B(2)∞Theproofiscomplete.Theorem5.9.H∞()+CisaclosedSchursubalgebraofB().2∞2Proof.Lemma5.8showsthatthecanonicalinclusionC∞/T∞→B()/H∞()isanisometryand,therefore,C/Tisaclosedsubspace22∞∞ofB()/H∞().22Letρ:B()→B()/H∞()bethecanonicalmap.Itfollowsthat222H∞()+C=ρ−1(C/T)isaclosedsubspaceofB().2∞∞∞2NowweshouldshowthatH∞()+CisaSchursubalgebraofB().2∞2But,infact,(A1+B1)∗(A2+B2)=(A1∗A2+B1∗A2+A1∗B2)+B1∗B2,forallA,A∈H∞()andB,B∈C.12212∞ThefirstexpressionontherighthandsideoftheaboveequalitybelongstoH∞()by[11].OntheotherhandB∗B∈C,sincelimPB=B,212∞nniii=1,2,againby[11].NowletusdenotebySthelinearandboundedoperatoronC2given⎛⎞⎛⎞a1a1a1...a1a1a1...−101012⎜..⎟⎜..⎟⎜a1a2a2.⎟⎜a1a2a2.⎟⎜−2−10⎟⎜−101⎟byS(A)=⎜......⎟forA=⎜....⎟.⎜a1...⎟⎜a1a2..⎟⎝−3⎠⎝−2−1⎠........................Thenwehavethefollowing:Lemma5.10.LetK:T2→(T2)−beacompactoperatorandS:T2→T2.Thenlim||KSn||=0.n→∞

131October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013ThematrixversionofBMOA117Proof.ItisenoughtoprovethelemmaforrankoneoperatorsK.LetK(A)=C,whereB∈T2,C∈(T2)−.ThenwehavethatKSn(A)=C=C,where⎛⎞⎛⎞a1a1a1...a1a1a1...123012⎜..⎟⎜..⎟⎜0a2a2.⎟⎜0a2a2.⎟∗⎜12⎟⎜01⎟S(A)=⎜....⎟forA=⎜....⎟∈T2.⎜⎝00..⎟⎠⎜⎝00..⎟⎠........................Hence||KSn||=||S∗nB||||C||→0.T2T2Lemma5.11.LetΦ∈C∞.ThenHΦ:T2→(T2)−isacompactoperator.Proof.LetF=eij,forfixedi,j≥1suchthati−j≥1.ThenHF(A)=i−j−1jj∞k=0akei,k+1,whereAhastheentries(ak)j=1onthekth-diagonal,fork≥0.ThereforeHFisacompactoperatorand,consequently,itiseasytoseethatHPnΦisacompactoperatortoo,foralln≥0.ByTheorem5.6,andusingthefactthatΦisacompactoperator,wehavethat||HΦ−HPnΦ||≤||Φ−PnΦ||B(2)→0.Consequently,HΦisacompactoperator.NowwearereadytostatethematrixversionoftheHartmantheorem(see[69],[77]):Theorem5.12.LetΦ∈B(2).Thefollowingconditionsareequivalent:(i)HΦisacompactoperator.(ii)Φ∈H∞()+C.2∞(iii)ThereexistsΨ∈C∞withHΦ=HΨ.Proof.Obviously(ii)⇒(iii).Since(iii)impliesthatΦk=Ψkforallk<0,theassertion(i)followsinthesamewayas(b)⇒(a)inTheorem5.6.(i)⇒(ii).Wehavethatn−1n∞||HΦS||=||HSnΦ||=(byCorollary5.7)=distB(2)(Φ,(S)H(2)).Then,forall>0andforalln≥0thereexistBn∈H∞()anda2matrixAnoffinitebandtypesuchthatAn=−n(An);An∈B()j=−1j2and,byusingTheorem5.6,nnn||HΦS||B(2)≥||Φ−A−B||B(2)−≥||HΦ−HAn||−

132October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013118MatrixspacesandSchurmultipliers:Matricealharmonicanalysis≥||PmHΦ−PmHAn||−forallm≥1.Hence,n||HΦS||≥||HΦ−PmHAn||−||HΦ−PmHΦ||−forallm,n.ButPmHAn=HPm(An),and,therefore,n||HΦS||≥||HΦ−HPm(An)||−||HΦ−PmHΦ||−∀m,n.Moreover,byhypothesis,HΦisacompactoperator,and,consequently,thereexistmandC∈H∞()suchthat||H−PH||<,and02Φm0Φn||HΦS||≥||HΦ−HPm(An)||−2≥(byCorollary5.7)≥||Φ−P(An)−C||−3,foralln.m0B(2)ByLemma5.10wegetthat≥dist(Φ,H∞()+C)−3.B(2)2∞Thereforedist(Φ,H∞()+C)=0and,byTheorem5.9,itfol-B(2)2∞lowsthatΦ∈H∞()+C.Theproofiscomplete.2∞IfwedenotebyVMOF(2):={Φ|Φ=A+B,A,B˜∈C∞}equippedwiththenorminducedbythatofBMOF(2),then,asintheproof(b)⇒(c)ofTheorem5.6,wecanderivethefollowingstatement:Theorem5.13.LetΦbeaninfinitematrix.ThenHΦisacompactoper-atorifandonlyifP−Φ∈VMOF(2).BMOAF(2)isthespaceofalluppertriangularmatricesA∈BMOF(2).ThisspaceisofinterestbecausetheproofofTheorem5.6-(b)⇒(c),Hahn-Banachtheoremandthewell-knownfact[34]thatC∗=B()via12thebilinearmap=trAB∗forA∈B()andB∈C21showusthatT∗=BMOA()1F2bythepreviousbilinearmap.SeealsoTheorem2.3[77].

133October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013ThematrixversionofBMOA1195.3NuclearHankeloperatorsandthespaceM1,2WeshallherederiveasufficientconditiontoguaranteethattheHankeloperatorHAintroducedinSection5.2,whereAisanuppertriangularmatrix,isnuclear.Moreover,ifthematrixAhasonlyafinitenumberofrowsthisconditionisalsonecessary.Let1≤p<∞.WedenotebySptheSchattenclassofalloperatorsfromT2intoT2−.WedenotebyMp,2thespaceofalluppertriangularinfinitematricesA=(al)suchthatkk≥0,l≥1⎛⎛⎞p/2⎞1/p∞∞∞||A||:=⎜⎝⎝|al|2⎠⎟⎠<∞.(5.5)p,2nl=1j=0n=jOurnextresultreads:Theorem5.14.LetAbeanuppertriangularinfinitematrixsuchthat1,2A∈M.ThentheHankeloperatorHA∗:T2→T2−isnuclearand,moreover,||HA∗||S1≤||A||M1,2.Proof.Weeasilyseethat⎛⎞1/2∞∞∞⎝k2⎠kklHA∗(B)=|aj|·A,(5.6)l=0k=1j=l+1−1/2whereB∈T,Akl=∞akEk+l∞|ak|2,theseries2j=l+1j−j+lj=l+1j(5.6)convergesinthenormofCandEk,wherej∈Z,andk≥1,isthe2jinfinitematrixhavingasentries1onthekthplaceonthejth-diagonal,and0otherwise.Consequently,⎛⎞1/2∞∞∞⎝k2⎠kklHA∗=|aj|·El⊗Al=0k=1j=l+11,2isanuclearoperatorforA=A∈Mand||HA∗||S1≤||A||M1,2.Theproofiscomplete.

134October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013120MatrixspacesandSchurmultipliers:MatricealharmonicanalysisNotesInSection5.1weintroducethematrixversionsoftheclassicalspacesBMOandBMOA,BMO(2)andBMOA(2).ThesespacesarepropersubspacesofthatintroducedpreviouslybyO.Blascoin[13].ThesedefinitionswereinventedalsobyV.Lieandhekindlycommunicatedhisresultstous.WealsomentionakindofH¨older’sinequality(Proposition5.2).TheseresultswillbeusedinChapter8.AnotherkindofmatrixversionofaBMOspaceisintroducedinSec-tion5.2.Thisspace,denotedbyBMOF(2),isusefulinthestudyofthematrixversionofHankeloperatorHΦandinamoregeneralcontextofnestalgebrasitwasintroducedbyS.Powerin[77].Theorem5.14seemstobenew.ItisalsoeasytoobservethataconverseofthistheoremholdsformatricesAwithafinitenumberofrows.

135October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Chapter6MatrixversionofBergmanspaces6.1SchattenclassversionofBergmanspacesWeintendnowtointroducetheSchattenclassversionofBergmanspaces.InordertodothisweapplyProposition1.1toobtainthat,foraw∗-measurableB(2)-valuedfunctionf,thefunctiont→||f(t)||B(2)isLebesguemeasurableonD.Weintroducealsothefollowingmatrixspaces:L∞(D,):={r→A(r)beingaw∗-measurablefunctionon[0,1)|2esssup||A(r)||B(2):=||A(r)||L∞(D,2)<∞},0≤r<1L=∞(D,),isthesubspaceofL∞(D,)consistingofallstrongmeasurable22functionson[0,1)and∞La(D,2):={Ainfiniteanalyticmatrix|||A||L∞(D,2):=asup||C(r)∗A||B(2)=||A(r)||L∞(D,2)<∞}.0≤r<1Definition6.1.Let1≤p<∞.WedenotebyLp(D,)thespace2ofallstrongmeasurableCp-valuedfunctionsdefinedon[0,1),suchthat1/p1p||A(r)||Lp(D,2):=20||A(r)||Cprdr<∞,whereCpistheSchat-tenclassoforderp,andwedefineL˜p(D,)asthespaceofallfunc-a2tionsA(r):=A∗C(r),whereAisanuppertriangularmatrixwith||A||Lp(D,2)<∞.HereofcourseC(r)meanstheToeplitzmatrixassociatedtotheCauchykernel1,for0≤r<1and∗standsforSchurproduct.1−rL˜p(D,)isasubspaceofLp(D,).a22ByLp(D,)wemeanthespaceofalluppertriangularmatricessucha2that||A(·)||Lp(D,2)<∞.121

136October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013122MatrixspacesandSchurmultipliers:MatricealharmonicanalysisWeidentifyL˜p(D,)andLp(D,)andcallLp(D,)theBergman-a2a2a2Schattenclasses.WeintendtointroducetheconceptofBergmanprojectionandweprovefirstsomeintroductoryresults:Lemma6.2.LetAbeanuppertriangularmatrixbelongingeithertoCp,1≤p≤∞,ortoB(2).Thenthefunctionr→A(r)isacontinuousfunctionon[0,1]takingvaluesinCp,1≤p≤∞,orinB(2),too.Proof.IfA∈B(2),thenwehave,whenrn→r∈[0,1],that||(C(rn)−C(r))∗A||B(2)≤(byTheorem2.14)|rn−r|≤||A||B(2)n→∞−→0.|1−rn||1−r|UsingthedualityandinterpolationbetweenCpwehavethatlim||(C(rn)−C(r))∗A||Cp=0,1≤p≤∞,n→∞ifA∈Cp.ByLemma6.2wehave:Corollary6.3.Let1≤p≤∞andletAbeanuppertriangularmatrix.IfA∈Cp(respectivelyifA∈B(2)),thensup||C(r)∗A||Cp=sup||C(r)∗A||Cp0≤r<10≤r≤1andsimilarlywith||||B(2)insteadof||||Cp.Proof.Wehavetoshowthat,for1≤p≤∞,||C(1)∗A||Cp≤sup||C(r)∗A||Cp,0≤r<1(respectivelythesimilarinequalityfor||||B(2).)AccordingtoLemma6.2wehavethat||A||Cp=||C(1)∗A||Cp=lim||C(r)∗A||Cp≤sup||C(r)∗A||Cpr→1r<1andsimilarforB(2).InthesequelwedenoteL∞(D,)byH∞(D,),orsimplybyH∞().a222Corollary6.4.H∞()isaBanachsubspaceinB().22

137October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces123Proof.Wehavethat||A||L∞(D,2)=sup||C(r)∗A||B(2)=(byCorollary6.3)a0≤r<1=sup||C(r)∗A||B(2)0≤r≤1=supsup|tr(C(r)∗A)B|r∈[0,1]||B||C1≤1,Balowertriangularmatrix≤||A||B(2)sup||C(r)∗B||C1||B||C1≤1,rank(B)<∞,Blowertriangular≤(byLemma6.2)≤||A||B(2).Ontheotherhand||A||L∞a(D,2)=sup||C(r)∗A||B(2)≥||A||B(2),0≤r≤1and,consequently,||A||L∞(D,2)∼||A||B(2).aProposition6.5.Let1≤p<∞.ThenL˜p(D,)isaclosedsub-a2spaceinLp(D,),and,thus,Lp(D,)maybeidentifiedbythemap2a2A→A(r),r∈[0,1),withaclosedsubspaceofLp(D,).Consequently,2theBergman-SchattenspaceLp(D,)isaBanachspace.a2Proof.LetAn∈Lp(D,).IfAn(r)→A(r)∈Lp(D,),then(An(r))a22nisaCauchysequenceinLp(D,)and,hence||C(r)∗(An−Am)||2Cp−→0a.e.withrespecttotheLebesguemeasureon[0,1].Consequently,n,m→∞byLemma6.2,itfollowsthatlim||C(r)∗(An−Am)||=0foralln,m→∞Cpr∈[0,1],thatisthesequence(C(r)∗An)isaCauchysequenceinCfornpallr∈[0,1],whichinturnimpliesthatlim(C(r)∗An)(i,j)=A(r)(i,j)n→∞foralli,j∈Nandforall0≤r≤1.WeconcludethatA(r)isanuppertri-angularmatrixforallr≤1andsince(An∗C(r))(i,j)=anrj−iitfollowsijthatthereisliman=aforalli,j∈NandA(r)=C(r)∗A,wheren→∞ijijA=(a).ThusA∈Lp(D,).iji,ja2Remark.Wenotethatintheaboveproofwehavethatlimr→1||A−A(r)||Lp(D,=0.Itisalsoclearthatthematricesoffinite-bandtypearea2)denseinLp(D,).Seealso[14]-Proposition2.3forrelatedresults.a2TherearesomegeneralresultsaboutBergman-Schattenclassesessen-tiallyduestoO.Blasco[14].

138October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013124MatrixspacesandSchurmultipliers:MatricealharmonicanalysisProposition6.6.Let1≤p<∞and⎛⎞a1a1a1...012⎜..⎟⎜0a2a2.⎟⎜01⎟A=⎜..⎟⎜00a3.⎟⎝0⎠............beanuppertriangularinfinitematrixThenA∈Lp(D,)ifandonlyifa2⎛⎞a1a1a1...123⎜..⎟⎜0a2a2.⎟⎜12⎟pB:=⎜..⎟∈La(D,2).⎜00a3.⎟⎝1⎠............Moreover,thereexistsapositiveconstantK<1suchthatK(||A0||Cp+||B||Lp(D,))≤||A||Lp(D,)≤||A0||Cp+||B||Lp(D,).a2a2a2Proof.SinceA=A0+S(B),whereSistheunilateralshift,that⎛⎞⎛⎞0c1c1c1...c1c1c1...012012⎜..⎟⎜..⎟⎜00c2c2.⎟⎜0c2c2.⎟⎜01⎟⎜01⎟isS(C):=⎜..⎟forC=⎜..⎟,wefindthat⎜000c3.⎟⎜00c3.⎟⎝0⎠⎝0⎠...........................||A||Lp(D,≤||A0||C+||B||p.Togettheotherinequality,notea2)pLa(D,2)12πitn−intrthatAn=(n+1)00A(re)reπdtdrforn≥0.Thisimpliesthe∞nestimate||An||C≤(n+1)||A||Lp(D,.Hence,sinceB(r)=An+1r,pa2)n=0weobtainthat1(||B(r)||p2dr)1/pCp01/21≤(||B(r)||p2rdr)1/p+(||B(r)||p2rdr)1/pCpCp01/2∞11(n+1)p1/p≤()||A||Lp(D,+2((||A0||Cp+||A(r)||Cp)rdr).41/p2na2)n=01/2Thisgivesthat||B||Lp(D,)≤L||A||Lp(D,),andtakingK=1/(L+1),a2a2theproofiscomplete.

139October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces125AnotherinterestingresultaboutBergman-Schattenclassesisasfollows[14]:Theorem6.7.LetAbeanuppertriangularmatrix,n∈N,1≤p<∞.ThenA∈Lp(D,)ifandonlyifthefunctionr→(1−r2)nA(n)(r)∈a2Lp(D,).2Proof.WeprovethatforanyuppertriangularmatrixAandk≥0,thefunction(1−r2)kA(r)belongstoLp(D,)ifandonlyif(1−r2)k+1A(r)2alsodoes.Thenarecurrentargumentgivesthestatement.Notethat(1−r2)k+1A(r)∈Lp(D,)ifandonlyif21(1−r2)pk+p||rA(r)||p2dr<∞.Cp0∞nWedenoteB(r)=rA(r)=n=0nAnrandobservethatforeachr<1wehavethatB(r2)=A(r)∗Λ(r),whereΛistheToeplitzmatrixassociatedtothefunctionλ(r)=r.(1−r)2Since2πitdtrM1(λ,r):=|λ(re)|=02π1−r2and1/p2π22itpdtMp(B,r):=||B(re)||Cp≤M1(λ,r)Mp(A,r),02πweobtainthat112π2pk+pp2pk+pititp1(1−r)||rA(r)||2dr=(1−r)||reA(re)||dtdrCpCpπ00011=4r3(1−r4)pk+pMp(B,r2)dr≤Cr(1−r2)pkMp(A,r)drpp0012π12pkitp12pkp=C(1−r)||A(re)||dtrdr=C(1−r)||A(r)||2rdr.CpπCp000Conversely,wetakeAsuchthat(1−r2)k+1M(A,r)∈pp1L((0,1),dr).Wemay,withoutlossofgenerality,assumethat(1−0r)(k+1)pMp(A,r)dr=1andalsothatA=0.p0rSinceMp(A,r)≤0Mp(A,s)dswehavethat11(1−r2)kp||A(r)||p2rdr=(1−r2)kpMp(A,r)2rdr≤Cpp00

140October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013126MatrixspacesandSchurmultipliers:Matricealharmonicanalysis1r1r2r(1−r2)kp(M(A,s)ds)pdr≤C(1−r)kp(M(A,s)ds)pdr.pp0000Forp=1wegetthat11r(1−r2)k||A(r)||2rdr≤C(1−r)k(M(A,s)ds)drC110001Ck+1=(1−s)M1(A,s)ds=C.k+10Forp>1wewrite,foreacht∈(0,1),trI=(1−r)kp(M(A,s)ds)pdr.tp00rLetu(r)=−1(1−r)pk+1andv(r)=(M(A,s)ds)p.pk+10pSinceu(t)v(t)<0andv(0)=0,wehavethatttI=u(r)v(r)dr≤−u(r)v(r)dr.t00Thus,trppk+1p−1It≤(1−r)Mp(A,r)(Mp(A,s)ds)drpk+100trpk+1(p−1)kp−1=(1−r)Mp(A,r)(1−r)Mp(A,s)ds)dr.pk+1001/pThentheassumptionandH¨older’sinequalityshowsthatIt≤CIt.Hence,It≤Cforalltandtheproofiscomplete.Remark.WeobservethatforA∈L1(D,)andn∈Nwehavethata212πitn−int1A(re)rerdtdr=00π12π2itn−1−i(n−1)t1An(1−r)A(re)rerdtdr=.00πn+1See[14]-Proposition2.6.Usingtheaboveremarkitiseasytoprovethefollowingresult(see[14]-Proposition2.7):Proposition6.8.IfA∈L1(D,),thena212πiuitA(se)s1A(re)=duds=00(1−reitse−iu)2π1π2iu(1−s)A(se)s12dudsforall0≤r<1.00(1−reitse−iu)3π

141October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces127Lemma6.9.LetA∈L2(D,),0≤r<1andB∈C.Thenthelineara22functionalF(A)=trA(r)B∗iscontinuousonL2(D,).r,Ba2Proof.IfAisanuppertriangularmatrixoffiniteorderandA(r)=C(r)∗A,thenweconsiderthefunctionfA(r,θ)onDgivenintheintroduction.ItisclearthatthefunctionaboveisanholomorphicC2-valuedfunctiononD,∞kikθand,consequently,thefunctionz→||k=0Akre||C2issubharmonic.Thus,for0L2(D,2)=2trA(s)[Kr,B(s)]sdsa0forallB∈C,0≤r<1andA∈L2(D,).2a2Leti,j∈NandletBbethematrixwhoseentriesb(k,l)areb(k,l):=δkiδlj.Thentheaboveformulayieldsthat:11A(r)(i,j)=2trA(s)K(s)∗sds=2tr[A(s)(K∗∗P(s))]sdsr,i,jr,i,j00foralli,j∈N,wherebyKr,i,jwedenotethematrixKr,BfortheabovematrixB,P(s)istheToeplitzmatrixassociatedtothePoissonkernel,thatis⎛⎞1ss2s3...⎜..⎟⎜s1ss2.⎟⎜⎟P(s)=⎜..⎟.⎜s2s1s.⎟⎝⎠...............

142October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013128MatrixspacesandSchurmultipliers:MatricealharmonicanalysisSinceA(r)isananalyticmatrix,wehavethat1(P(r)∗A)(i,j)=tr(P(s)∗A)[P(s)∗Kr,i,j](2s)ds0forallj≥iandall0≤r<1.Itiseasytoseethat,j−i∞(j−i+1)rδi,mδj,li≤j,Kr,i,j=l,m=1∞(0)i>j.l,m=1Definition6.10.Letr→A(r)beanelementofL2(D,).SinceL˜2(D,)2a2isaclosedsubspaceintheHilbertspaceL2(D,),thereisanuniqueor-2thogonalprojectionP˜onL˜2(D,),calledBergmanprojection.Wedenotea2byPthecorrespondingoperatorfromL2(D,)ontoL2(D,).2a2Proposition6.11.Forallfunctionsr→A(r)fromL2(D,)andforall2i,j∈Nwehavethat,2(j−i+1)rj−i1a(s)sj−i+1dsifi≤j,{[P(A(·))](r)}(i,j)=0ij0ifi>j.Proof.Wehavethat[P(A(·))](r)(i,j)=Fr,i,j(P(A(·)))==(sincePisaselfadjointprojection)==(sinceKr,i,jisananalyticmatrix)=L2(D,2)a11=2tr[A(s)K∗(s)]sds=2(j−i+1)rj−ia(s)·sj−i+1ds,r,i,jij00ifj≥iandmeanstrAB∗.Ifj−1.Thenweput⎧∞⎨Γ(j−i+2+α)rj−iδδi≤j,(j−i)!Γ(2+α)i,lj,mKr,i,j,α=l,m=1⎩0i>j

143October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces129andwehavethat,forananalyticmatrixA(s)=P(s)∗A,1∞arj−i=(α+1)2s2k+1(1−s2)αds∀i,j,ijkr,i,j,αk0k=0whereA=(a)∞.iji,j=1Then,(α+1)Γ(j−i+2+α)rj−i(21a(s)sj−i+1(1−s2)αds)ifj≥i[PA(·)](r)=(j−i)!Γ(α+2)0ijα0ifj:=tr(A(s)[B(s)]∗)2sds,0whereA(·)∈L∞(D,),B(·)∈L1(D,).22NowwearelookingfortheadjointP∗ofP:111∞∞=2(P∗A(·))(r)(i,j)b(r)rdr11ij0i=1j=1∞∞1=(P∗A(·))(r)(i,j)b(r)(2r)dr.1iji=1j=10Ontheotherhand1==trA(r)(PB)∗(r)(2rdr)1110∞∞1=A(r)(i,j)(P1B)(r)(i,j)(2rdr)i=1j=10∞∞Γ(j−i+3)1=[A(s)](i,j)sj−i(2sds)×(j−i)!Γ(2)0i=1j=i1b(s)sj−i(1−s2)(2sds).ij0

144October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013130MatrixspacesandSchurmultipliers:MatricealharmonicanalysisWetakeB(s)(i,j)=χIk(s)/(μ(Ik))andB(s)(l,k)=0,(l,k)=(i,j),∀(i,j)∈N×N,whereIkrisasequenceofintervalssuchthatlimμ(Ik)=k→∞0,anddμ=2sds.ByLebesgue’sdifferentiationtheoremwehavethat,Γ(j−i+3)rj−i(1−r2)1A(s)(i,j)sj−i(2sds)ifj>i,(P∗A(·))(r)(i,j)=(j−i)!Γ(2)010ifj≤i,a.e.forallr∈[0,1).WeshowthatP∗:L∞(D,)→L∞(D,)isaboundedoperator.In122ordertoprovethiswehavetoremarkthat||A(r)||2=esssup||A(r)||2=L∞(D,2)B(2)0≤r<1∞∞esssupsup|a(r)h|2.ijj0≤r<1∞|hj|2≤1j=1i=1j=1Consequently,sinceL1[0,1]hascotype2,thereisaconstantK>0suchthat||P∗A(·)||2=1L∞(D,2)∞∞1Γ(j−i+3)esssupsup|hrj−i(1−r2)a(s)sj−i(2sds)|2=jij0≤r<1||h||2≤10(j−i)!Γ(2)i=1j=i∞1∞Γ(j−i+3)esssupsup(1−r2)2|(a(s)((rs)j−i)h)(2sds)|2ijj0≤r<1||h||l2≤1i=10j=i(j−i)!≤Kesssup(1−r2)2sup×0≤r<1||h||l2≤11∞∞[(|a(s)[(rs)j−i(j−i+2)(j−i+1)]h|2)1/2(2sds)]2.ijj0i=1j=iSincetheToeplitzmatrixC(rs)givenbythesequenceoffunctions(c)(rs)∞,whereiji,j=1(rs)j−i(j−i+2)(j−i+1)ifj≥icij(rs):=cj−i(rs)=0ifj

145October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces131∞kikθisaSchurmultiplier(weremarkthatu(k+2)(k+1)e=k=02),then,byBennett’sTheorem,themultipliernormofthema-(1−ueiθ)3trixC(rs)isexactlytheL1(T)-normof2,thatisitisequalto(1−rseiθ)3∞Γ(n+3/2)22(rs)2n.n=0(n!)2Γ(3/2)2Thus,⎛⎞1/2∞∞sup⎝|a(s)(rs)j−i(j−i+2)(j−i+1)h|2⎠ijj∞|h2j|≤1j=1i=1j=i∞Γ(n+3/2)2=||A(s)∗C(rs)||≤||A(s)||·(rs)2n.B(2)B(2)(n!)2Γ(3/2)2n=0Consequently,||P∗A(·)||2≤1L∞(D,2)∞21Γ(n+3/2)2Kesssup(1−r2)2||A(s)||·r2ns2n+1(2ds)≤B(2)(n!)2Γ(3/2)2r<10n=02∞Γ(n+3/2)2r2nKesssup(1−r2)2||A(·)||2∼L∞(D,2)22r<1(n!)(n+1)Γ(3/2)n=02212(byStirling’sformula)∼esssup(1−r)(1−r2)2||A(·)||L∞(D,2)r<1∼||A(·)||2,L∞(D,2)whichshowsinturnthatP∗:L∞(D,)→L∞(D,)isbounded.The122proofiscomplete.Wehavethefollowingdualitytheorem:Theorem6.13.Let1=tr[A(r)B∗(r)](2rdr).0Proof.WeusetheboundednessoftheprojectionP:Lp(D,)→2Lp(D,).SincetheproofissimilartothatofTheorem7.11weleaveouta2thedetails.Cf,also[14]Theorem3.6.

146October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013132MatrixspacesandSchurmultipliers:Matricealharmonicanalysis6.2SomeinequalitiesinBergman-SchattenclassesThefollowingHardy-Littlewoodinequalitiesareusefulinthestudyofclas-sicalHardyspacesHp,with1≤p≤∞.See[23].Hardy-LittlewoodTheorem.(i)If0

147October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces133Multiplyingtheaboveinequalitybyrandintegratingover(0,1)wehavethat∞1−1n+1(n+1)||An||C1rdr≤||A||L1,an=00or∞−2(n+1)||An||C1≤C||A||L1.an=02.LetA∈Ap,uncand1

148November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013134MatrixspacesandSchurmultipliers:MatricealharmonicanalysisletT:B(D,`2)→`∞(`∞);begivenbyT(A)=(An)n≥0.WenotethatTmapsL2(D,`)continuouslyinto`(`,w),wherew(n)=1/(n+1),fora222alln≥0,and(`2,w)isthespaceofallsequencesx=(xn)n≥0suchthatP|x|2w(n)<∞,withthenaturalnorm.n≥0nIndeed,similarlytoTheorematpage84-[28]wegetthatX1||T(A)||2=||A||2≤C(2)||A||2.n+1n∞L2a(D,`2)n≥0Hence,T:Lp(D,`)→[`(`,w),`(`)]=`(`,w),whereθ=a222∞∞θpp1−2/p,isaboundedoperator.(See[93]forthelastequation.)Hence,1/pX1||A||p≤C(p)||A||p.n+1nCpLa(D,`2)n≥05.ByTheorem6.13wefindthat[Lp(D,`)]∗=Lq(D,`),1/p+1/q=1.a2a2Ontheotherhand,[`(`,w)]∗=`(`,w),where1/p+1/q=1.Therefore,ppqqtheconclusionfollowsfrom4.6.Foreachnandr∈(0,1)wehavethatZ2πn1it−intAnr=A(re)edt.2π0Thisimpliesthat,foranyn∈Nand00suchthat!1/pX∞pK1sup||An||Cp2k−1≤n<2kk=0

149October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces135⎛⎞1/p∞≤||A||p≤K⎝(||A||)p⎠.La(D,2)2nCpk=02k−1≤n<2kWearenowreadytopresenttheHausdorff-YoungTheoremforBergman-SchattenclassesextendingTheorem2-[28]atpages81-82.pTheorem6.15.Let1≤p≤∞andletq=beitsconjugateexponent.p−1(i)If1≤p≤2,thenA∈Lp(D,)impliesthata2⎛⎞1/q⎝(n+1)1−q||A||q⎠≤||A||p.nTpLa(D,2)n≥01−qq(ii)If2≤p≤∞,thenn≥0(n+1)||An||Tp<∞impliesthat1/qp∞1−qqA∈L(D,2)and||A||Lp(D,≤(n+1)||An||.aa2)n=0TpProof.Incase(ii),letμbethediscretemeasureonthesetNwhichas-signsthemassμ(n)+1totheintegern=0,1,2,...Considerthelinearop-eratorTthatmapsthesequence{1A}totheformalseriesAzn.n+1nn≥0nWewanttoshowthatTisboundedasanoperatorfromLq(N,dμ;T)topLp(D,dσ;T),withnorm||T||≤1,whereLp(D,dσ;T)isthespaceofallppp-LebesgueBochnerintegrableTp-valuedfunctionsdefinedontheunitdiskDwithrespecttotheareameasuredσonD.Inthecasep=2thisfollowsfrom||T({A/(n+1)})||2=||{A/(n+1)}||2.nL2(D,dσ;T2)nL2(N,dμ;T2)Forp=∞itistheclearthatsup||Azn||≤||A||.|z|≤1nnT∞nnT∞Theresultfollowsbyusingtheseestimatesandcomplexinterpolationofvector-valuedLpspaces.Case(i).LetA(z)∈Lp(D,dσ;T).DefinethelinearoperatorT(A)=p{b},whereb=A(z)zndσ.IfA∈Lp(D,),thenb=A/(n+1).nnDa2nnWiththemeasureμdefinedasbefore,wewanttoshowthatTisaboundedoperatorfromLp(D,dσ;T)toLq(N,dμ;T),withnorm||T||≤1.Forp=1ppthisisthetrivialinequality||bn||T1≤||A||L1(D,dσ;T1)foralln∈N.Forp=2itfollowsfromtherelation∞∞(n+1)||b||2=(n+1)||A(z)zndσ||2nT2L2(D,2)an=0n=0D∞∞1∞=A(z)lA(ζ)l(n+1)(zζ)ndσ(z)dσ(ζ)k+1kkk=0l=1DDn=0

150October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013136MatrixspacesandSchurmultipliers:Matricealharmonicanalysis∞∞1A(z)lA(ζ)l=kkdσ(z)dσ(ζ)=,k+1DD(1−zζ)2k=0l=1wherePdenotestheBergmanprojection.Since2||≤||PA(·)||L2(D,2)||A(·)||L2(D,2)≤||A(·)||L2(D,2),aaathisgivesthedesiredinequalityforp=2.Byusingtheseestimatesandcomplexinterpolationasbeforeweobtainthat∞1/q1/pqpp(n+1)||bn||Tp≤||A(z)||Tpdσ,A(·)∈L(D,dσ;Tp).n=0DSpecializingtoA∈Lp(D,)andrecallingthatb=A/(n+1),wea2nnarriveatthedesiredresult.6.3AcharacterizationoftheBergman-SchattenspaceWegiveacharacterizationofthespaceL1(D,)completelysimilartoa2thoseobtainedbyMateljevicandPavlovicin[61].FirstweprovesomenecessaryLemmas,whicharealsoofindependentinterest.nLemma6.16.LetA=k=mAk,wherem≤n.Then||A||rn≤||A(r)||≤||A||rm,C1C1C1forall01andthefunctionseit→||B[seit]||,for0

151October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces137Letσ(A)=n1−kAbetheCesaromeanoftheordernofnk=0n+1ktheuppertriangularmatrixA,and||σn(A)||1:=sup0

152October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013138MatrixspacesandSchurmultipliers:MatricealharmonicanalysisLetφbeanon-negativeincreasingfunctiondefinedon(0,1]forwhichφ(tr)≤Ctαφ(r),00.Lemma6.19.Letψ(r)=φ(r)qr−,whereq<∞,qα−>−1,andletαsatisfythecondition(6.2).Then1C−1x−1ψ(1/x)≤ψ(1−r)rx−1dr≤Cx−1ψ(1/x),x≥1.0Proof.Wehavethat1xI(x):=ψ(1−r)rx−1dr=x−1ψ(t/x)(1−t/x)x−1dt.00Sinceφsatisfiestheconditions(6.2)and(6.3)ityieldsthatφ(t/x)≤C(tα+tβ)φ(1/x),00.(6.4)n=0TheproofofTheorem6.22,whichfollows,isbasedonLq-behaviourofthefunctionsnF(r)=(1−r)−1/qφ(1−r)sup{λr2:n≥0}1n

153October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces139and∞nF(r)=(1−r)−1/qφ(1−r)λr2,2nn=0where(λn)isasequenceofnon-negativerealnumbers.Proposition6.21.LetF=F1orF=F2.Then−1−nC||F||Lq≤||(φ(2)λn)||q≤||F||q.Proof.Weconsideronlythecaseq<∞.Inthecaseq=∞theproofissimilarandisbasedonLemma6.20.Letq<∞.Thenqq−1qq2kqF(r)≥F1(r)≥(1−r)φ(1−r)λkrforallkand,by(6.4),∞nkF(r)q≥C−1φ(1−r)q2nr2λr2q.kn=0Hence,∞nF(r)q≥C−1φ(1−r)q2nλqr2(1+q).(6.5)nn=0Ontheotherhand,fromLemma3.4andhypothesis(6.3)itfollowsthat1nφ(1−r)qr2(q+1)dr≥C−12−nφq(2−n).0Combiningthiswith(6.5),weobtaintheright-handsideinequalityinProposition3.6.Toprovetheleft-handsideinequality,letn−1n−1η=2nδr2,θ=2−nδλr2,nnnwhereδ=α/2andαsatisfies(6.2).Then∞q∞q∞q∞nλr2=ηθ≤ηθqnnnnnn=0n=0n=0n=0∞n−1≤C(1−r)−qδ2−nqδλqr2q,nn=0wherewehaveused(6.4).Hence,∞n−1F(r)q≤F(r)q≤Cψ(1−r)2−nqδλqr2q,2nn=0whereψ(r)=φ(r)qr−qδ−1.NowthedesiredresultfollowsfromLemma6.19.

154October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013140MatrixspacesandSchurmultipliers:MatricealharmonicanalysisOurmainresultinthissectionreads:Theorem6.22.LetAbeanuppertriangularmatrix.ThenA∈L1(D,)a2ifandonlyif∞||σn(A)||1<∞.(n+1)2n=1Proof.FirstweremarkthatA∈L1(D,)ifandonlyifa21||A(r)||C1dr<∞.0LetA∈L1(D,).Then,bythelefthandsideinequalityofLemmaa26.17,∞||A(r)||=(1−r)||A(r)||rn≥C1C1n=0∞(1−r)||σ(A)||r2n.n1n=0Consequently,anintegrationwithrespecttoryields11∞∞>||A(r)||dr≥(1−r)||σ(A)||r2ndr=C1n100n=0∞11∞1||σn(A)||1≥2||σn(A)||1.(2n+1)(2n+2)8(n+1)n=0n=0Conversely,supposethat∞1||σn(A)||1<∞.(n+1)2n=0Letnxn=(k+1)(n−k+1)||σk(A)||1.k=0Then,summingbypartsasin(4.4)in[61],wegetthat∞∞||σ(A)||(n+1)rn=(1−r)2xrn.(6.6)n1nn=0n=0Ontheotherhand,usingLemma6.18,weseethatx≤C(n+1)3||σ(A)||nn1

155October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces141and,therefore,∞1∞1(n+1)5xn≤C(n+1)2||σn(A)||1<∞.n=0n=0ByLemma4.8in[61],withq=1,φ(r)=r,r∈(0,1],wegetthat1∞(1−r)4xrndr<∞.n0n=0Using(6.6)wearriveat1∞(1−r)2||σ(A)||(n+1)rndr<∞n10n=0and,bytherighthandsideinequalityinLemma6.17,finallyweobtainthat1||A(r)||C1dr<∞0i.e.A∈L1(D,).Theproofiscomplete.a26.4UsualmultipliersinBergman-SchattenspacesInthissectionwecharacterizethemultiplierswithrespecttousualproductofmatricesfortheBergman-Schattenspacesofindex2,L2().a2Morespecific,letAbeaninfiniteuppertriangularmatrix.AiscalledanusualmultiplierforL2()andonewritesA∈M(L2),ifforallB∈L2(),a2aa2itfollowsthatAB∈L2(),orequivalently(bytheoremofclosedgraph)a2thereisthesmallestconstantM(A)<∞suchthat||A·B||L2(2)≤M(A)||B||L2(2).(6.7)aaNowletb∈N.Letusdenotebyn(w)={x=(x,x,...,x)withthe212ndef21/2n|xj|norm||x||2,w=j=1n−j+1}.Then,Abeingasabovewedenoteby||A||n=||A||B(n(w),2(w)),n∈N.22Wehavethefollowingresult:Theorem6.23.AisanusualmultiplierforL2()ifandonlyifa2sup||A||n<∞andM(A)=sup||A||n.nn

156October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013142MatrixspacesandSchurmultipliers:MatricealharmonicanalysisProof.FirstwedenotebyP(r)theToeplitzmatrix⎛⎞1rr2r3...⎜r1rr2...⎟⎜⎟⎜r2r1r...⎟,0≤r<1⎝⎠...............andwedenotebyA(r)=A∗P(r),where,asusualinthisbook∗meanstheSchurproductofmatrices.LetusremarkthatA(r)·B(r)=(A·B)(r)forallBuppertriangularmatricesandforallr∈[0,1).Then,foranusualmultiplierAforL2(),wehavea21∞1∞||(AB)rk||2(2rdr)≤M(A)2||Brk||22rdr,(6.8)kC2kC200k=0k=0foralluppertriangularmatricesB.TakenowB=B(n),thenthcolumnofB.Then,substitutingin(6.8)wehave,forA=ajandB(n)=kk≥0,j≥1⎛1⎞00...0bn−10...⎜00...0b20...⎟⎜n−1⎟⎜.......⎟⎜⎜..............⎟⎟,⎝00...0bn0...⎠000...000...AB(n)⎛⎞⎛⎞a1a1a1a1...a1...00...0b10...0123n−1n−1⎜⎜0a20a21a22...a2n−2...⎟⎟⎜⎜00...0b2n−20...⎟⎟⎜333⎟⎜3⎟⎜00a0a1...an−3...⎟⎜00...0bn−30...⎟=⎜⎜.......⎟⎟·⎜⎜.......⎟⎟..............⎜.......⎟⎜.......⎟⎜⎟⎜⎟⎝00.........an...⎠⎝00...0bn0...⎠0000.........0...00...000...⎛00...0a1b1+a1b2+···+a1bn0...⎞0n−11n−2n−10⎜00...0a2b2+a2b3+···+a2bn0...⎟⎜0n−21n−3n−20⎟⎜.......⎟=⎜⎜..............⎟⎟.⎝00...0anbn0...⎠0000...000...Hence||AB(n)||2=L2(2)a

157October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces143⎛⎞1n−1n−2⎝|a1bj+1|2r2n−2+|a2bj+2|2r2n−4+···+|anbn|2⎠(2rdr)jn−1−jjn−2−j000j=0j=0n−11n−21=|a1bj+1|2·+|a2bj+2|2·+···+|anbn|2,jn−1−jnjn−2−jn−100j=0j=0and(n)2121221n2||B||L2(2)=|bn−1|·+|bn−2|·+···+|b0|.ann−1Hence⎛⎞1/2n−11n−21⎝|a1bj+1|2·+|a2bj+2|2·+···+|anbn|2⎠≤jn−1−jnjn−2−jn−100j=0j=0⎛⎞1/2n−11n⎝n−j2⎠jM(A)|bj|·=||bn−j||n(w)n−j2j=1j=0or||A||n≤M(A)foralln.Consequentlysup||A||n≤M(A).nConversely,ifsupn||A||n=M<∞,ifkisarbitrarybutfixedandB=n−1B(k)∈L2(),wehave,denotingbyAB(k)(k)thesequencek=0a2placedonthekthcolumnbeginningfromthetop,nn(k)||AB||2=||AB(k)||2=||AB(k)||2L2(2)L2(2)k(w)aa2k=1k=1nn≤M||B(k)||2≤M||B(k)||2=M||B||2k(w)L2(2)L2(2)2aak=1k=1forallk.HenceAB∈L2()∀B∈L2()andM(A)=sup||A||.a2a2nn

158October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013144MatrixspacesandSchurmultipliers:MatricealharmonicanalysisNowwecancharacterizethematricesAsuchthatsupn||A||n<∞.Theorem6.24.IfAisanuppertriangularmatrix,thenitfollowsthatsupn||A||n<∞ifandonlyifA∈B(2).Proof.LetA=alsuchthatsup||A||=1<∞.kl≥1,k≥0nnThen,foracolumnmatrixBsuchthat⎛⎞b1n−1⎜b2⎟⎜n−2⎟n(B)=⎜.⎟∈2(w)⎝..⎠bn0with|b1|2|b2|2||(B)||2=n−1+n−2+···+|bn|2=12,w0nn−1wehave|n−1a1bk+1|2|n−2a2bk+2|2||(AB)||2=k=0kn−k−1+k=0kn−k−1+···+|anbn|2=2,w00nn−1n−1√bk+1n−2√bk+2|(a1n−k)√n−k−1|2|(a2n−k−1)√n−k−2|2k=0kn−kk=0kn−k−1nn2++···+|a0b0|.nn−1k+1bIfwedenotebyy=√n−k−1,wherek=0,1,...,n−1wehavekn−k||(B)||2=|y|2+···+|y|2=||(y)||2.2,w0n−1k2Hence00n−1n−kn−2n−k−1||(AB)||2=|a1y|2+|a2y|2+···+|any|22,wknkkn−1k+10n−1k=0k=0=||Ay||2,n2where⎛>>1⎞a1a1n−1a1n−2...a√n−10...⎜01n2>nn⎟2⎜21n−2√an−2⎟⎜⎜0a0a2n−1...n−10...⎟⎟⎜a3⎟An=⎜00a3...√n−30...⎟.⎜0n−2⎟⎜......⎟⎝...............⎠000...an0...0

159October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces145Hence||A||≤||A||,whichinturnimpliesthatsup||A||≤nB(2)nnnB(2)sup||A||=1,inotherwordsthesequence(A)∞belongstotheunitnnnn=1ballB(B(2))ofB(2).ItiseasytoseethatA(k,l)→A(k,l)forallk,l∈N.SincetheunitnballB(B(2))isσ(B(2),C1)-compactitfollowsthatA∈B(B(2)),thatis||A||B(2)≤1=supn||A||n.Theconverseimplicationiseasy,itisenoughtoverifythatsuchamatrixisanusualmultiplieronL2()andapplytheprevioustheorem.a2EachmatrixΦ∈L2()issaidtogenerateasubspace[Φ],theclosureofa2thesetofmultiplesofΦbymatricesoffinitebandtype.WhenΦ∈H∞(),2itisimportanttoobservethat[Φ]doesnotnecessarilycoincidewithΦL2()={ΦF:F∈L2()},a2a2sincethelattersetneednotbeclosed.ThecrucialrequirementisthattheoperatorofmultiplicationbyΦbeboundedbelow:inotherwords,thatthereexistaconstantc>0suchthat||ΦF||L2(2)≥c||F||L2(2)forallaaF∈L2().a2Theorem6.25.LetΦbeaninvertiblematrixfromH∞().Thentheset2ΦL2()isclosedinL2()ifandonlyiftheoperatorofmultiplicationMa2a2ΦisboundedbelowonL2().a2Proof.SupposefirstthatMisboundedbelow.ThenifAn∈L2()Φa2and{ΦAn}convergestosomeG∈L2(),itfollowsthatAn→Afora2someA∈L2().HenceΦAn→ΦAbecauseΦ∈H∞(2)isanusuala2multiplieronL2().HenceG=ΦA∈ΦL2(),whichisconsequentlyaa2a2closedsubspace.Consequently,wedefineT(G)=Φ−1GwiththedomainD(T)={G∈[Φ]:Φ−1G∈L2()}.a2ItiseasytoremarkthatTisaclosedoperator.IndeedifGn→G∈L2(),whereGn∈D(T)andT(gn)→F,thatisΦ−1Gn→F∈L2(),a2a2thenitfollowsthatGn→M(F)=ΦFinL2().ConsequentlyG=ΦFΦa2andΦ−1G=F∈L2(),thatisG∈D(T)andTisaclosedoperator.a2ObservefirstthatΦL2()isclosedifandonlyifΦL2()=[Φ],becausea2a2thematricesoffinitebandtypearedenseinL2().a2SincebyhypothesisΦL2()isaclosedsubspaceandD(T)={G=a2ΦA,A∈L2(),Φ−1G∈L2()}=ΦL2(),byclosedgraphtheoremita2a2a2followsthatTisaboundedoperator.Hence||A||L2(2)≤C||ΦA||L2(2)forsomeconstantC>0andforallaaA∈L2().HenceMisaboundedbelowoperator.a2Φ

160October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013146MatrixspacesandSchurmultipliers:MatricealharmonicanalysisNowwegivesomeinterestingexamplesofmatricesΦsuchthatMΦbeboundedbelowoperatoronL2().a2Letα∈Cwith|α|<1,α=0.Let0≤r0<|α|andΦ=Φα∗C(r0),whereΦαistheToeplitzmatrixassociatedtotheMoebiustransformφ(z)=α−z,|z|<1.α1−αzThenMisboundedbelowlinearandboundedoperatoronL2().Φa2Indeedwehave⎛2222⎞α|α|−1(|α|−1)α(|α|−1)α...⎜0α|α|2−1(|α|2−1)α...⎟⎜⎟⎜..⎟⎜00α|α|2−1.⎟Φα=⎜⎟.⎜..⎟⎜⎝000α.⎟⎠...............Taking0≤r<1andB∈L2()wehavea2⎛α(|α|2−1)rr(|α|2−1)αr2r2(|α|2−1)α2r3r3...⎞000⎜0α(|α|2−1)rr(|α|2−1)αr2r2...⎟⎜00⎟⎜..⎟⎜00α(|α|2−1)rr.⎟[MΦ(B)](r)=⎜0⎟·⎜..⎟⎜⎝000α.⎟⎠...............⎛b1b1rb1r2b1r3...⎞0123⎜0b2b2rb2r2...⎟⎜012⎟⎜..⎟⎜00b3b3r.⎟⎜01⎟=⎜..⎟⎜000b4.⎟⎝0⎠...............αB(r)−(1−|α|2)rτ[(B−B1)(r)]−(1−|α|2)αr2τ[(B−B1−B2)(αrr)]−01020(1−|α|2)α2r3τ[(B−B1−B2−B3)(r)]−...03HereBkisthekthrowofBandτistheuppertranslationwithkrowskofthecorrespondingmatrix.Thenitfollowsthat1/21)22||MΦ(B)||L2(2)=||MΦ(B)(r)||C2(2rdr)≤|α|+(1−|α|)r0+a0

161October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013MatrixversionofBergmanspaces147*22232(1−|α|)r0|α|+(1−|α|)r0|α|+...||B||L2(2)=a"#21|α|+(1−|α|)r0||B||L2(2)≤[|α|+(1+|α|)r0]||B||L2(2)=1−r|α|aa0C(α)||B||L2(2),ahencethemultiplicationoperatorMisboundedonL2()(evenifrisΦa20notlessthan|α|).NowwewillshowthatM,if0≤r<|α|isboundedbelowonL2().Φ0a2Indeed||Φ−1∗C(r)||≤K(α).α0B(2)Morespecific⎛22⎞11−|α|1−|α|...αα2α3⎜1−|α|2⎟⎜01...⎟Φ−1=⎜αα2⎟.α⎜001...⎟⎝α⎠............Let0≤r0<|α|.Then⎛22⎞11−|α|(r0)1−|α|(r0)2...ααααα⎜1−|α|2r⎟⎜01(0)...⎟Φ−1∗C(r)=⎜ααα⎟.α0⎜001...⎟⎝α⎠............Hence$⎛⎞$$0r0(r0)2...$2$αα$−111−|α|$⎜00r0...⎟$||Φα∗C(r0)||B(2)≤+$⎝α⎠$=|α||α|$........$$....$B(2)11−|α|2r111−|α|2r100|α|+|α||α|sup|1−r0eit|≤|α|+|α||α|1−r0=t∈R|α||α|1−|α|r01==K(α,r0).|α|−r0φ|α|(r0)Since)*Φ−1∗C(r)[Φ∗C(r)]=Φ−1Φ∗C(r)=I∗C(r)=I,α0α0αα00

162November15,20139:58WorldScientificBook-9inx6invers*11*oct*2013148MatrixspacesandSchurmultipliers:Matricealharmonicanalysisitfollowsthat−1−1Φα∗C(r0)=[Φα∗C(r0)].Butno−1B∗C(r)=[Φα∗C(r0)]∗C(r){Φα∗C(r0)∗C(r)}{B∗C(r)}=−1Φα∗C(r0)ΦB∗C(r).HenceZ1||B||2=||B∗C(r)||22rdr=L2(`2)C2a0Z1||Φ−1∗C(r)ΦB∗C(r)||22rdr=α0C20||Φ−1∗C(r)ΦB||2≤(byTheorems4.2and4.3)≤α0L2a(`2)||Φ−1∗C(r)||2||ΦB||2≤K(α,r)2||ΦB||2.α0B(`2)L2a(`2)0L2a(`2)NotesInSection6.1weintroduceaversionofmatrixvaluedBergmanspacesstudiedpreviouslyindependentlybyO.Blasco[14].WecallthesespacesBergman-Schattenspaces.TheyareappropriatespacesinordertodevelopatheorysimilartoclassicalHarmonicAnalysis.ForinstanceseeTheorem6.7whichisaperfectanalogueofTheorem4.2.9[94].Moreover,amatrixversionofBergmanProjectionisintroducedandadualitytheorembetweenBergman-Schattenspaces(Theorem6.13)isgiven.InSection6.2weprovesomeinequalitiessimilarwiththosefromthemonograph[28].Inparticular,Theorem6.15istheHausdorff-YoungThe-oremforBergmanSchattenclasses.InSection6.3wederiveacharacterizationoftheBergman-Schattenspace(seeTheorem6.22)whichiscompletelysimilarwiththatobtainedbyMateljevicandPavlovicin[61].InSection6.4wecharacterizetheusualmultipliersontheBergman-Schattenspaceofindex2.

163October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Chapter7AmatrixversionofBlochspaces7.1ElementarypropertiesofBlochmatricesTheBlochfunctionsandtheBlochspacehavealonghistorybehindthem.TheywereintroducedbytheFrenchmathematicianAndr´eBlochinthebe-ginningofthelastcentury.Manymathematicianspaidattentiontothesespacese.g.thefollowing:L.Ahlfors,J.M.Anderson,J.Clunie,Ch.Pom-merenke,P.L.Duren,B.W.RombergandA.L.Shields.Correspondingly,therearealotofinterestingresultsinthisarea(seeforexample[27],[3],andthefollowingrecentmonographs[94]and[28]).OuraimistointroducetheconceptofBlochmatrix,whichextendsthenotionofBlochfunctionandtoprovesomeresultsgeneralizingthoseoftheearliercitedpaper[3].ThebasicideabehindourconsiderationsistoconsideraninfinitematrixAastheanalogueoftheformalFourierseriesassociatedtoa2π-periodicdistribution,thediagonalsAk,k∈Z,beingtheanaloguesoftheFouriercoefficientsassociatedtotheabovedistribution.Inthismannerwegetaone-to-onecorrespondencebetweeninfiniteToeplitzmatricesandformalFourierseriesassociatedtoperiodicdistributions.Hence,aninfinitematrixappearsinanaturalwayasamoregeneralconceptthanthoseofaperiodicdistributiononthetorus.Definition7.1.ThematricealBlochspaceB(D,2)isthespaceofallan-alyticmatricesAwithA(r)∈B(2),0≤r<1,suchthat||A||:=sup(1−r2)||A(r)||+||A||<∞,B(D,2)B(2)0B(2)0≤r<1whereB(2)istheusualoperatornormofthematrixAonthesequence∞k−1space2,andA(r)=k=0Akkr.149

164October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013150MatrixspacesandSchurmultipliers:MatricealharmonicanalysisAmatrixA∈B(D,2)iscalledaBlochmatrix.ItisclearthattheToeplitzmatrices,whichbelongtotheBlochspaceofanalyticmatricesB(D,2)coincidewithBlochfunctions.Hence,B(D,2)appearsasanextensionoftheclassicalBlochspaceoffunctionsB.AnimportantclassofBlochmatricesconsistsofthespaceL∞(D,)a2asthefollowingpropositionshows:Proposition7.2.TheBanachspaceL∞(D,)isasubspaceofB(D,),a22||A||B(D,2)≤6||A||L∞(D,2)aandL∞(D,))B(D,).a22Moreprecisely,theanalyticToeplitzmatrixAgivenbythesequence{1}∞isnotinL∞(D,),butA∈B(D,).k+1k=0a22Proof.Wenotethat(1−r2)A(r)=C(r)∗A1(r),where1(1−r2)(j−k)r(j−k−1)/2ifj≥k+1,C1(r)(k,j)=0ifj

165October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces151ThematrixversionofBlochspacecanbeconsideredasthelimitcaseofLp(D,)asp→∞.a2FirstwenotethatifA∈L∞(D,),thenr→A(r)isaw∗-measurable2functionand,consequently,eachfunctionaij(r)isaLebesguemeasurablefunctionon[0,1)foralliandjandwemayintroducePA(·)asinPropo-sition6.11.Theorem7.3.P:L∞(D,)→B(D,)andP|:L=∞(D,)→22L∞(D,22)B(D,2)areboundedsurjectionoperators.Proof.Clearlyitisenoughtoproveonlythefirstassertion.LetA(·)∈L∞(D,)andB=PA(·).WeshowthatB∈B(D,).22Ityieldsthat||B(r)||2≤B(2)⎡⎛⎞⎤21∞∞sup⎢⎜a(s)rj−i−1sj−i(j−i+1)(j−i)h⎟⎥⎣⎝ijj⎠(2sds)⎦||h||2≤10i=1j=i+112≤A(s)∗C(r,s)(2sds),B(2)0where(j−i+1)(j−i)(rs)j−i−1sifj>i,C(r,s)(i,j)=0ifj≤i.Thus,1π21/2sdθ||B(r)||B(2)≤C||A(·)||L∞(D,2)·iθ3sds0−π|1−rse|π1∼(byLemma1.3)∼C||A(·)||L∞(D,2)·2.1−r∞Consequently,||B||B(D,2)≤C||A(·)||L∞(D,2),thatis,P:L(D,2)→B(D,2)isaboundedoperator.InordertoshowthatPisontowetakeB∈B(D,)andputB1=2B−B−B.EasycalculationsshowusthatP[(B1)(r)(1−r2)]=B1∗T,014|k|(|k|+1)whereT=(tj−i)i,j∈Z,withtk=(2|k|−1)(2|k|+1).ClearlyT1=(t1),wheret1=(2|k|−1)(2|k|+1),isaSchurmulti-j−ii,j∈Zk4|k|(|k|+1)plier.

166October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013152MatrixspacesandSchurmultipliers:MatricealharmonicanalysisHence,itfollowsthatB2(r):=T1∗(B1(r))(1−r2)∈L∞(D,)and2P[B2(·)]=B1.WeonlyneedtoprovethatB+Br∈L∞(D,).012SinceP[B0+B1r]=B0+B1,itsufficestoshowthatr→B1r∈L∞(D,),(sinceB∈B()bythehypothesisB∈B(D,)).2022Then,since||B||=sup(1−r2)||B||=||B||,1B(D,2)1B(2)1B(2)0≤r<1itfollowsthatB1∈B(2).Thusr→Br∈L∞(D,).Theproofiscomplete.12Remark7.4.NotethatB(D,2)endowedwiththenorm||·||B(D,2)isaBanachspaceand,bytheopenmappingtheorem,itfollowsthatB(D,2)isisomorphictothequotientspaceL∞(D,)/KerP,endowedwithquotient2norm.Theorem7.5.TheprojectionPisaboundedoperatorfromL∞(D,)12(respectivelyfromL∞(D,2))ontoB(D,2).Proof.TheproofisaneasyadaptationoftheproofofTheorem6.12andthusweleaveoutthedetails.ByusingTheorem7.5andTheorem6.12weeasilygetthefollowingcorollary:Corollary7.6.Lp(D,)=[L1(D,),B(D,)]withequivalentnorms,a2a22θfor1

167October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces153∞n∞nLemma7.7.LetA∈B(D,2);A=n=0Anzandg(z)=n=0bnz∈I.Thenh(z)=∞Abzn:D→B()isacontinuousfunctioninn=0nn2|z|≤1and,moreover,h(z)B(2)≤2AB(D,2)gI(7.1)forallz≤1.Inparticular,itfollowsthatthereexists∞2πn1−ıθıθA,g=limAnbnρ=limA(ρe)g(ρe)dθ,ρ→1−ρ→1−2π0n=0forallf∈B(D,2),g∈I.Proof.Let|ζ|<1.Wehavethat∞A(z)=f(z)=nAzn−1Ann=1and∞dn[z(g(z)−b0)]=(n+1)bnz.dzn=1Then,wegeteasilythat,forz=reıθandζ∈D,112πd∞(1−r2)A(ζz)[z(g(z)−b)]e−ıθdθdr=Abζn−1.0nnπ00dzn=1ByapplyingH¨older’sinequalityandintegratingtermbytermwefindthat∞AbζnnnB(2)n=112π21≤sup(1−|z|)A(ζz)B(2)(|g(z)−b0|+r|g(z)|)dθdr.|z|<1π00Furthermore,wehavethat12π12πr|g(reiθ)−b|dθdr≤|g(teiθ)|dtdθdr=0000002π112π1dr|g(teiθ)|dtdθ=(1−t)|g(teiθ)|dtdθ.00t00

168October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013154MatrixspacesandSchurmultipliers:MatricealharmonicanalysisSincez→A(z)isasubharmonicfunction,wegetthatB(2)∞Abζn≤AbnnB(2)00B(2)n=012π21iθ+sup(1−|z|)A(z)B(2)|g(te)|dθdt.z∈Dπ00Hence,h(ζ)B(2)≤2AB(D,2)gI,for|ζ|<1.Inordertoshowthecontinuityofhin|z|≤1,wetakeζ1,ζ2∈Dandnotethat∞h(ζ)−h(ζ)=A(bζn−bζn)≤12B(2)nn1n2B(2)n=02AB(D,2)g(ζ1)−g(ζ2)I.Butitisknownthatthelastnormconvergesto0as|ζ1−ζ2|→0.(SeeTheorem2.2[3].)Hence,hcanbeextendedbycontinuitytoDandweget(7.1).Theproofiscomplete.Theorem7.8.LetA=kAkbeaBlochmatrix.Thenthefollowinginequalityholds:∞∞A∞|w|2μ+ν+1ν||wμwν||B(2)≤K,(7.2)μ+ν+12ν+1μ=0ν=0ν=0wherewν,ν=0,1,2,...arecomplexnumbersandK≤2||A||B(D,2).Conversely,(7.2)impliesthatA∈B(D,2)and||A||B(D,2)≤2K.Proof.Itisclearthatthedoubleseriesconvergesiftherighthandseriesconverges.Thereforewehavethat∞∞A∞Anμ+ν+1n+1wμwν=wνwn−ν=A,g,(7.3)μ+ν+1n+1μ=0ν=0n=0ν=0where∞1ng(z)=wwzn+1,z∈Dνn−νn+1n=0ν=0

169October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces155andbyA,gwemeanthat∞2πn1−ıθıθA,g=limAnbnρ=limA(ρe)g(ρe)dθ,ρ→1−ρ→1−2π0n=0forA(ρe−ıθ)=∞Aρne−ınθandg(ρeıθ)=∞bρneınθ,wherethen=0nn=0nseriesare||·||B(2)-convergent.Moreover,∞n∞g(z)=(ww)zn=(wzn)2.νn−νnn=0ν=0n=0Hence,bytheParsevalformula,wegetthat112π112π∞g=|g(z)|dθdr=|wzn|2dθdr=In2π0002π0n=01∞∞|w|2|w|2r2ndr=n.n02n+1n=0n=0Therefore,byLemma6.7,wehavethat∞|w|2nB(2)≤2AB(D,2),2n+1n=0thatis(7.2)holdsforA∈B(D,2).b)Conversely,if(7.2)holdswetakeζ∈Dandfindwnsuchthatg(z)=∞nζn−1zn=z=∞1(nww)zn.(SeeTheoremn=1(1−ζz)2n=0n+1ν=0νn−ν3.5-[3].)Usingthecomputationsdonein[3]atpage17itfollowsthat∞|w|22n=gI≤.2n+11−|ζ|2n=0By(7.2)and(7.3)wegetthatz∞|w|22KnA(ζ)B(2)=B(2)≤K2n+1≤1−|ζ|2.n=0ThereforeAB(D,2)≤2K.Theproofiscomplete.

170October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013156MatrixspacesandSchurmultipliers:MatricealharmonicanalysisWecangiveanelementaryclassofBlochmatrices.(SeealsoProposition1.5in[14].)∞Theorem7.9.LetA=k=0A2k.ThenA∈B(D,2)ifandonlyifsupk||Ak||B(2)<∞.Proof.ByTheorem7.8itfollowsthatthereisaconstantC>0suchthatC||A||B(D,2)≥supk||Ak||B(2),forallinfinitematricesA.WeconsideralacunarymatrixAasinthestatementofthetheorem.Then$$||zf(z)||∞$∞$AB(2)n$k2k$=|z|$A2k2z$≤sup||A2k||B(2)·1−|z|$$kn=0k=0B(2)⎛⎞∞∞2|z|⎝2k⎠|z|n≤2sup||An|z|n=sup||A.2k||B(2)22k||B(2)k(1−|z|)kn=12k≤nn=1Consequently,(1−r2)||A(r)||≤4sup||A,B(2)2k||B(2)kwhich,obviously,impliesthat||A||B(D,2)≤4sup||Ak||B(2).kItwasremarkedin[3]thattheclassicalBlochspaceoffunctionsBisaBanachalgebrawithrespecttoconvolution,or,equivalently,toHadamard∞ikθ(Schur)compositionoffunctions,thatis,forf=k=0ake∈Bandg=∞beikθ∈B,f∗g=∞abeikθ∈B.(See[3].)k=0kk=0kkNowweextendthisremarkintheframeworkofmatriceswithrespecttotheSchurproduct.ThisresultwaskindlycommunicatedtousbyV.Lie.Theorem7.10.ThespaceB(D,2)isacommutativeBanachalgebrawithrespecttoSchurproductofmatrices.Proof.Let⎛⎞a0a1......11⎜0a0a1...⎟∞⎜22⎟itkkiktA=⎜00a0...⎟,fj(re)=ajre,⎝3⎠.....k=0.......

171October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces157where0≤r<1,t∈[0,2π)and||A||=sup(1−r2)||A(r)||.B(D,2)B(2)r<1Then,denotingbyfthepartialderivativeoffwithrespecttor,jj||A||isgivenby:B(D,2)∞2π2itijt−itdt21/2||A||B(D,2)=sup(1−r){sup(|fj(re)eh(e)|)}.r<1||h||2≤102πj=1(See[18].)Hence,(||A∗B||)2=B(D,2)∞2π22itijt−itdt2sup{sup(1−r)|(fj∗gj)(re)eh(e)|},r<1||h||2≤102πj=1where(fj)jcorrespondstoAasaboveand(gj)jcorrespondstoB.Then,wehavethat√r2πiti(θ+t)−iθdθr(fj∗gj)(re)=2fj(se)gj(se)sds,002πforallj.BytheCauchy-Schwarzinequalityitfollowsthat∞2π2(f∗g)(reit)eijth(e−it)dt=jj2πj=10∞√r2π2π21ssi(θ+t)ij(t+θ)1dtdθ4r2gj(eiθ)eijθ(fj(se)eh(eit)2π)2πdsj=1000∞√r2πdθ≤4r−2|g(se−iθ)|2sds×j002πj=1√r2π2π2dtdθsf(sei(θ+t))eij(t+θ)h(e−it)ds:=I.j2π2π000Moreover,2π22g−it2dθ2supsup(1−s)j(se)≤(||B||B(D,2))j≥1s<102π

172October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013158MatrixspacesandSchurmultipliers:Matricealharmonicanalysisandfor||h||2=1wealsohavethat∞2π2(1−s2)2f(seit)eijth(e−it)dt≤(||A||)2.j2πB(D,2)j=10Consequently,√2√2rs(||B||)rs(||A||)−2B(D,2)B(D,2)I≤4rdsds=0(1−s2)20(1−s2)2(1−r)−2(||A||)2(||B||)2,B(D,2)B(D,2)thatis||A∗B||≤C||A||||B||.B(D,2)B(D,2)B(D,2)Theproofiscomplete.Theorem7.11.TheBanachspaceL1(D,)∗(dualofL1(D,))maybea2a2identifiedwithB(D,).Namely,letA∈L1(D,)andB∈B(D,).Then2a22wehavethat1∗||=|tr[A(r)B(r)](2rdr)|≤C||A||L1(D,2)·||B||B(D,2),a0whereC>0isaconstant.Proof.SinceC1,theSchattentraceclassofoperators,isaseparableBanachspacewithC∗=B(),by=tr(AB∗),wehave,in12viewTheorem1.2,thatL1(D,)∗=L∞(D,),usingthedualitymap221∗=tr[A(r)B(r)](2rdr).0ThenwehavethatL˜1(D,)∗=L∞(D,)/(L˜1(D,))⊥.a22a2UsingthefactthatL1(D,)iscanonicallyisomorphictoL˜1(D,),wea2a2havetoshowthatKerP=KerP˜=(L˜1(D,))⊥inL∞(D,).a22ButKerP˜⊂(L˜1(D,))⊥,sinceforA(r)∈L∞(D,)suchthatPA˜(·)a22=0,wehave,atleastforfinitelyordermatricesA(·),B(·),that=,andifB∈L˜1(D,)),thena2===0,and,consequently,A(·)∈(L˜1(D,))⊥.a2

173October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces159Conversely,letA(·)∈(L1(D,))⊥,thatis=0∀B∈a2L1(D,)).TakingB(r)(i,j)=rj−iforj>i,withfixedj,ianda21B(r)(i,j)=0otherwise,wegetthataij(r)(2rdr)=0forallj>i.0Thus(PA˜)(r)(i,j)=0foralli,j,thatisA(·)∈KerP.˜ForB(r)∈L∞(D,)andA∈L1(D,),weeasilygetthat2a211|tr[A(r)B∗(r)](2rdr)|≤|tr[A(r)B∗(r)]|(2rdr)≤001||A(r)||C1·||B(r)||B(2)2rdr≤||A||L1a(D,2)·||B(·)||L∞(D,2),0sousingRemark7.4wegettherequiredinequality,sinceforA∈L1(D,),a2B∈B(D,2)wehaveobviouslythat||=||≤||A||L1(D,2)||B(·)||L∞(D,2)a∀B(·)∈L∞(D,)suchthatPB(·)=B.Theproofiscomplete.2nLemma7.12.LetAbeamatrixoffiniteband-type,thatisA=k=1Ak,suchthatAk∈C1fork=1,2,...andletB∈B(D,2).Then∞1=tr(Ak∗Bk).k+1k=01Proof.Werecallthat=tr[A(r)B∗(r)]2rdr.Wedenotethe0entriesofthediagonalmatrixA,wherek∈Z,by(al)∞.Thenitiseasykkl=1toseethatn∞trA(r)B∗(r)=alblr2kkkl=1k=0and,consequently,1n∞1∞n=alblr2k2rdr=2r2k+1albldrkkkk00l=1k=0k=0l=1∞1n∞1=albl=tr(A∗B).k+1kkk+1kkk=0l=1k=0Theproofiscomplete.

174October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013160MatrixspacesandSchurmultipliers:MatricealharmonicanalysisNextwepresentaresult,whichshowsustheintrinsicconnectionbe-tweenthematricialBlochspaceandtheBergmanmetric.(SeeThm.5.1.6[94]fortheclassicalToeplitz(function)case.)Theorem7.13.IfA∈B(D,2),then2||fA(z)−fA(w)||B(2)sup(1−r)||A(r)||B(2)=sup.0≤r<1z,w∈Dz=wβ(z,w)Proof.TheproofissimplyatranslationoftheproofofTheorem5.1.6[94],replacingfbyfA.Weomitthedetails.Inparticular,weconcludethat,infact,theBlochmatricesarethoseuppertriangularmatricessuchthatB∗C(z)areLipschitzB(2)-valuedfunctionswithrespecttoBergmanmetriconD.WecanusethisresultforprovingthatB(D,2)isaBanachspace.WenotethatbyTheorem7.11thisisclear,butthenextproofismuchsimpler.Theorem7.13hasasanimmediateconsequencethefollowinginequality:1+|z|||fA(z)||B(2)≤||A||B(D,2)log1−|z|forall|z|≥1,whichimpliesthatthepointevaluationofamatrixA,that2isthelinearoperatorΔz:B(D,2)→B(2),givenbyΔz(A)=fA(z),isboundedforallz∈D.This,initsturn,impliesthatifasequenceAnofmatricesconvergesintheBlochnorm,thenΔ(An)doessolocallyzuniformlywithrespecttoz.Inparticular,wecanusetheinequalityabovetoprovethecompletenessofB(D,2).Proposition7.14.ThespaceofBlochmatricesB(D,2)isaBanachspace.Proof.DenotingthedilationsoffAby(fA)r(z)=fA(rz)=fA∗C(r)(z),wehavethat||A∗C(r)||=sup(1−|z|2)||(f)(z)||+||(f)(0)||=B(D,2)ArB(2)ArB(2)z∈Drsup(1−|z|2)||(f)(rz)||+||f(0)||,AB(2)AB(2)z∈Dwhichincreasesto||A||B(D,2)asrincreasesto1.Usingthisfact,wecannowshowthatB(D,2)iscomplete.

175October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces161Let{An}beaCauchysequenceinB(D,).By(∗),thisimpliesthat2{fAn}isauniformCauchysequenceoneachcompactsubsetofD,and,hence,itconvergeslocallyuniformlytosomevector-valuedanalyticfunc-tionf.Itremainstoshowthat||An−A||→0.Given>0,chooseAB(D,2)Nsuchthat||An−Am||<whenn,m≥N.Then,forr<1,B(D,2)||An∗C(r)−A∗C(r)||≤||An∗C(r)−Am∗C(r)||+B(D,2)B(D,2)||Am∗C(r)−A∗C(r)||≤||An−Am||B(D,2)B(D,2)+||Am∗C(r)−A∗C(r)||<+||Am∗C(r)−A∗C(r)||.B(D,2)B(D,2)Observethatthelasttermapproaches0asm→∞,since(Am∗C(r))convergesto(A∗C(r))uniformlyonD.Thus||An∗C(r)−A∗C(r)||≤2forn≥Nandallr<1.B(D,2)Finally,weletr→1toarriveatthedesiredconclusion.Theproofiscomplete.7.2MatrixversionoflittleBlochspaceNowweintroduceanotherspaceofmatrices,theso-calledlittleBlochspaceofmatrices.Definition7.15.ThespaceB0(D,2)isthespaceofalluppertriangular2infinitematricesAsuchthatlimr→1−(1−r)||(A∗C(r))||B(2)=0,whereC(r)istheToeplitzmatrixassociatedwiththeCauchykernel.ClearlyB0(D,2)isaclosedsubspaceofB(D,2)iftheformerspaceisendowedwiththenormofB(D,2).LetA∈B(D,2)andAr(s)=A(rs)=A(r)∗P(s)forall0≤r<1and0≤s<1,whereP(s)istheToeplitzmatrixassociatedtothePoissonkernel,thatis⎛23⎞1sss...⎜..⎟⎜s1ss2.⎟⎜⎟⎜..⎟P(s)=⎜s2s1s.⎟.⎜⎟⎜..⎟⎜s3s2s1.⎟⎝⎠...............

176October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013162MatrixspacesandSchurmultipliers:MatricealharmonicanalysisThenitfollowsthatArisamatrixbelongingtoB0(D,2),forall0≤r<1,since2π2(1−s)dθslim→1(1−s)||Ar(s)||B(2)≤slim→11−r2s2||A||B(D,2)·r|1−seiθ|−πr21∼||A||B(D,2)·1−r2slim→1(1−s)log1−s2=0.Theorem7.16.LetA∈B(D,2).ThenA∈B0(D,2)ifandonlyiflim||Ar−A||B(D,2)=0.r→1−Proof.BytheremarkaboveitfollowsthatAr∈B0(D,2)andweusetheobviousfactthatB0(D,2)isaclosedsubspaceofB(D,2)inordertoconcludethattheconditionissufficient.Conversely,letA∈B0(D,2).Then∀>0thereis0<δ<1suchthat(1−s2)||A(s)||<∀δ2

177October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces163s(1−r)δ≤||A()||·.δB(2)(1−rδ)(1−δ)WerecallthatEistheToeplitzmatrixhavingallitsentriesequalto1.Therefore,sup(1−s2)||A(rs)−A(s)||≤B(2)0≤s≤δss1−s2δ(1−r)sup(1−()2)||A()||··≤B(2)s20≤s≤δ<δδδ1−2(1−δ)(1−rδ)δ1−δ2δ(1−r)||A||B(D,2)·δ2·(1−δ)(1−rδ),1−δ2forallt≥0.Consequentlylimsup(1−s2)||A(s)−A(s)||=0.rB(2)r→1−s≤δTheproofiscomplete.Corollary7.17.B0(D,2)istheclosureofallmatricesoffinitebandtypeintheBlochnorm.Inparticular,thisimpliesthatB0(D,2)isaseparablespace.nnProof.LetA∈B0(D,2)andA=k=0Ak..Then,byTheorem7.16,wehavethat∀>0thereisar0<1suchthat||Ar−A||B(D,2)</2.Wenotethatr→A(r)forr∈[0,1)isacontinuousB(2)-valuedfunctionon[0,s]fors<1.Indeed,let0

178October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013164MatrixspacesandSchurmultipliers:Matricealharmonicanalysisforallanalyticmatrices.Thus,forr

179October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces165Thus,bythedefinitionofBergmanprojectionP,wegetthat[PB2(·)](r)(i,j),ta(j−i+1)(j−i)rj−i1forj−i≥2,ijij33(j−i)+1[(j−i+1)]=220ifj−i<2.Consequently,bytaking,3[3(j−i)+1]forallj=i,i,j≥1tij=4(j−i)9ifj=i,i≥1,4itfollowsthatTisaSchurmultiplier,andP[B2(·)]=A1(·).LetnowB(r)=2(1−r2)A+3(1−r2)rA+B(r).012Itisclearthat[PB(·)](r)=A(r).But,sinceA∈B0(D,2),itfollowsthatB2(r),and,consequently,B(r),isacontinuousB(2)-valuedfunction,andlimr→1B(r)=0.Thus3)holdsandwehaveprovedthat1)implies3).Itisobviousthat3)implies2).Itremainstoprovethat2)implies1).Let2)holdandchooseB(r)∈B(2)suchthat[PB(·)](r)=A(r)forr∈[0,1].Assumethatr→B(r)isacontinuousB(2)-functionon[0,1]andletM=sup0≤r≤1||B(r)||B(2)<∞.Let0≤r0<1befixedandconsiderAr0(r)givenbytheformula,(j−i+1)(rr)j−i(21b(s)sj−i+1ds)ifj−i≥0,A(r)(i,j)=00ijr00otherwise.Consequently,accordingtoTheorem7.3,wefindthatAr0=P[P(r0)∗B(·)]=P[Br0(·)]∈B(D,2),whereP(r0)istheToeplitzmatrixassociatedtothePoissonkernel.LetC(D,2)denotethespaceofallcontinuousB(D,2)-valuedfunctionsdefinedon[0,1].Nextweprovethatthefunctions→P(P(r0)∗B(s))belongstoC(D,2)ifBisacontinuousB(2)-valuedfunctionsuchthatlimsup||Ar(s)−A(s)||B(2)=0.(7.4)r→1s∈[0,1]This,initsturn,impliesthatlimr→1Ar=AinB(D,2).Therefore,bytheTheorem7.16,itfollowsthatA∈B0(D,2).

180October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013166MatrixspacesandSchurmultipliers:MatricealharmonicanalysisLets,s0∈[0,1].Then|k|ikθ||P(r0)∗[B(s)−B(s0)]||B(2)≤||r0e||M(T)·||B(s)−B(s0)||B(2)→0k∈Zfors→s0andB(s)isacontinuousfunctionon[0,1].HerewehaveusedBennett’sTheoremandthefactthat1−r2||r|k|eikθ||=||||≤1.M(T)|1−reiθ|2M(T)k∈ZThus,thefunctions→P[P(r)∗B(s)]belongstoC(D,2).0Hence,itonlyremainstoprovethat(7.4)holds.Infact,|k|ikθ||P(r0)∗B(s)−B(s)||B(2)≤||(r0−1)e||M(T)·||B(s)||B(2)k∈Z≤M·||(r|k|−1)eikθ||foralls∈[0,1].M(T)k∈ZDenotingbyμ(θ)themeasure(r|k|−1)eikθ,then,foratrigono-rk∈Zmetricpolynomialφ(θ)=maeinθ,wehavethatn=−mnmμ(φ)=(r|n|−1)aand|μ(φ)|≤|φ(r)−φ(1)|≤2||φ||,rnrn=−mwhereφ(r)isthevalueofthePoissonextensionofφinthepointr.Consequentlyμrisameasurewiththenormlessthan2.Butlimμ(φ)=0foralltrigonometricpolynomesφ.Thus,w∗-limμ=r→1rr→1r0inM(T)andthenitisclearthatlimr→1||μr||=0and,accordingtoThe-orem7.3,therelation(7.4)isproved.Thus,alsotheimplication2)⇒1)isprovedandtheproofiscomplete.Now,usingthenotationwhichpreceedsTheorem6.12,weget:Theorem7.19.P2isacontinuousoperator(preciselyacontinuouspro-jection)fromL1(D,)ontoL1(D,).2a2Proof.ByTheorem1.2thetopologicaldualofL1(D,))isL∞(D,)22withrespecttothedualitypair:1:=tr(A(s)[B(s)]∗)2sds,0whereA(·)∈L∞(D,),B(·)∈L1(D,).Byusingadualityargumentit22issufficienttoprovethatP∗:L∞(D,)→L∞(D,)isbounded.222

181October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces167WearelookingfortheadjointP∗ofP:221∞∞=2(P∗A(·))(r)(i,j)b(r)rdr22ij0i=1j=1∞∞1=(P∗A(·))(r)(i,j)b(r)(2r)dr.2iji=1j=10Ontheotherhandityieldsthat1==trA(r)(PB)∗(r)(2rdr)2220∞∞1=A(r)(i,j)(P2B)(r)(i,j)(2rdr)i=1j=10∞∞1Γ(j−i+4)j−i=[A(s)](i,j)s(2sds)×(j−i)!Γ(3)0i=1j=i1b(s)sj−i(1−s2)2(2sds).ij0Nowweconsider{Ik},asequenceofintervalssuchthatlimμ(Ik)=0,dμ=2sdsandr∈Ik.k→∞Foreveryk,wetakeB(s)(i,j)=χIk(s)/(μ(Ik))andB(s)(l,k)=0,(l,k)=(i,j)forevery(i,j)∈N×N.ByLebesgue’sdifferentiationtheoremwehavethat,Γ(j−i+4)rj−i(1−r2)21A(s)(i,j)sj−i(2sds)ifj≥i,(P∗A(·))(r)(i,j)=(j−i)!Γ(3)020ifj

182October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013168MatrixspacesandSchurmultipliers:MatricealharmonicanalysisConsequently,becauseL1[0,1]hascotype2,thereisaconstantK>0,suchthat||P∗A(·)||2=2L∞(D,2)∞∞1Γ(j−i+4)j−i22j−i2||sup|hjr(1−r)aij(s)s(2sds)|||L∞=||h||l2≤1i=1j=10(j−i)!Γ(3)∞1∞Γ(j−i+4)24j−i2||sup(1−r)|(aij(s)((rs))hj)(2sds)|||L∞≤||h||l2≤1i=10j=i(j−i)!Γ(3)1∞∞(j−i+3)!K||(1−r2)4sup[(|a(s)[(rs)j−ih|2)1/2(sds)]2||.ijj||h||l2≤10i=1j=i(j−i)!SincetheToeplitzmatrixC(rs)=(c)(rs)∞,whereiji,j=1(rs)j−i(j−i+3)(j−i+2)(j−i+1)ifj≥icij(rs):=cj−i(rs)=0ifj

183October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces169WedenotebyC0(D,2)thespaceofallcontinuousB(2)-valuedfunc-tionsB(r)on[0,1]suchthatlimr→1B(r)=0inthenormofB(2).Lemma7.20.LetV=(P)∗,thatis2(P)∗(A(r))(i,j)=2,(j−i+3)(j−i+2)(j−i+1)rj−i(1−r2)21a(s)sj−i(2sds)ifj−i≥0,20ij0otherwise.ThenVisanisomorphicembeddingofB0(D,2)inC0(D,2).Proof.AccordingtoTheorem7.18,forB∈B0(D,2)wecanfindsomeA(·)∈C0(D,2)suchthat[PA(·)](r)=B(r).Clearly,wehavethatP∗B=P∗PA=T1∗(P∗A),2221whereA1(r)=T∗A(r),forT=(tj−i)i,jwith,2(j−i+1)forj−i=−2,tj−i=j−i+20otherwise.TisaSchurmultiplierandthesameistrueforT1=(t1),wherej−ii,j,j−i+2forj−i=−1,t1=2(j−i+1)j−i0otherwise.Thus||A1(r)||B(2)∼||A(r)||B(2)forallr∈[0,1].Hence,weobtainthat||P∗A(r)||221B(2)∞∞Γ(j−i+4)1=sup|hrj−i(1−r2)2a1(s)sj−i(2sds)|2≤jij||h||2≤1i=1j=i(j−i)!Γ(3)01∞∞Γ(j−i+4)Ksup(1−r2)4[(|a1(s)(rs)j−ih|2)1/2(2sds)]2.ijj||h||2≤10j=1j=i2(j−i)!TheToeplitzmatrixC(r,s)=(cj−i(r,s))i,j,where(rs)j−i(j−i+3)(j−i+1)2ifj≥icj−i(r,s)=0otherwise,isaSchurmultiplier,since∞k2ikθ1(rs)(k+3)(k+1)e∼(1−rseiθ)4k=0

184October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013170MatrixspacesandSchurmultipliers:Matricealharmonicanalysisandπ∞1dθ22n∼n(rs).−π|1−rseiθ|42πn=0Therefore,wehavethat1∞||P∗A(r)||≤C(1−r2)2||A(s)||·(n2(rs)2n)(2sds).21B(2)B(2)0n=0Sincelims→1||A(s)||B(2)=0,forall>0,thereisδ>0suchthat||A(s)||<forall1≥s≥δand,consequently,(1−r2)2δ2∗P2A1(r)B(2)≤C(+A(·)C(D,2)·222).(1−rδ)∗1Itfollowsthatlimr→1P2A1(r)B(2)=0,and,sinceTisaSchurmul-∗2∗tiplier,itfollowsthatP2B∈C0(D,)and||P2B||B(2)≤C||A(·)||C(D,2).Moreover,inviewoftheproofofTheorem7.18,wecanfindanA(·)∈C0(D,2)suchthat||A(·)||C0(D,2)≤C(||B0||B(2)+||B(·)||B(D,2)),whereC>0isanabsoluteconstant.BynowalsousingtheargumentsintheproofofTheorem7.19itfollowsthatP∗:B(D,)→C(D,)is20202bounded.Ontheotherhand,ifA∈B0(D,2),then,sinceA(r)isananalyticmatrix,itisobviousthatA(r)=[PA(·)](r)=(P[P∗A(·)])(r)forallr∈2[0,1).Thus,byusingTheorem7.19,weconcludethatthereexistsaconstantC>0suchthat||A(·)||≤C||P∗A(·)||,whichimpliesthatB(D,2)2C(D,2)P∗:B(D,)→C(D,))isanisomorphicembedding.Theproofis20202complete.FromnowonweidentifyB(D,2)withthespaceB(D,2)ofallana-00lyticmatricesA∗C(r),forA∈B(D,2).0WeintroduceB0,c(D,2)astheclosedBanachsubspaceofB0(D,2)consistingofalluppertriangularmatriceswhosediagonalsarecompactoperators.WearenowreadytoprovethatthisspaceB0,c(D,2)isinfactthepredualoftheBergman-Schattenspace.Moreexactly,ourlastmainresultinthisSectionisthefollowingdualityresult:Theorem7.21.ItyieldsthatB(D,)∗=L1(D,)withrespecttothe0,c2a2usualduality,wheneverB0(D,2)hasthenorminducedbyB(D,2).

185October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces1711Proof.LetA∈L1(D,).ThenB→tr[B(s)A∗(s)](2sds)definesaa20linearandboundedfunctionalonB0,c(D,2)duetoTheorem7.11.Con-versely,letusassumethatFisaboundedlinearfunctionalonB0,c(D,2).ThenweshallshowthatthereisamatrixCfromL1(D,)suchthata21F(B)=tr[B(r)C∗(r)](2rdr),(7.5)0forBfromadensesubsetofB0(D,2).InwhatfollowsweidentifythematricesA∈B0(D,2)withthefunctionnknr→A∗C(r).Inparticular,weidentifyk=0rAkwithk=0Ak.ByLemma7.20itfollowsthatP∗:B(D,)→C(D,),isaniso-20202morphicembedding.ThusX=P∗(B(D,))isaclosedsubspacein20,c2C(D,C)andF◦(P∗)−1:X→C,isaboundedlinearfunctionalonX,0∞2whereC0(D,C∞)isthesubsetinC0(D,2)whoseelementsareC∞-valuedfunctions.ByHahn-BanachtheoremF◦(P∗)−1canbeextendedtoa2boundedlinearfunctionalonC0(D,C∞).LetΦ:C0(D,C∞)→Cdenotethisfunctional.ItfollowsthatC0(D,C∞)=C0[0,1]⊗ˆC∞,Thus,Φisabilinearintegralmap,i.e.,thereisaboundedBorelmeasureμon[0,1]×UC1,whereUC1istheunitballofthespaceC1withthetopologyσ(C1,C∞),suchthatΦ(f⊗A)=f(r)tr(AB∗)dμ(r,B)∀f∈C[0,1]andA∈C.0∞[0,1]×UC1nThus,forthematrixk=0Ak∈B0,c(D,2),identifiedwiththeanalyticmatrixnArk,wehavethatk=0knnnF(A)=F(rkA)=[F◦(P∗)−1][P∗(rkA)]kk22kk=0k=0k=0n(k+3)(k+2)k22=Φ(r(1−r)Ak)2k=0n(k+3)(k+2)k22∗=tr[(r(1−r)Ak)B]dμ(r,B)[0,1]×UC1k=02n(k+3)(k+2)k∗22:=<μ(r,B),tr(rAk)B(1−r)>.2k=0

186October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013172MatrixspacesandSchurmultipliers:MatricealharmonicanalysisOntheotherhand,wewishtoprovethatn1n1nF(A)=tr(skA)(C(s)∗)(2sds)=tr(s2kAC∗)(2sds)kkkk00k=0k=0k=0nC∗=trA(k).kk+1k=0Now,lettingA=ei,i+k,whereei,i+kisthematrixhaving1asthesinglenonzeroentryontheith-rowandthe(i+k)th-column,fori≥1andj≥0,wefindthat(k+1)(k+2)(k+3)k22Ck=<μ(r,B),r(1−r)Bk>,k=0,1,2,....2Therefore,denoting[0,1]×UC1byV,wehave1||C(s)||C1(2sds)01n(k+3)!k22=||(sr)(1−r)Bkdμ(r,B)||C1(2sds)≤0V2k!k=01n(k+3)!k22[||(rs)(1−r)Bk||C1)(2sds)]d|μ|(r,B)≤V02k!k=01n(k+3)!k22ik(·)[||(rs)(1−r)e||L1(T)||B||C1)(2sds)]d|μ|(r,B)V02k!k=012π22(1−r)dθ∼[(2sds)]d|μ|(r,B)V00|1−rseiθ|42π1∞=(1−r2)2(n+1)2(sr)2n(2sds)d|μ|(r,B)V0k=0∞=(1−r2)2(n+1)r2nd|μ|(r,B)≤||μ||<∞.Vn=0Consequently,C∈L1(D,)andwegettherelation(7.5),usingthea2nfactthatthesetofallmatricesk=0AkisdenseinB0,c(D,2).Theproofiscomplete.

187October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013AmatrixversionofBlochspaces173NotesTheideabehindourstudyofmatrixversionsofdifferentkindsoffunctionspacesistoconsiderthediagonalsofanuppertriangularinfinitematrixasananalogueofFouriercoefficientsofananalyticfunctionordistribution.UsingthisideaweconsideraBlochspaceofmatrices(see[72])andproveresultssimilartothosewhichcanbefoundinthewell-knownpaper[3].WementiontheremarkablefactthatthisBlochspaceisacommutativeBanachalgebraundertheSchurproductandthatitisthetopologicaldualoftheBergman-Schattenspace.Section7.2isdedicatedtointroduceandstudythelittleBlochspaceofmatrices.Inparticular,itisprovedthatthedualspaceofthesubspaceofthelittleBlochspaceconsistingofcompactmatricesistheBergman-Schattenspace.(SeeTheorem7.21.)

188May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

189October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Chapter8SchurmultipliersonanalyticmatrixspacesAninterestingtopicinmatricealharmonicanalysisisthestudyofSchurmultipliersondifferentclassesofBanachspacesofinfinitematrices.FirstwedescribetheSchurmultipliersfromB(2)intoB(D,2).Theorem8.1.AnuppertriangularmatrixAbelongsto(B(2),B(D,2))∞k−1ifandonlyifsup0≤r<1(1−r)||k=1kAkr||M(2)<∞,or,equivalently,ifandonlyiftheM()-valuedfunctionf(z)=∞AzkisaLipschitz2Ak=0kfunctionwithrespecttotheBergmanmetricβ(z,w).Proof.LetB∈B(2)andA∈(B(2),B(D,2))with||A||(B(2),B(D,2))=C.Then∞sup(1−r)||kA∗Brk−1||≤C·||B||kkB(2)B(2)0≤r<1k=1anditfollowsthat∞sup(1−r)||kArk−1||≤C.kM(2)0≤r<1k=1Conversely,if∞sup(1−r)||kArk−1||=C<∞,kM(2)0≤r<1k=1then,foranarbitraryB∈B(2),wegetobviouslythat∞sup(1−r)||kA∗Brk−1||≤C·||B||,kkB(2)B(2)0≤r<1k=1and,consequently,A∈(B(2),B(D,2))and||A||(B(2),B(D,2))≤C.175

190October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013176MatrixspacesandSchurmultipliers:MatricealharmonicanalysisOntheotherhand,itisclearthattheproofofTheorem5.1.6in[94]holdsalsoifweconsiderM(2)-analyticfunctionsinsteadofusualanalyticones.Thus,A∈(B(2),B(D,2))ifandonlyifthefunctionfA(z)isaLipschitzfunctionwithrespecttotheBergmanmetric.Theproofiscomplete.Remark.TheprevioustheoremmayberegardedasamatrixextensionofTheorem4.3ofJevticandPavlovic[47],namely:TheoremofJevticandPavlovic.(H∞,B)=H1,∞,1.1NextweconsiderthepossibilitytofindthespaceofallSchurmultipliersfromB(D,2)intoitselfandexpectthatthefollowingstatementholds:1,∞,1Theorem8.2.(B(D,2),B(D,2))=H1(2)={Auppertriangularmatricessuchthatsup(1−r)||A(r)||<∞}.0

191October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Schurmultipliersonanalyticmatrixspaces177Proof.LetAbeanuppertriangularmatrixsuchthatsup(1−r)||A(r)||<∞M(2)r<1andBsuchthatsup(1−r)||B(r)||B(2)<∞.r<1Wehavetoshowthatsup(1−r)||(A∗B)(r)||B(2)<∞.0

192October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013178MatrixspacesandSchurmultipliers:MatricealharmonicanalysisNextwewanttopresentanapplicationofBlochmatricestodescribingtheSchurmultipliersfromamatricealversionoftheHardyspacetoamatrixversionoftheBMOspace,extendinganiceresultofMateljevicandPavlovic[60].FirstwegiveaformulaforthenorminB(2)duetoV.Lie[50](seealsoChapter2):∞1/21||B||=sup|LB(x)h(x)dx|2.B(2)k||h||≤10L2(0,1)k=1Moreover,weusethisformulatoproveaninterestinginequalityalsoduetoV.Lie.Proposition8.5.IfA∈H1()andB∈B(D,),thenwehavethat221/212||(A∗B)(r)||B(2)(1−r)dr≤C||A||H1(2)||B||B(D,2),0whereC>0isaconstantindependentofthechoiceofAandB.Proof.Fromtheaboveformulaappliedto(A∗B)wemayfindafixedh∈L2([0,1])with||h||=1suchthat2∞1||(A∗B)(r)||2∼|LA∗LB(re2πit)e2πijth(e2πit)dt|2.B(2)jjj=10Weusenowthefollowingrelation,whichholdsiffandgareanalyticfunctionsontheunitdiskDandifz=qe2πiζ,where1≥|z|=q>0:√q1z(f∗g)(z)=2f(re2πi(θ+ζ))g(re−2πiθ)re2πiζdθdr,00andweget,byduality,thatthereisax=(xj)j≥1∈2(N),with||x||2≤1,suchthat∞1||(A∗B)(r)||≈|xLA∗LB(re2πit)e2πijth(e2πit)dt|∼B(2)jjjj=10∞√r11|(LA)(se2πiθ)x((LB)(se−2πi(θ−t))e2πijth(e2πit)dt)sdsdθ|jjjj=1000√⎛⎞1/2r1∞≤⎝|x|2|LA(se2πiθ)|2⎠×jj00j=1

193October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Schurmultipliersonanalyticmatrixspaces179⎛⎞1/2∞1⎝|(LB)(se−2πi(θ−t))e2πijth(e2πit)dt|2⎠sdsdθ≤jj=10√11/2r∞|x|2|(LA)(se2πiθ)|2dθ0j=1jjC||B||B(D,2)×ds.01−sFromthisrelationwededucethat1||(A∗B)(r)||2(1−r)dr≤C||B||2×B(2)B(D,2)01√r1∞2A2πiθ21/220(j=1|xj||(Lj)(se)|)dθsup(1−r)dsdr.||x||2≤1001−sUsingnowProposition4.4wefindthat1||(A∗B)(r)||2(1−r)dr≤C||B||2||A||2.B(2)B(D,2)H1(2)0Theproofiscomplete.NowweprovetheextensionoftheresultofMateljevicandPavlovic:Theorem8.6.Wehavethefollowingrelation:B(D,)=H1(),BMOA()222withequivalenceofnorms.Proof.FirstweprovethatH1(),BMOA()⊃B(D,).222Itisenoughtoshowthat||A∗B||BMOA(2)≤K||A||H1(2)||B||B(D,2),forallA∈H1(),andB∈B(D,),whereK>0isaconstantnot22dependingonA,thatis,accordingtoCorollary5.4,wehavetoprovethat:1/212||(A∗B)||B(2)(1−r)dr≤C||A||H1(2)||B||B(D,2),0whereC>0isaconstantindependentofthechoiceofAandB.ButthisisjustProposition8.5.

194October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013180MatrixspacesandSchurmultipliers:MatricealharmonicanalysisTheoppositeinclusioniseasy.Wehavetoprovethat(H1(),BMOA())⊂B(D,).222Inordertodothis,weassumethatthefollowingrelationholds:(H1(),B(D,))⊂B(D,).(8.1)222Thenitisenoughtoprovethat:BMOA(2)⊂B(D,2).Weremarkthat,foranuppertriangularmatrixA,wehavethatA||A||L2(2)=sup||Lk||L2(T),(8.2)kwhereTistheunidimensionaltorus.Withthesamenotationsasabove,by[18],wehavethat∞1/21A2A||A||B(2)=sup|Lk(s)h(s)ds|≤sup||Lk||L∞(T),||h||2≤10kk=1which,initsturn,by[33]and(8.2),impliesthat||A||≤sup||LA||≤sup||LA||≤||A||.B(D,2)kB(D)kBMOABMOA(2)kkHence,wehavetoproveonly(8.1).ThisistheassertionofthenextPropositionsotheproofiscompletewhenthisresultisproved.Proposition8.7.Wehavethat(H1(),B(D,))⊂B(D,).222Proof.LetBbeanuppertriangularmatrixsuchthatA∗B∈B(D,2),forallA∈H1().BytheclosedgraphtheoremthereisaconstantC>02suchthat1||A∗B||B(D,2)≤C||A||H1(2),forallA∈H(2).iθNowtaketheToeplitzmatrixAassociatedwithfunctionθ→e.(1−reiθ)2Thus,2π1dθ1||A||H1(2)=iθ2=2,2π0|1−re|1−rwhichinturnimpliesthatC||A∗B||B(D,2)≤2.1−r∞k−1OntheotherhanditisclearthatA∗B=k=0kBkr=B(r),and,thus,||B(r)||≤Cforall0

195October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Schurmultipliersonanalyticmatrixspaces181Moreover,usingthedefinitionofB(D,2),wehavethatC||B∗C(r)∗C(ρ)||B(2)≤(1−r2)(1−ρ2),forall00dependingonlyonB.Lettingr=ρwegetthatC||B∗C(r)∗C(r)||B(2)≤(1−r2)2,∀0

196October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013182MatrixspacesandSchurmultipliers:MatricealharmonicanalysisThefollowingresultextendsCorollary1in[60]:Corollary8.9.Ityieldsthat(H1(),VMOA())=B(D,).222Proof.LetA∈H1()andB∈B(D,).Then,byTheorem8.6,221||X∗B||BMOA(2)≤c||X||H1(2)forallX∈H(2),(8.4)wheretheconstantcdoesnotdependonX.IfwesubstituteX=A(s)−Ain(8.4),wegetthat||A∗B(s)−A∗B||BMOA(2)≤c||A(s)−A||H1(2).Sinceobviouslythetermontheright-handsideofthelastinequalityap-proaches0whens→1,itfollowsthatA∗B∈VMOA(2).Theproofiscomplete.Letn1,n2<...bealacunarysequenceofintegersinthesensethatnk+1/nk≥q>1.∞nItiswell-knownthatg(z)=zk∈B,(seeforinstancethemorek=1generalTheorem7.9),sothatwehavebythepreviouscorollary:Corollary8.10.IfA=A∈H1(),then,foreverylacunarysequencen2{nk},∞A˜:=Ank∈VMOA(2).k=1Remark.ItfollowsbythedefinitionofH1()thatthespaceofall2uppertriangularHilbert-SchmidtmatricesTiscontainedinH1(),so22∞thatifA∈T2,thenwehavethatk=1A2k∈VMOA(2).SinceclearlyVMOA()⊂BMOA()⊂H2(),wegetthematrix222versionofPaley’stheorem:1∞Theorem8.11.IfA=An∈H(2),thenP(A):=k=1A2k∈H2().2AnotherconsequenceofCorollary8.10isthefollowingextensionofaremarkof[60]:∞Corollary8.12.ItyieldsthatA=k=1A2k∈VMOA(2)ifandonlyifA∈H2(),thatis,ifandonlyif2∞sup|ak2<∞,2j|kj=1whereA=(ak).2jj,k≥1

197October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Schurmultipliersonanalyticmatrixspaces183Proof.Since,clearly,H2()⊂H1()theassertionofCorollary8.1222followsfromTheorem8.11andtheabovediscussion.WementionalsotheextensionofCorollary2in[60],whichfollowsbyTheorems7.16andtheproof(seerelation(8.1))ofTheorem8.6:Corollary8.13.ThespaceofallSchurmultipliers(H1(),B(D,))co-202incideswithB(D,2).NotesVariousresultsaboutSchurmultipliersondifferentanalyticmatrixspacesaregiveninthischapter.ForinstanceinTheorem8.1itisshowedthat∞A∈(B(),B(D,))ifandonlyifsup(1−r)||kArk−1||<∞.22kM(2)0≤r<1k=1AnotherimportantresultisTheorem8.2,whichgivesacharacterizationofthespace(B(D,2),B(D,2)).ButthemostimportantresultofthissectionisananswertotheproblemtofindamatrixversionofthebeautifultheoremofMatjelevicandPavlovic[60]aboutthespaceofallFouriermultipliersfromH1intoBMOA.ThisisTheorem8.6mainlyduetoV.LieanditsproofusesresultsfromChapters4and5.Thestatementis:B(D,)=(H1(),BMOA()).222AsoneofitsconsequenceswementiononlyamatrixversionofPaley’stheorem:∞1∞Theorem8.11IfA=k=1A2k∈H(2),thenP(A)=k=1A2k∈H2().2

198May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

199October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Bibliography[1]J.Arazy,Someremarksoninterpolationtheoremsandtheboundnessofthetriangularprojectioninunitarymatrixspaces,Int.Eq.Oper.Th.1,(1978),453-495.[2]J.Arazy,Onthegeometryoftheunitballofunitarymatrixspaces,Int.Eq.Oper.Th.,4(1981),151-171.[3]J.M.Anderson,J.ClunieandCh.Pommerenke,OnBlochfunctionsandnormalfunctions,J.ReineAngew.Math.,270(1974),12-37.[4]J.M.AndersonandA.Shields,CoefficientmultipliersofBlochfunctions,Trans.Amer.Math.Soc.,224(1976),255-265.[5]J.Arazy,S.D.FisherandJ.Peetre,M¨obiusinvariantfunctionspaces,J.ReineAngew.Math.363(1985),110-145.[6]A.B.Alexandrov,EssaysonnonlocallyconvexHardyclasses,LectureNotesinMath.864(1981),1-89,Springer-Verlag,Berlin-Heidelberg-NewYork.[7]A.B.Alexandrov,Ontheboundarydecayinthemeanofharmonicfunctions,St.PetersburgMath.J.5(1996),507-542.[8]A.B.AlexandrovandV.V.Peller,HankelandToeplitz-Schurmultipliers,Math.Ann.,324(2002),277-327.∗[9]W.Arveson,SubalgebrasofC−algebras,ActaMath.123(1969),144-224.[10]W.Arveson,Interpolationproblemsinnestalgebras,J.Funct.Analysis20(1975),208-233.[11]G.Bennett,Schurmultipliers,DukeMath.J.,44(1977),603-639.[12]O.Blasco,AcharacterizationofHilbertspacesintermsofmultipliersbetweenspacesofvector-valuedanalyticfunctions,MichiganMath.J.,42(1995),537-543.[13]O.Blasco,Vector-valuedanalyticfunctionsofboundedmeanoscillationandgeometryofBanachspaces,IllinoisJ.Math.,41(1997),532-558.[14]O.Blasco,IntroductiontovectorvaluedBergmanspaces,Functionspacesandoperatortheory,9–30,Univ.JoensuuDept.Math.Rep.Ser.,8,Univ.Joensuu,Joensuu,2005.[15]O.BlascoandA.Pelczynski,TheoremsofHardyandPaleyforvectorvaluedanalyticfunctionsandrelatedclassesofBanachspaces,Trans.Amer.Math.Soc.323(1991),335-367.185

200October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013186MatrixspacesandSchurmultipliers:Matricealharmonicanalysis[16]M.Bozejko,Littlewoodfunctions,Hankelmultipliersandpowerboundedop-eratorsonaHilbertspace,ColloquiumMath.51(1987),35-42.[17]S.Barza,L.E.PerssonandN.Popa,AmatricealanalogueofFejer’stheory,Math.Nachr.,260(2003),14-20.[18]S.Barza,V.D.LieandN.Popa,ApproximationofinfinitematricesbymatricealHaarpolynomials,Ark.Mat.43(2005),no.2,251–269.[19]L.BerghandJ.Lofstrom,Interpolationspaces.AnIntroduction,SpringerVerlag,Berlin,1976.[20]S.Barza,D.KravvaritisandN.Popa,MatricealLebesguespacesandHoelderinequality,J.Funct.SpacesAppl.3(2005),no.3,239–249.p[21]R.R.Coifman,ArealvariablecharacterizationofH,StudiaMath.,51(1974),269-274.[22]R.G.Cooke,Infinitematricesandsequencespaces,Macmillan1950.p[23]P.L.Duren,TheoryofHSpaces,AcademicPress,1970.[24]R.G.Douglas,Banachalgebratechniquesinoperatortheory,AcademicPress,NewYork,1972.[25]M.D´echamps-Godim,F.Lust-PiquardandH.Queffelec,OntheminorantpropertiesinCp(H),PacificJ.Math.119(1985),89-101.[26]P.DoddsandF.Sukochev,RUC-decompositionsinsymmetricoperatorspaces,Integr.Equat.Oper.Th.,29(1997),269-287.p[27]P.L.Duren,B.W.RombergandA.L.Shields,LinearfunctionalsonHspaceswith0

201October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Bibliography187eralizationsofatheoremofPaley,Quart.J.Math.,OxfordSer.,8(1937),161-171.[40]G.H.HardyandJ.E.Littlewood,Anewproofofatheoremonrearrange-ments,J.LondonMath.Soc.23(1948),163-168.[41]K.Hoffman,BanachSpacesofAnalyticFunctions,PrenticeHall,EnglewoodCliffs,1962.[42]J.R.Holub,OnthemetricgeometryofIdealsofOperatorsonHilbertSpace,Math.Ann.201(1973),157-163.[43]C.Horowitz,ZerosoffunctionsintheBergmanspaces,DukeMath.J.41(1974),693-710.[44]C.Horowitz,FactorizationtheoremsforfunctionsintheBergmanspaces,DukeMath.J.44(1977),201-213.[45]U.HaagerupandG.Pisier,Factorizationofanalyticfunctionswithvaluesinnon-commutativeL1−spacesandapplications,Can.J.Math.,XLI,(1989),882-906.[46]D.RJocicandD.Krtinic,Schur-Laurentmultipliersforblockmatricesandgeometriccharacterizationofcontinuousmatrices,LinearandMultilinearAlgebra,vol.58,(2010),523-534.[47]M.JevticandM.Pavlovic,Coefficientmultipliersonspacesofanalyticfunc-tions,ActaSci.Math.(Szeged),64(1998),531-545.[48]D.Krtinic,AmatricialanalogueofFejer’stheoryfordifferenttypesofcon-vergence,Math.Nachr.,280(2007),1-6.[49]S.KwapienandA.Pelczynski,Themaintriangleprojectioninmatrixspacesanditsapplications,StudiaMath.,34,(1970),43-68.[50]V.Lie,Integraloperatorsininfinitematrixtheory;thestudyofma-tricealRieszprojection,(Romanian),Master’sdissertationatUniversityofBucharest,49p,2003.[51]F.Lust-Piquard,In´egalit´esdeKhintchinedansCp(1

202October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013188MatrixspacesandSchurmultipliers:Matricealharmonicanalysis◦matrices,Doctoralthesis,LuleaUniversityofTechnology,Sweden,2010.2[59]M.Marsalli,NoncommutativeHspaces,Proc.Amer.Math.Soc.,125(1997),779-784.p[60]M.MateljevicandM.Pavlovic,MultipliersofHandBMOA,PacificJ.Math.146(1990),71-84.p[61]M.MateljevicandM.Pavlovic,L−behaviouroftheintegralmeansofanalyticfunctions,StudiaMath.77,(1984),219-237.[62]L.G.Marcoci,L.E.Persson,I.PopaandN.Popa,AnewcharacterizationofBergmanSchattenspacesandadualityresult,J.Math.Anal.Appl,360(2009),67-80.[63]N.Nikolskii,Treatiseontheshiftoperator,SpringerVerlag,Berlin,1986.p[64]D.M.Oberlin,Translation-invariantoperatorsonL(G),00,aninves-tigationoftherateofrationalapproximationandotherapplications,Mat.Sbornik,122(1983),481-510.[67]V.Paulsen,Completelyboundedmapsanddilations,PitmanResearchNotesinMath.146,Longman,Wiley,New-York,1986.[68]M.Pavlovi´c,Introductiontofunctionspacesonthedisk.MatematickiInsti-tutSANU,Beograd,2004.[69]L.B.Page,BoundedandcompactvectorialHankeloperators,Trans.Amer.Math.Soc.,150(1970),529-539.[70]S.Parrott,OnaquotientnormandtheSz.-Nagy-Foia¸sliftingtheorem,J.Funct.Anal.,30(1978),311-328.[71]J.Peetre,NewthoughtsonBesovspaces,DukeUniv.Press.,Durham,N.C.,1976.[72]N.Popa,MatricealBlochandBergman-Schattenspaces,Rev.RoumaineMath.PuresAppl.,52(2007),459-478.[73]N.Popa,Acharacterizationofuppertriangulartraceclassmatrices,C.R.Acad.Sci.Paris,Ser.I347(2009),59-62.[74]N.Popa,SchurmultipliersbetweenBanachspacesofmatrices,ProceedingsoftheSixthCongressofRomanianMathematiciansBucharest,2007,vol.1,373-380[75]I.PopaandN.PopaMatricesofboundedvariation,inProceedingoftheIn-ternationalConference:MathematicalAnalysisanditsApplications,Athens2002,pp.318-323.[76]I.PopaandN.PopaInequalitiesinmatrixspaces,Math.Rep.,12(62)(2010),169-180.[77]S.Power,CommutatorswiththetriangularprojectionandHankelformsonnestalgebras,J.LondonMath.Soc.32(1985),272-282.[78]I.I.Privalov,RandeigenschaftenanalytischerFunktionen,VerlagderWiss.,Berlin,1956.[79]A.PelczynskiandF.Sukochev,SomeremarksonToeplitzmultipliersand

203October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Bibliography189Hankelmatrices,StudiaMath.,175,(2006),175-204.1[80]G.Pisier,MultipliersoftheHardyspaceHandpowerboundedoperators,Preprint,2000.[81]G.Pisier,Similarityproblemsandcompletelyboundedmaps,LNM1618,SpringerVerlag,Berlin,1995.[82]A.Pietsch,OperatorIdeals,VEBDeutscherVerlagderWissenschaften,Berlin,1978.[83]A.PietschandH.Triebel,Interpolationstheorief¨urBanachidealevonbeschr¨anktenlinearenOperatoren,StudiaMath.,31,(1968),95-109.[84]A.L.Shields,AnanalogueofaHardy-Littlewood-Fejerinequalityforuppertriangulartraceclassoperators,Math.Z.,182(1983),473-484.∞[85]D.Sarason,GeneralizedinterpolationinH,Trans.Amer.Math.Soc.,127(1967),179-203.[86]I.Schur,BemerkungenzurTheoriederbeschra¨nktenBilinearformenmitunendlichvielenVera¨nderlichen,J.ReineAngew.Math.,140(1911),1-28.[87]B.Simon,Traceidealsandtheirapplications,LondonMath.SocLectureNotesSeries,no.35,Cambridge,1979.1[88]W.T.SleddandD.A.Stegenga,AnHmultipliertheorem,Ark.Mat.,19(1981),no.2,265-270.1[89]B.Smith,AstrongconvergencetheoremforH(T),Banachspaces,harmonicanalysisandprobabilitytheory,(Storrs,Conn.,1980/1981),LectureNotesinMath.,vol.995,Springer-Verlag,Berlin,1983,169-173.[90]E.M.SteinandR.Shakarchi,Fourieranalysis;AnIntroduction,PrincetonUniversityPress,Princeton,2003.∗[91]W.Stinespring,PositivefunctionsonC-algebras,Proc.Amer.Math.Soc.,6(1966),211-216.[92]A.L.ShieldsandD.L.Williams,Boundedprojections,dualityandmulti-pliersinspacesofanalyticfunctions,Trans.Amer.Math.Soc.,162(1971),287-302.[93]H.Triebel,Interpolationtheory,functionspaces,differentialoperators.Sec-ondedition.JohannAmbrosiusBarth,Heidelberg,1995.[94]K.Zhu,OperatortheoryinBanachfunctionspaces,MarcelDekker,New-York,1990.[95]K.Zhu,AnalyticBesovspace,J.Math.Anal.Appl.,157,(1991),318-336.[96]A.Zygmund,Trigonometricseries,CambridgeUniversityPress,Cambridge,1959.[97]N.Wiener,ThequadraticvariationofafunctionanditsFouriercoefficients,Massachusett’sJ.Math.,3(1924),72-94.[98]G.Wittstock,EinOperatorenwertigenHahn-BanachSatz,J.Funct.Anal.,40(1981),127-150.

204May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank

205October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013Index1H(H−1)-multiplier,80ΦmatrixversionofHankel(X,Y)thespaceofallSchuroperator,1131∞multipliersfromXintoY,1L(D,2),L(D,2),5p,unc(·,·)thescalarproductinaHilbertLa(D,2),132space,3M(2)thespaceofallSchur(p,q)-bounded,80multipliersonB(2),2A∗BtheSchurproductoftheM(T)theconvolutionalgebraofmatricesAandB,1BorelmeasuresonT,101Akthekth-diagonalmatrix,1P,PBergmanprojection,128B(2)thespaceofallboundedlinearPTthetriangularprojection,69operatorson2,2Pα,12922BMOAthespaceofanalyticTp(R),Tp(C)spacesofupperfunctionsofboundedmeantriangularmatrices,70oscillation,4Tp,0

206October20,20138:49WorldScientificBook-9inx6invers*11*oct*2013192MatrixspacesandSchurmultipliers:Matricealharmonicanalysisp,2M,119T,2ρ(z,w)thepseudo-hyperbolicdistance,7σn(A),137∞iktfA(r,t)=Ak(r)e,2k=−∞mXtheoperatorinducedbythemultiplierm,80∗w-measurablefunction,5VMOA(2),109Bergman-Schattenclasses,122Analyticmatrices,21Banachspaceof(H−1)-Fouriertype,81Bennett’sTheorem,2Blochmatrix,150crudelyfinitelyrepresentableBanachspace,86MatricealHausdorff-YoungTheorem,75MatrixversionofNeharitheorem,xi,114Pavlovi´cTheorem,97SchmidtTheorem,3Shieldsinequality,70singularvaluesofanoperatorT,3stronglymeasurablefunction,5theHankelmatrixHa,90pThematrixspacesLr(2),resp.pLc(2),1≤p≤2,66Thenoncommutativefactorizationtheorem,87theprojectivetensorproduct,88Toeplitzmatrices,2VectorialNeharitheorem,113