On The Interpolation of Injective or Projective Te

arXiv:math/0401337v1[math.FA]24Jan2004OnTheInterpolationofInjectiveorProjectiveTensorProductsofBanachSpacesOmranKoubaDepartmentofMathematicsHigherInstituteforAppliedSciencesandTechnologyP.O.Box31983,Damascus,Syria.E-mail:[e-mail]:Weproveageneralresultonthefactorizationofmatrix-valuedanalyticfunctions.Wededucethatif(E0,E1)and(F0,F1)areinterpolationpairswithdenseintersections,thenundersomeconditionsonthespacesE0,E1,F0andF1,wehave[E0b⊗F0,E1b⊗F1]θ=[E0,E1]θb⊗[F0,F1]θ,0θ1.WefindalsoconditionsonthespacesE0,E1,F0andF1,sothatthefollowingholds[E0∨⊗F0,E1∨⊗F1]θ=[E0,E1]θ∨⊗[F0,F1]θ,0θ1.Someapplicationsoftheseresultsarealsoconsidered.1.Introduction,notationandbackgroundAllBanachspacesconsideredinthispaperarecomplex.Byann-dimensionalBanachspace,wemeanCnequippedwithanorm.IfXandYareBanachspaces,thenL(X,Y),X∨⊗YandXb⊗Ydenote,respectively,theBanachspaceofboundedoperatorsfromXintoY,TheclosureofX⊗YinL(X∗,Y)equippedwiththeinducednorm,andthecompletionofX⊗Ywithrespecttotheprojectivetensornormdefinedby:∀u∈X⊗Y,||u||∧=inf(mXk=1||xk||X||yk||Y:u=mXk=1xk⊗yk)X∨⊗YandXb⊗YarecalledrespectivelytheinjectiveandprojectivetensorproductofXandY.InthecasewhenXandYarebothfinite-dimensional,wehave(Xb⊗Y)∗=L(X,Y∗)=X∗∨⊗Y∗.Usingtheprecedingduality,weseethattheresultsannouncedintheabstractaresimilar,inthefinite-dimensionalcontext.1LetE0,E1,F0andF1befinite-dimensionalBanachspaces.Theusualinterpolationtheoremassertsthat||.||L(Eθ,F∗θ)≤||.||[L(E0,F∗0),L(E1,F∗1)]θ,whereXθdenotes[X0,X1]θfor0θ1.Thequestionweareinterestedin,isthefollowing:Underwhatconditionsonthespacescanonefindaconstantc,independentofthedimensionoftheconsideredspaces,sothat||.||[L(E0,F∗0),L(E1,F∗1)]θ≤c||.||L(Eθ,F∗θ).Wewillseethattheconstantccanbemajorizedusingthetype2constantsofthespacesE0,E1,F0andF1(orthe2-convexityconstantsintheBanachlatticecase).Wefirstrecallsomedefinitionsandnotation.En,mdenotesthespaceofcomplexn×mmatrices.A∗andtAare,respectively,theadjointandthetransposedmatrixofA.AmatrixA∈En,mwillbeidentifiedwithalinearoperatorfromCmintoCnusingthecanonicalbases.LetδbeanormonEn,m,thedualnormisdefinedbyδ∗(A)=suptr(tB.A):B∈En,mandδ(B)≤1 .LetD,Dand∂Ddenote,respectivelytheopenunitdiscinC,itsclosureanditsboundary.Forz∈Dandt∈∂DwedenotebyPz(t)thePoissonkernel:Pz(t)=1−|z|2|t−z|2andletdmbetheHaarmeasuronT≡∂D.WeusethenotationA(D,En,m)(resp.H∞(D,En,m))todenotethesetofanalyticfunctionsonDvaluedinEn,mwhicharecontinuousonD,(resp.boundedonD).Wesaythatanoperatoru:X→Yisp-summingforsomep≥1ifthereisaconstantcsuchthatforallfinitesequencesx1,...,xninXwehavenXk=1||u(xk)||pY!1/p≤csupnXk=1|ξ(xk)|p!1/p:ξ∈X∗,||ξ||X∗≤1.Wedenotebyπp(u:X→Y)thesmallestconstantcsatisfyingthisproperty,andbyΠp(X,Y)thespaceofallp-summingoperatorsfromXintoY.Thisspaceequippedwiththep-summingnormπpisaBanachspace.WedenotebyΓ2(X,Y)thesetofalloperatorswhichfactorthroughaHilbertspace.Foranoperatoru∈Γ2(X,Y),wedefinethenormγ2(u:X→Y)=inf{||A:X→H||||B:H→Y||}2wheretheinfimumistakenoverallfactorizationsofuoftheformu=B.AandallHilbertspacesH.ThespaceΓ2(X,Y)epuippedwiththeprecedingnormisaBanachspace.Let1≤p≤+∞.WedenotebyℓnpthespaceCnequippedwiththenorm||x||p=(Pn1|xk|p)1/p,andwiththehabitualchangeforp=+∞.Let{gn}n≥1beasequenceofindependentidenticallydistributedGaussianrealvaluednormalrandomvariablesonsomeprobabilityspace(Ω,A,P).ABanachspaceXiscalledoftype2(resp.Gaussiancotype2),ifthereisaconstantcsuchthatforallx1,...,xninXwehavenXk=1gkxkL2(Ω;X)≤cnXk=1||xk||2!12(resp.≥c−1(P||xk||2)1/2).WedenotebyeT2(X)(resp.eC2(X))thesmallestconstantcforwhichthisholds.Let(e1,...,en)bethecanonicalbasisofCn.Foranyoperatoru:ℓn2→Xwedefineℓ(u:ℓn2→X)=nXk=1gku(ek)L2(Ω;X).FormoredetailsonoperatoridealsandthegeometryofBanachspaces,wereferthereaderto[P],[Pi1]and[LT1].ConcerningBanachlatticeswereferthereaderto[LT2].Weonlyrecallthefollowingdefinition:ABanachlatticeXiscalled2-convex(resp.2-concave),ifthereexistsaconstantMsothatforeverychoiceofvectorsx1,...,xninXnXk=1|xk|2!1/2≤MnXk=1||xk||2!1/2(resp.≥M−1(P||xk||2)1/2).ThesmallestpossiblevalueofMisdenotedM(2)(X)(resp.M(2)(X)).WeassumethereaderfamiliarwiththecomplexinterpolationmethodeofCalder´on,seeforinstance[C]or[BL].Werecallsomedefinitionsandresultsconcerningtheinterpolationoffamiliesoffinite-dimensionalBanachspaces.Afamilyofnorms{δλ}λ∈Λ,whereΛisanytopologicalspace,onEn,missaidtobecontinuousifforeveryA∈En,mthefunctionλ7→δλ(A)iscontinuous.Acontinuousfamilyofnorms{δz}z∈DonEn,missaidtobesubharmonicifforeveryEn,m-valuedanalyticfunctionFdefinedonsomedomainΩ⊂Dthefunctionz7→Logδz(F(z))issubharmoniconΩ.If{δt}t∈∂DisacontinuousfamilyofnormsonEn,m,thereexistsauniquecontinuousfamilyofnorms{δz}z∈DonEn,mwhichcoincideswiththeoriginaloneontheboundary3andsuchthatboth{δz}z∈Dand{δ∗z}z∈Daresubharmonic.Thisfamily(orthefamilyofspaces{(En,m,δz)}z∈D)iscalledtheinterpolationfamilywithboundarydata{δt}t∈∂D.If{αz}z∈DisaninterpolationfamilyofnormsonCm,{βz}z∈DisasubharmonicfamilyofnormsonCnandT∈A(D,En,m)thenthefunctionz7→Log||T(z):(Cm,αz)→(Cn,βz)||issubharmonic.Concerningtheprecedingtw