The relations between volume ratios and new concep

TheRelationsbetweenVolumeRatiosandNewConceptsofGLConstantsY.GordonM.JungeN.J.NielsenyAbstractInthispaperweinvestigateapropertynamedGL(p;q)whichiscloselyrelatedtotheGordon-Lewisproperty.OurresultsonGL(p;q)arethenusedtoestimatevolumeratiosrelativeto‘p,1p1,ofunconditionaldirectsumsofBanachspaces.IntroductionInthispaperweinvestigateapropertynamedGL(p;q),1p1,1q1,closelyrelatedtotheGordon-LewispropertyGL,andthebehaviorofp-summingnormsofoperatorsdefinedondirectsumsofBanachspacesinthesenseofanunconditionalbasis.Theseresultsarethenusedtoestimatethevolumeratiosvr(X;‘p),1p1,whereXisafinitedirectsumoffinitedimensionalspaces.ABanachspaceissaidtohaveGL(p;q),1p1,1q1,ifthereisaconstantCsothatiq(T)Cp(T)foreveryfiniterankoperatorTfromanarbitraryBanachspacetoX.Herepdenotesthep-summingnormandiqtheq-integralnorm.ThispropertywasalsoconsideredbyReisner[27],notehowevertheslightdifferenceinthenotation:OurGL(p;q)correspondstohisq0;p0-GL-space.Wenowwishtodiscussthearrangementandcontentsofthispaperingreaterdetail.InSection1ofthepaperweinvestigatethebasicpropertiesofGL(p;q)andprovesomeinequalitiesforp-summingoperators,respectivelyq-integraloperators,definedon,respectivelySupportedinpartbythefundforthepromotionofresearchintheTechnionandbytheVPRfund.ySupportedinpartbytheDanishNaturalScienceResearchCouncil,grants9503296and9600673.1withrangein,adirectsumofBanachspacesinthesenseofanunconditionalbasis.Theseinequalitiesarethenusedtoprovethatif(Xn)isasequenceofBanachspaceswithuniformlyboundedGL(p;q)-constantsandXisthedirectsumoftheXn’sinthesenseofap-convexandq-concaveunconditionalbasis,thenXhasGL(p;q)aswell.MoregenerallyweobtainthatifYisaBanachspacewithGL(p;q)andLisap-convexandq-concaveBanachlattice,thenL(Y)hasGL(p;q).Kp(L)andKq(L)denotethep-convexityandq-concavityconstantsofLrespectively.InSection2wecombinetheresultsofSection1withthoseof[6]toobtainsomeestimatesofvolumeratios.Oneofourresults,Theorem2.5,hasthefollowinggeometricconsequence:LetLbeap-convexandq-concaveBanachlatticehavingann-dimensionalBanachspaceY=(Rn;kk)asanisometricquotient.Let1p;q1,1p+1p0=1,thentherearen-dimensionallinearquotientsVpandVqofB‘pandB‘qrespectively,sothatVqBYVpforwhichjVpjjVqj1ncqp0glp;q(L)cqp0Kp(L)Kq(L):IfXisafinitedirectsumofnk-dimensionalBanachspacesXk,1kminthesenseofafinite1-unconditionalbasis,thenweprovethatmYk=1vr(Xk;‘p)nk)1=nvr(X;‘p)for1p1,wheren=Pmk=1nk.0NotationandPreliminariesInthispaperweshallusethenotationandterminologycommonlyusedinBanachspacetheoryasitappearsin[17],[18][26]and[31].IfXandYareBanachspaces,B(X;Y)(B(X)=B(X;X))denotesthespaceofboundedlinearoperatorsfromXtoYandthroughoutthepaperweshallidentifythetensorproductXYwiththespaceof!-continuousfiniterankoperatorsfromXtoYinthecanonicalmanner.Furtherif1p1weletp(X;Y)denotethespaceofp-summingoperatorsfromXtoYequippedwiththep-summingnormp,Ip(X;Y)denotesthespaceofallp-integraloperatorsfromXtoYequippedwiththep-integralnormipandNp(X;Y)denotesthespaceofallp-nuclearoperatorsfromXtoYequippedwiththep-nuclearnormp.Werecallthatif1p1thenanoperatorTissaidtofactorthroughLpifitadmitsafactorizationT=BA,2whereA2B(X;Lp())andB2B(Lp();Y)forsomemeasureandwedenotethespaceofalloperatorsfromXtoY,whichfactorthroughLpbyp(X;Y).IfT2p(X;Y)wedefinep(T)=inffkAkkBkjT=BA;AandBasaboveg;pisanormonp(X;Y)turningitintoaBanachspace.Allthesespacesareoperatoridealsandwerefertotheabovementionedbooksand[13],[24]and[14]forfurtherdetails.Toavoidmisunderstandingwestressthatinthispaperap-integraloperatorTfromXtoYhasap-integralfactorizationendinginYwithip(T)definedaccordingly;insomebooksthisisreferredtoasastrictlyp-integraloperator.Intheformulasbelowweshall,asiscustomary,interpret1astheoperatornormandi1asthe1-norm.Ifn2NandT2B(‘n2;X)thenfollowing[31]wedefinethe‘-normofTby‘(T)=(ZRnkTxk2d(x))12whereisthecanonicalGaussianprobabilitymeasureon‘n2.ABanachspaceXissaidtohavetheGordon-Lewisproperty(abbreviatedGL)[7]ifevery1-summingoperatorfromXtoanarbitraryBanachspaceYfactorsthroughL1.ItiseasilyverifiedthatXhasGLifandonlyifthereisaconstantKsothat1(T)K1(T)foreveryBanachspaceYandeveryT2XY.InthatcaseGL(X)denotesthesmallestconstantKwiththisproperty.WeshallsaythatXhasGL2ifithastheabovepropertywithY=‘2andwedefinetheconstantgl(X)correspondingly.AneasytracedualityargumentyieldsthatGLandGL2areselfdualpropertiesandthatGL(X)=GL(X),gl(X)=gl(X)whenapplicable.Itisknown[7]thateveryBanachspacewithlocalunconditionalstructurehasGL.IfEisaBanachspacewitha1-unconditionalbasis(en)and(Xn)isasequenceofBanachspacesthenweput(1Xn=1Xn)E=fx21Yn=1Xnj1Xn=1kx(n)kenconvergesinEg3andifx2(P1n=1Xn)Ewedefinekxk=k1Xn=1kx(n)kenkthusdefininganormon(P1n=1Xn)EturningitintoaBanachspace.IfXn=Xforalln2NweputE(X)=(P1n=1Xn)E.If(Yn)isanothersequenceofBanachspacesandTn2B(Xn;Yn)foralln2NwithsupnkTnk1thenwedefinetheoperator1n=1Tn:(PnXn)E!(P1n=1Yn)Eby(1n=1Tn)(x)=(Tnx(n)).Clearlyk1n=1TnksupnkTnk.Weshallneeda“continuous”versionoftheabovedirectsumssohenceletXbeaBanachspaceandLaBanachlattice.IfPnj=1xjyj2XLthenitfollowsfrom[18,SectionId)]thatsupkxk1jPnj=1x(xj)