; TeX output 2001.06.15:1537 s( L:"V G cmbx10ComplemenrtedSubspacesofSpaces4QObtainedbryInterp Oolation.+ 1b"V cmbx10D.J.H.TGarlingah't}\ cmti7St.)John-'sColi?lege,CambridgeCB21TP,England.G` VS.J.TMon9tgomery-SmithdDepartment)ofMathematics,UniversityofMissouri, mColumbia,)MO65211,U.S.A.蛍-+4ٓR cmr7AZ cmr5BSTRAӍCT:uIf' 0er cmmi7ZzisaquotientofasubspaceofaseparableBanachspaceX J,andV-+4isranyseparableBanachspace,9thenthereisaBanachcouple(A 0 ;]A 1)rsuchthatA 0-+4and@A 1'areisometrictoX=O! cmsy76V,d~andanyintermediatespaceobtainedusingthereal-+4orcomplexinterp7olationmethodcontainsacomplementedsubspaceisomorphictoZ}.-+4Thusmanyprop7ertiesofBanachspaces,$includinghavingnon-trivialcotyp7e,$havingthe-+4Radon{Niko7dymRpropertyZ,eandRhavingtheanalyticunconditionalmartingaledierence-+4sequenceprop7ertyZ,donotpasstointermediatespaces.p K`y cmr10ThereGaremanyBanachspacepropGertiesthatpasstospacesobtainedbythecomplex%methoGdofinterpolation.HIF*orexample,itisknownthatifacouple(b> cmmi10A0|s;A1)issuchthatA0andH:A1ĭbGothhaveH:theUMDH7(unconditionalmartingaledierencesequence)property*,JandifAL'isthespaceobtainedusingthecomplexinterpGolationmethodwithparameter,thenA-ܫhastheUMDypropGertywhenever0<5<1.7AnotherexampleistypGeofBanachspaces:ifԽA0Q0hastypGep0andA1hastypGep1|s,vthenAZhastypGep7,vwhere1=pMO=(17!", cmsy10 )=p0&+=p1|s.v Similar+resultsaretruefortherealmethoGdofinterpolation.cIfwedenotebyA7;p0thespaceobtainedusingtherealinterpGolationmethodfromacouple(A0|s;A1)withparametersʵandp,(thenA7;p!hastheUMDpropGertywheneverA0 andA1havetheUMDpropGerty*,0<5<1,>Xand81
XthenA7;p>"hastypGeUUp7,where1=pMO=(18 )=p0S+=p1ȫandUUp=pی(see[5]2.g.22). However,yStherer areotherpropGertiesforwhichithasbeenhithertounknownwhethertheySpasstotheintermediatespaces.mExamplesincludetheRadon{NikoGdymproperty*,theAUMD(analyticunconditionalmartingaledierencesequence)propGerty*,1andhavingnon-trivialUUcotypGe. ff xs Thesecondnamedauthorwassupp7ortedinpartbyN.S.F.GrantDMS9001796. A.M.S.)(1980 )subjectclassication B:ϧ46B99. * w[SPZACESOBTAINEDBYINTERPOLATION s ThisVpapGerdealswiththeseproperties,Vshowingthattheydonotpasstotheinter-%mediatespaces.{Indeed,(weshowthesurprisingfactthatthereisacouple(A0|s;A1)suchthatA0 iandA1arebGothisometrictol1|s,butalltherealorcomplexintermediatespacescontainracomplementedsubspaceisomorphictoc0|s.eThisimprovesaresultduetoPisier,who3gaveanexampleofacouple(A0|s;A1)forwhichA0isisometrictoL1,9A1isisometricto"adensesubspaceofc0|s,UMandc0isnitelyrepresentedineveryintermediatespaceAobtainedUUbythecomplexinterpGolationmethod(see[3]).NotationHereweoutlinethenotationwewilluseabGoutinterpGolationcouples./ThereaderisreferredtoUU[1]or[2]fordetails. A$ ':
cmti10BanachUc}'oupleis;apairofBanachspaces(A0|s;A1);suchthatA0AandA1bGothembedintoXacommontopGologicalvectorspace,
,whichXweshallcalltheambientŘsp}'ace.Givensuch1acouple,8]wedeneBanachspacesA0l+uA1(withnormk1 xkU=infƷfkx0|sk9A 0"9^+kvx1|sk$A 1#UΫ:x0W
2ښA0|s,cx12A1|s,cx0!+@x1=xg)a
andA0!\@A1}(withnormkaxk«=maxwfk xkA 0v;kxkaƟA 1g).Amap0bGetweentwocouplesT*:(A0|s;A1)!(B0;B1)0isalinearmapT*:A0l;+ȱA1C!B0+ȱB1suchUUthatTc(A0|s)B0ȫandUUT(A1|s)B1,UUandsuchthatkUVTckA 0 !B 00j,kUVTckA 1 !B 13<1. Ani)interp}'olationmethod,I ,isafunctorthattakesaBanachcouple(A0|s;A1)toasingleUUBanachspaceAI,suchthatA0S\8A1CAI]A0+8A1ȫwithq