; TeX output 2001.02.15:1608 hGS 9K`y UU cmr10AnnalesAcademiScienĽtiarumFN8ennic MathematicaVN8olumen26,2001,233{248$&CD7"V cmbx10WEAKLٚYCOMPACTCOMPOSITIONō^<;OPERAٚTORSONANALYTIC;|VECTOR-VwxALUEDFUNCTIONSPٚACES9,"V cmbx10Jos«}WeTBonet,P9awe :lTDomaQ;nski,andMik\raelLindstr@om~UnivĽersidadPolitecnicadeVN8alencia,E.T.S.Arquitectura,Dpto.deMatemUaticaAplicada dES-46071VN8alencia,Spain;jb;Bonet@pleiades.upĽv.es :FA.MicĽkiewiczUniversityN8,FacultĽyofMathematicsandComputerScience@ul.%Matejki48/49,PL-60-769PĽozna;Cn,Poland;domanski@amu.edu.pl\ BAb;BoAk{ademiUnivĽersityN8,DepartmentofMathematicsvFI-20500 BAb;Bo,Finland;mlindstr@abo.bD"V UU cmbx10DAbstract.hLetX<b> UU cmmi10X b;BeXaBanacĽhspace.ItisproĽvedXthatthecomp;BositionoperatoronXXg- v{aluedHardyspaces,wĽeightedBergmanspacesandBlo;BcĽhspacesisweaklycompactorRosenthalifCandonlyifb;BothCid 5:X@!", UU cmsy10@!PXUandthecorrespondingcompositionoperatoronscalarv{aluedspacesarewĽeaklycompactorRosenthal,resp;BectivelyN8.w 1.pIn9troQductionYO K`y cmr10Let}b> cmmi10':UPD5!", cmsy10!DMbGe}ananalyticselfmapofthecomplexunitdiscDG.WItcanbGe easilyprovedthatifthecompGositionoperatorC 0er cmmi7':UPfڧ7!f''onvector-vqalued(i.e.withV_vqaluesinaBanachspaceV_X)Hardy*,^BergmanorBloGchspacesbGelongstosomeopGeratorideal,thenbothitsscalarversionandtheidentityopGeratoronX belongtorthesameideal.F*ortheidealofweaklycompactopGeratorsLiu,SaksmanandTylliA[[LST]AVprovedtheconverseforvector-vqaluedHardyspacesA[HٓR cmr71|s(X ) ,EZBergmanspacesB1|s(X )andBO! cmsy71x(X)g=H ^1 H(X)aswellasforBloGchspacesusinganalyticmethoGds.ōIfavector-vqaluedspaceofanalyticfunctionsE [X ]canbGerepresentedasthespaceQL(^E ;X )ofQalllinearbGoundedoperatorsfromthepredualofthescalarversion@of@E [X ]intoX,Dthen@wegiveaverysimplefunctionalanalyticargumentwhich]Qreplacesthemoreanalyticonesin[LST].Inthiswayweobtaintheresultsfor?BloGchspacesandextendtheresultsof[LST]!toweightedBergmanspacesofinniteorderB^ qv፺1x(X ) ."InthatpartofthepapGerourmainideaistousethefollowingՎresultduetoSaksmanandTylliin[ST],seealso[R],[LS].$ ': cmti10L}'et ޱE,'[Fc,]y ff xsUU1991MathematicsSubvjectClassication:%Primary47B38,47B10,46E40,46E15.Then8researcĽhoftheauthorswaspartiallysupp;BortedbyDGESICn"provjectno.PB97-0333, theCommitteeofScienĽticResearch(KBN),Poland,#grant2P03A05115andAcademyofFinland,resp;BectivĽelyN8. * 234MZ!p0J cmsl10J.UUBonet,P*.DomaGnski,andM.LindstromhGE1|s,&F1b}'eYBanachspacesandletYR߷22}>y rsfs10L@(E ;Fc)andBG2L@(E1|s;F1)Yb}'eYtwoweakly c}'ompactUtoperators.>IfUtBorRi;isc}'ompact,thenthemapUtTI7!%R!T(Bfr}'omL@(F1|s;E )intoL(E1|s;Fc)iswe}'aklycompact.UnfortunatelytheopGeratorrepresentationmentionedabGovedoGesnotholdingeneral,c@forinstance,forHardyspacesH1|s(X )orBergmanspacesB1(X ) .ThusthemainpartofthepapGerisdevotedtothatcase.W*eareabletoextendthe;methoGdsandtheresultsof[LST]:totheclassicalweighted;BergmanspacesB^ ql13S(X ) ,YA 1,YaclasswhichincludesbGoth䌱H1|s(X)andB1|s(X) .kAnessentialimprovement!isdoneinaformuladerivedfromtheso-calledStantonformula(seeLemmaUU3).Letusobservethatfor1x