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1、SUBMODULESOFTHEHARDYMODULEOVERPOLYDISCJAYDEBSARKARDedicatedtoRonaldG.Douglasontheoccasionofhis75thbirthdayAbstract.WesaythatasubmoduleSofH2(Dn)(n>1)isco-doublycommutingifthequotientmoduleH2(Dn)/Sisdoublycommuting.Weshowthataco-doublycommutingsubmoduleo
2、fH2(Dn)isessentiallydoublycommutingifandonlyifthecorrespondingonevariableinnerfunctionsarefiniteBlaschkeproductsorthatn=2.Inparticular,aco-doublycommutingsubmoduleSofH2(Dn)isessentiallydoublycommutingifandonlyifn=2orthatSisoffiniteco-dimension.Weobtainan
3、explicitrepresentationoftheBeurling-Lax-HalmosinnerfunctionsforthosesubmodulesofH2(D)whichareco-doublycommutingH2(Dn−1)submodulesofH2(Dn).Finally,weprovethatapairofco-doublycommutingsubmodulesofH2(Dn)areunitarilyequivalentifandonlyiftheyareequal.1.Intr
4、oductionLet{T1,...,Tn}beasetofncommutingboundedlinearoperatorsonaseparableHilbertspaceH.Thenwecanturnthen-tuple(T1,...,Tn)onHintoaHilbertmodule[16]HoverC[z]:=C[z1,...,zn],theringofpolynomials,asfollows:C[z]×H→H,(p,h)7→p(T1,...,Tn)h,forallp∈C[z]andh∈H.T
5、hemodulemultiplicationoperatorsbythecoordinatefunctionsonHaredefinedbyMzih=zi(T1,...,Tn)h=Tih,forallh∈Handi=1,...,n.Therefore,aHilbertmoduleisuniquelydeterminedbytheunderlyingcommutingoperatorsviathemodulemultiplicationoperatorsbythecoordinatefunctionsa
6、ndviceversa.LetS,Q⊆HbeclosedsubspacesofH.ThenS(Q)issaidtobeasubmodule(quotientmodule)ofHifMS⊆S(M∗Q⊆Q)foralli=1,...,n.NotethataclosedsubspaceQisaquotientzizimoduleofHifandonlyifQ⊥∼arXiv:1304.1564v2[math.FA]18Oct2013=H/QissubmoduleofH.TheHardymoduleH2(Dn
7、)overthepolydiscistheHardyspaceH2(Dn)(cf.[17]and[27]),theclosureofC[z]inL2(Tn),withthestandardmultiplicationoperatorsbythecoordinatefunctionsz(1≤i≤n)onH2(Dn)asthemodulemaps.iThemodulemultiplicationoperatorsonasubmoduleSandaquotientmoduleQofaHilbertmodu
8、leHaregivenbytherestrictions(Rz1,...,Rzn)andthecompressions(Cz1,...,Czn)ofthemodulemultiplicationsofH,respectively.Thatis,Rzi=Mzi
9、SandCzi=PQMzi
10、Q,2010MathematicsSubjectClassification.47A13,47A15,47A20,47A45,47A80,46E20,30H10.Keywordsandp
