简介：LetB(E,F)bethesetofallboundedlinearoperatorsfromaBanachspaceEintoanotherBanachspaceF,B+(E,F)thesetofalldoublesplittingoperatorsinB(E,F)andGI(A)thesetofgeneralizedinversesofA2B+(E,F).InthispaperweintroduceanunboundeddomainW(A,A^+)inB(E,F)forA2B+(E,F)andA^+2GI(A),andprovideanecessaryandsufficientconditionforT2W(A,A^+).Thenseveralconditionsequivalenttothefollowingpropertyareproved:B=A^+(IF+(T??A)A^+)??1isthegeneralizedinverseofTwithR(B)=R(A^+)andN(B)=N(A^+),forT2W(A,A^+),whereIFistheidentityonF.Alsoweobtainthesmooth(C￥)diffeomorphismMA(A^+,T)fromW(A,A^+)ontoitselfwiththefixedpointA.LetS=fT2W(A,A^+):R(T)\N(A^+)=f0gg,M(X)=fT2B(E,F):TN(X)R(X)gforX2B(E,F)g,andF=fM(X):8X2B(E,F)g.UsingthediffeomorphismMA(A^+,T)weprovethefollowingtheorem:SisasmoothsubmanifoldinB(E,F)andtangenttoM(X)atanyX2S.ThetheoremexpandsthesmoothintegrabilityofFatAfromalocalneighborhooldatAtotheglobalunboundeddomainW(A,A^+).Itseemstobeusefulfordevelopingglobalanalysisandgeomatricalmethodindifferentialequations.