Banach Spaces Closed Linear Operator Group Inverse Of Disturbance And Representation Theorem

It is well known that the generalized inverses and group inverses in Banach spaces for bounded linear operators is very important in applications in diverse fields such as singular differential, difference equation, Markov chain, many-body dynamics and so on. One core of the theory of generalized inverse is its perturbation and expression theories which studies the problem whether the operator, after a minor perturbation, still has generalized inverse and whether the generalized inverse (in some sense) convergence to the original one or have some expression.The expression T+(I+δTT+)-1may be the simplest possible for generalized inverse of the perturbed operator. Many equivalent conditions for group inverses and generalized inverses to have this expression have been obtained in the case of bounded operators.As we all know, a large number of the operators which arise naturally in applications (e.g. mathematical physics, quantum mechanics and partial differential equations) are unbounded. However, many of them have bounded inverses or bounded generalized inverse. To solve problems involving such unbounded case, we always deal with a class of important ones, i.e., closed linear operators. It is worthy to point out that the differential operators or partial differential operators are always closed linear operators.In this paper, we first use the gap of spaces and the reduced minimum module to prove that the local boundedness imply the continuity of generalized inverse in some sense in Banach space. Thus the equivalence in Moore-Penrose inverse circumstance in [29] is hold.Theorem Let X and Y be Banach spaces. Let T∈B(X,Y) with a generalized inverse T+∈B(Y, X), if there exist M, Δ>0, such that T=T+δT has a generalized inverseT∈B(Y,Z)and‖T+‖≤M for all δT∈B(X,Y)with‖δT‖<â–³.ThenT+â†'T+,as δTâ†'0.Secondly,we exploring the perturbation problem of group inverse in Banach spaces for closed linear operators:Let X and Y be two Banach spaces,T be a closed linear operator from into Y such that its domain is dense in X and T has the bounded group inverse Tg,what condition on the small perturbation δT can guarantee that the group inverse of T+δT has the simplest expression T+(I+δTT+)-1?Theorem Let X be a Banach space.Let T∈C(X)with a group inverse Tg∈B(X).Let δT∈L(X)be T-bounded with T-bound b<1and δTT+satisfy‖δTT+y‖≤λ1‖y‖+λ2‖(I+δTT+)y‖,(?)y∈X, where λ1,λ2∈[0,1),then the following statements are equivalent:(1)B=Tg(I+δTTg)-1=(I+TgδT)-1Tg:Xâ†'X is a group inverse ofT=T+δTï¼›(2)R(T)=R(T)且N(T)=N(T)ï¼›(3)R(T)(?)R(T)且N(T)cN(T)ï¼›(4)T:TTgT=TTgTï¼›(5)δT=δTTgT=TTgδTï¼›(6)R(δT)(?)R(T)å'ŒN(T)(?)(δT).Furthermore,are there some other expressions in weaker conditions?Theorem Let X be Banach space.Let T∈C(X)with a group inverse Tg∈B(X), R(T)∩N(Tg)={0}.Let δT∈L(X)be T-bounded with T-bound b<1and δTTg satisfy‖δTTgy‖≤λ1‖y‖+λ2‖(I+6TTg)y‖,(?)y∈X, where λ1,λ2∈[0,1),then B=T+(I+δTT+)-1=(I+T+δT)-1T+:Xâ†'X is a generalized inverse of T=T+δT.Furthermore if S=T+T+TT+-I,R(S)=D(T), S-1∈B(X),Then Tg=Tg(I+δTTG)-1(Tg(I+δTTg)-T+TTg(I+δTTg)-1-I)-1+(I-Tg(I+δTTg)-1T)(Tg(I+δTTg)-1T+TTg(I+δTTg)-1-I)-1Tg(I+δTTg)-1(Tg(I+δTTg)-T+TTg(I+δTTg)-1-I)-1.