Closed Linear Operators In Banach Space Perturbation Theorem Of Generalized Inverse

It is well known that the perturbation analysis of Moore-Penrose inverses in Hilbert spaces and generalized inverses in Banach spaces for bounded linear operators is very important in applications in diverse fields such as optimization, statistics, economics, programming, networks and so on. The expression T+(I+δTT+)-1maybe the simplest possible for generalized inverse of the perturbed operator. Many equivalent conditions for generalized inverses and Moore-Penrose 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 further explore the following general perturbation problem:Let X and Y be two Banach spaces, T be a closed linear operator from X into Y such that its domain is dense in X and T has the bounded generalized inverse T+, what condition on the small perturbation8T can guarantee that the generalized inverse (T+δT)+exists and it has the expression T+(I+δTT+)-1? Such problems in the case of stable perturbation and in the case that the perturbation does not change the null space have been studied when a‖T+‖+b‖TT+‖<1. It should be noted that the perturbation condition a‖T+‖+b‖TT+‖<1, which implies‖δTT+‖<1, is always assumed, then the invertibility of the operator I+δTT+and the boundedness of the inverse operator (I+δTT+)-1can be obtained directly from the well-known Banach Lemma. It is natural to ask whether this condition can be relaxed. Motivated by the idea in [6,11,42], we give a complete answer to the mentioned problem under a weaker perturbation condition. Utilizing this result, we also consider the expression problems for the Moore-Penrose inverses of closed EP operator in Hilbert space.As an illustration,we give some examples of the generalized inverses of closed operators and the Moore-Penrose inverses of closed EP operator. The main results of this paper improve and extend the results in[7-8,11-12,19,23,26,35,38-39,42]Theorem Let X and Y be two Banach spaces.Let T∈C(X,Y) with a generalized inverse T+∈B(Y,X).Let δT∈L(X,Y)be T-bounded with T-bound b<1and δTT+satisfy‖δTT+y‖≤λ1‖y‖+λ2‖(I+δTT+)y‖,(?)∈Y,where λ1,λ2∈[0,1),then the following statements are equivalent:(1)B=T+(I+δTT+)-1=(I+T+δT)-1T+:Yâ†'X is a generalized inverse of T=T+δT(2)R(T)∩N(T+)={0}ï¼›(3)Y=R(T)(?)(T+)ï¼›(4)X=N(T)(?)(T+)ï¼›(5)X=N(T)+R(T+)ï¼›(6)(I+δTT+)-1R(T)=R(T)ï¼›(7)(I+δTT+)-1TN(T)∈R(T)ï¼›(8)(I+T+δT)-1N(T)=N(T).In this case,R(T)is closed and‖B-T+‖≤‖T+‖·‖(I+δTT+)-1‖·‖δTT+‖.Theorem Let X and Y be two Hilbert spaces.Let T∈C(X,Y) with a generalized inverse T+∈B(Y,X).Let δT∈L(X,Y)be T-bounded with T-bound b<1and δTT+satisfy‖δTT+y‖≤λ1‖y‖+λ2‖(I+δTT+)y‖(?)y∈Y, where λ1,λ2∈[0,1),If R(T)∩N(T+)={0)then T=T+δT with a Moore-Penrose inverse T, andT+={I-[(T+(I+δTT+)-1T)**-(T+(I+δTT+)-1T)*]2}-1[T+(I+δTT+)-1T]*. T+(I+δTT+)-1[TT+(I+δTT+)-1]*{I-[TT+(I+δTT+)-1-(TT+(I+δTT+)-1)*]2}-1. Theorem Let X and Y be two Banach spaces.Let T∈C(X,Y) with a Moore-Penrose inverse T+∈B(Y,X).Let δT∈L(X,Y)be T-bounded with T-bound b<1and δTT+satisfy‖δTT+y‖≤λ1‖y‖+λ2‖(I+δTT+)y‖,(?)y∈Y, where λ1,λ2∈[0,1),then B=T+(I+δTT+)-1=(I+T+δT)-1T+:Yâ†'X is a Moore-Penrose inverse of T=T+δT if and only ifN(T)=N(T) and R(T)=R(T).