Banach Space Of Thick Closed Operator The Existence Of Generalized Solution Type In Advance

The theory of generalized inverse is a widely useful branch of modem mathematics, and it has many subjects, such as the generalized inverse of matrix, the generalized inverse of linear transformation in linear spaces, the linear generalized inverse of linear operator in Hilbert spaces, orthogonal generalized inverse, the linear generalized inverse of linear operator in Banach spaces, the metric generalized inverse, the linear generalized inverse of nonlinear operator. It is well known that the perturbation analysis of generalized inverses has wide applications and plays an important role in many fields, such as computation, optimization, control theory and nonlinear analysis. In this paper, we study the perturbation problem for the generalized inverse and the existence problem for generalized resolvents of closed linear operators in Banach spaces.We first provide some stability characterizations of the closed linear operators under T-bounded perturbations, which improve some well known results in the case of bounded linear operators, the case of closed linear operators under bounded perturbations and the case of closed linear operators that the perturbation dose not change the null space nor the range of the operator. Theorem Let X and Y be two Banach spaces. Let T be a densely defined closed operator with a bounded generalized inverse T+∈B(Y,X). Let δT∈L(X,Y) be T-bounded with the non-negative constants a,b, i.e.,‖δTu‖≤a‖u‖+b‖Tu‖,(?)u∈D(T). If a‖T+‖+b‖TT+‖<1, then the following statements are equivalent:(1) R(T)∩N(T+)={0};(2) B=T+(I+δTT+)-1:Y�'X is a generalized inverse of T=T+δT;(3) Y=R(T)⊕N(T+);(4) X=N(T)⊕R(T+);(5) X=N(T)+R(T+);(6)(I+δTT+)-1T maps N(T) into R(T). In this case,As we all know, the spectrum plays a fundamental role in operator theory. Corresponding to the generalized inverse, we can consider the generalized resolvent and generalized spectrum of the linear operator. Based on our stability characterizations of the closed linear operators under T-bounded perturbation, some sufficient and necessary conditions for the existence of the gene-ralized resolvents of closed linear operations are obtained. An explicit expression for generalized resolvent is also given.Theorem Let X be a Banach space and T be a densely defined closed operator with a bounded generalized inverse.(1) If T has an analytic generalized resolvent on a neighborhood of zero, then for any generalized inverse T+∈B(X) of T, there exists a neighborhood V of zero such that R(T-λI)∩N(T+)＝{0},(?)λ∈V(2) If T+∈B(X) is a generalized inverse of T and there exists a neighborhood U of zero such that R(T-λI)∩N(T+)＝{0},(?)λ∈U then T has an analytic generalized resolvent on a neighborhood of zero. In this case, Rg(T,λ)=T+(I-λT+)-1:X�'X is a generalized resolvent of T on a neighborhood of zero.As applications, the existence characterizations of the generalized resolvents of Fredholm operators and semi-Fredholm operators are also considered.Theorem Let X be a Banach space and T be a densely defined closed semi-Fredholm operator with the generalized inverse T+∈B(X). Then T has an analytic generalized resolvent on a neighborhood of zero if and only if there exists a neighborhood of U of zero such that for all λ∈U,either dim N(T-λI)=dim N(T)<∞or co dim R(T-λI)=co dim R(T)<∞. In this case. Rg(T,λ)=T+(I-λT+)-1:X�'X is a generalized resolvent of T on a neighborhood of zero.Theorem Let X be a Banach space and T be a densely defined closed Fredholm operator. Then T has an analytic generalized resolvent on a neighborhood of zero if and only if there exists a neighborhood of U of zero such that for all λ∈U. dim N(T-λI)=dim N(T)<∞or co dim R(T-λI)=co dim R(T)<∞. In this case,if T+∈B(X)is a genetalized inverse of T,then Rg(T,λ)=T+(I-λT+)-1:X�'X is a generalized resolvent of T on a neighborhood of zero.