In Banach Space Is Available Within The Operator Perturbation Theorem And The Application Of The Inverse

The theory of generalized inverses has become an important branch in modern mathematics and has substantial content, such as generalized inverses of matrix, the generalized inverses of linear transformations in linear space, the Moore-Penrose inverses of linear operator in Hilbert space, the generalized inverses of linear operator in Banach space. Generalized inverse is an indispensable tool in investigating the least squares problems, ill-posed problems, system identification problems, etc.One core of the theory of generalized inverse is its perturbation theory 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.It is well known that if a bounded linear operator T is invertible and T-1is its inverse, then T-1(I+δTT-1)-1is the inverse of T=T+δT for‖T-1‖·‖δT‖<1. It is natural to ask the following problem: Let T-be an inner inverse of T with‖T-‖·‖δT‖<1, is T-(I+δTT-)-an inner inverse of T=T+δT? If the answer is no, when is it?In this paper, we first give an example to illustrate that even in the case of matrices, T-(I+δTT-)-1may not be an inner inverse of T=T+δT in general. Next, we provide some characterizations for T-(I+δTT-)-1to be an inner inverse of T=T+δT for the bounded linear operators in Banach spaces.Theorem Let X and Y be Banach spaces. Let T∈B(X,Y) with an inner inverse T-∈B(Y,X) and δT∈B(X,Y) with‖T-‖·‖δT‖<1. Then B=T-(I+δTT-)-1=(I+T-δT)-1T-is an inner inverse of T=T+δT if and only if the following statements hold:(1) R(T)∩N(T-)={0};(2)(I+δTT-)-1TN(T)(?)N(T-TT--T-). Theorem Let X and Y be Banach spaces.Let T∈B(X,Y) with an inner inverse T-∈B(Y,X) and δT∈B(X,Y) with‖T-‖·‖δT‖<1.Then the following statements are equivalent:(1)B=T-(i+δTT-)-1=(I+T-δT)-1T-is an inner inverse of T=T+δT(2)R(T)∩N(TT-)={0}；(3)(I+δTT-)-1TN(T)(?)R(T).(4)(I+δTT-)-1R(T)=R(T)；(5)(I-T-δT)-1N(T)=N(T).Furthermore,we investigate the similar problems for generalized inverse,{1,3}-inverse,{1,4}-inverse,{1,5}-inverse,{1-2,3}-inverse,{1,2,4}-inverse,Moore-Penrose inverse,group inverse and Drazin inverse.The results obtained in this paper extend and improve many recent results in this area.