Banach Spaces Linear Operator And Disturbance Outside Inverse Theorem

It is well known that many important generalized inverses, such as the Moore-Penrose inverses、the weighted Moore-Penrose inverses、the Drazin inverses、 the weighted Drazin inverses、the group inversres can be reduced to outer inverses. Outer inverse plays a prominent part in numerical analysis、optimization、 mathematical statistics and so on. For example, outer inverse is an important tool in the iterative method of nonlinear operator equation (Newton method、Newton-like method) with singular Frechet derivative. The basic reason why outer inverse has important practical value is that any outer inverse of nonzero bounded linear operator always exists. Furthermore, the perturbation of outer inverse under the bounded linear operators is stable and with good properties.Let X and Y be Banach spaces. Let T∈B(X, Y) with an outer inverse T{2}. It is well known that T{2{(I+δTT{2})-1is the outer inverse of T=T+δT for‖T{2}‖.‖δT‖<1.But T{2}(I+δTT{2})-1may not be a generalized inverse of T=T+δT.It is natural to ask the following problem:when is T{2}(I+δTT{2})-1to be a generalized inverse of T=T+δT? N.Castro-Gonzalez and J.Y.Velez-Cerrada derive the perturbation theorem of Drazin inverse in Banach spaces on2008.In this paper, we first provide some necessary and sufficient conditions of T{2}(I+δTT{2})-1to be an generalized inverse of T=T+δT. Next, we give some characterizations for T{2}(I+δTT{2})-1to be a group inverse of T=T+δT for the bounded linear operators in Banach spaces. The results in this paper extend and improve the results in [4,5,29,31,34,36].Theorem Let X and Y be Banach spaces. Let T∈B(X,Y) with an outer inverse T{2}∈B(Y,X) and δT∈B(X,Y) with‖T{2}‖·‖δT‖<1. Then the following statements are equivalent: (1)B=T{2}(I+δTT{2})-1=(I+T{2}δT)-1T{2} is a generalized inverse of T=T+δT；(2)R(T)∩N(T{2})={0};(3)X=N(T)(?)R(T{2});(4)X=N(T)+R(T{2});(5)Y=R(T)(?)(T{2});(6)R(T)=R(TT{2})；(7)R(T)(?)R(TT{2});(8)N(T)=N(T{2}T)；(9)N(T{2}T)(?)N(T)；(10)(I+δTT{2})-1R(T)=R(TT{2});(11)(I+T{2}δT)-1N(T{2}T)=N(T)；(12)(I+δTT{2})-1TN(T{2}t)(?)r(TT{2}).Theorem Let X be Banach space.Let T∈B(X,Y)with an outer inverse T{2}∈B(X) and δT∈B(X,Y) with‖T{2}‖·‖δT‖<1.Then the following statements are equivalent:(1)B=T{2}(I+δTT{2})-1=(I+T{2}δT)-1T{2} is a group inverse of T=T+δT；(2)R(于)∩N(T{2})={0}and T{2}T=T{2}-TTT{2},TT{2}=T{2}TTT{2}；(3)X=N(T)(?)R(T{2}) and N(T{2})(?)N(T{2}T),R(TT{2})(?)R(T{2}).