The Stable Perturbations And Simplest Representations Of The Outer And Inner Inverses Of Bounded Linear Operators In Banach Spaces

The theory of generalized inverse has become an important research fields in modern mathematics and has many content,which includes the generalized inverse of matrix,the generalized inverse of linear transformation in linear space,the Moore-Penrose inverse of linear operator in Hilbert spaces and the generalized inverse of linear operator in Banach spaces etc.It is well known that the generalized inverse theory have lots of applications in many fields such as numerical analysis,nonlinear analysis,optimization theory,control theory,mathematical statistics,differential equations and so on.Let X and Y be Banach spaces and T{2} ?B(Y,X)be an outer inverse of T?B(X,Y).The simplest expression B=T{2}{I+?TT{2)-1=(I-T{2}? T)-1T{2}is an outer inverse of T=T+?T for all ?T ?B(X,Y)with ||?TT{2} ||<1.This is called the stability of outer inverse.It is nature to consider the following problem:What condition on the small perturbation ?T can guarantee that the simplest expression B=T{2}(I+?TT{2})-1[=(I+T{2}?T)-1T{2} is a {2,3}-inverse,{2,4}-inverse,or {2,5}-inverse of T?In Chapter 2,we first give an example to show that B is not the {2,3}-inverse,{2,4}-inverse and {2,5}-inverse of T.Some characteristics for B to be the {2,3}-inverse,{2,4}-inverse and {2,5}-inverse of T are also obtained.As corollaries,we get necessary and sufficient conditions for B to be the {1,2,3}-inverse,{1,2,4}-inverse and group inverse of T.Chapter 3 is devoted to the case of inner inverse.Let T?B(X,Y)have an inner inverseT{1}?B(Y,X).If ?T?B(X,Y)satisfies || ?TT{1} ||<1,whether the simplest expression B=T{1}(I+?TTf{1})-1=(I+T{1}?T)-1 T{1)is an inner inverse of T?It should be pointed out that inner inverse is totally different from outer inverse.For example,the outer inverse is stable while the inner inverse is unstable.In the case of outer inverse,TB and BT are idempotent operators,however,TB and BT are not necessarily idempotent and the direct sum may not necessarily hold for inner inverse.We present some necessary and sufficient conditions for the simplest expression B to be the inner inverse of T and then we get some characteristics for B to be the {1,3}-inverse,{1,4}-inverse and {1,5}-inverse of T-Our results extend and improve many previous results in this area.