The Weighted MP Inverse Of The Conjugate Operator On The Hilbert C*-module

The theory of generalized inverse has always been an important branch of research.which has important applications in numerical analysis,differential equations7 numerical linear algebra,optimization,cybernetics and other fields.This thesis focuses on the ex-istence and uniqueness of the weighted M-P inverse of the adjoint operator on Hilbert C~*-modules,the relationship between the different weighted M-P inverses and the norm estimations,and the reverse order law of weighted M-P inversesIn this thesis,under the framework of Hilbert C~*-modules,we first introduce the weight and the weighted space induced by a self-adjoint and invertible operator.When A is an adjoint operator and the weights M and N are self-adjoint and invertible,we give the necessary and sufficient condition for the existence of the weighted M-P inverse AMN(?).If AMN(?),exists,then it is unique.Then,the necessary and sufficient conditions for the existence of matrix weighted M-P inverse AMN(?),are obtained;When the weights M and N are positive definite operators,the necessary and sufficient conditions for the existence of the weighted M-P inverse AMN(?),on Hilbert C~*-modules are obtained;When the weights M and N both commute with A,the necessary and sufficient conditions for the existence of the weighted M-P inverse AMN(?) on Hilbert C~*-modules are obtained.Then,we study the relationship between the weighted M-P inverse AMN1(?) and AMN2(?),the relationship between and and the relationship between AM1,N1(?) and AM2,N2(?)while the adjoint operator A is fixed whereas and the weights M and N change.The necessary and sufficient conditions for the invertibility of the two operators RM;N1,N2 and LM1,M2;N are clarified.It is shown that when both and exist,this two operators are not necessarily invertible.In addition,the additive perturbations of the weighted M-P inverses of the adjoint operators on Hilbert C~*-modules are also studied.In the last chapter,we study the reverse order law(AB)(?)MP=BNP(?)A(?)MP for the weighted M-P inverse of the adjoint operators on Hilbert C~*-modules.First,we give an example to show that R(AB)=R(A)?R(B)may not hold even if some strong conditions are satisfied.Then we give a sufficient condition for it to be true.Finally,we give some sufficient conditions for the reverse order law(AB)(?)MP=B(?)NPA(?)MN.