Operator Moore-penrose Inverse Of The Operator Equation

In this article,we study the representations of the Moore-Penrose inverse of operators in an infinite dimensional Hilbert space and give the explict representations of the Moore-Penrose of 1×2 operators matrix.We also study two kinds of operator equations such as AXA~*=B and AX=XAX in an infinite dimensional space and give the characterization of solutions.This paper contains four chapters.Chapter 1 mainly introduces some notations, definitions and some well-known theorems.Firstly,we give some notations,and introduce the definitions of positive operator,Moore-Penrose inverse,Schur complement etc. Subsequently we give some well-known theorems.In Chapter 2,we generalize the results of[1].That is,we extend the results about matrices in a finite dimensional space to operators in an infinite dimensional Hilbert space. Using the way of space decomposition and the technique of block operator matrices,we study the explict representations of the Moore-Penrose inverse of 1×2 operators matrix.In Chapter 3,using the technique of block operator matrices,we characterise the sufficient and necessary conditions for the existence of solutions and positive operator solutions of the operator equation AXA~*=B and give the matrix representations.In three section,we prove the sufficient and necessary conditions for the existence of solutions of the operator equation AX=XAX by using the technique of block operator matrices.In Chapter 4,using the technique of block operator matrices,a kind of new proof of a result in[2]is given,which makes the structure of matrices be more clear.