Hermitian solutions to the system of operator equations T_iX=U_i.

Abstract

In this article we consider the system of operator equations<br /> T_iX=U_i for i=1,2,...,n<br /> and give necessary and suffcient conditions for the<br /> existence of common Hermitian solutions to this system of operator equations for arbitrary operators without the closedness condition. Also we<br /> study the Moore-penrose inverse of a<br /> ncross 1 block operator matrix and.<br /> then give the general form of common Hermitian solutions to this system<br /> of equations. Cosequently, we give the necessary and sffcient conditions<br /> for the existence of common Hermitian solutions to the system of operator<br /> equati<br /> and also present the necessary<br /> conditions for solvability of the equation sum_{i=1}{n}T_iX_i=U