Hermitian solutions to the system of operator equations T_iX=U_i

In this article, we consider the system of operator equations T_iX=U_i for i=۱,۲,...,n and give necessary and sufficient conditions for the existence of common Hermitian solutions to this system of operator equations for arbitrary operators without the closedness condition. Also, we study the Moore-Penrose inverse of a n\times ۱ block operator matrix and then give the general form of common Hermitian solutions to this system of equations. Consequently, we give the necessary and sufficient conditions for the existence of common Hermitian solutions to the system of an operator equation and also present the necessary conditions for the solvability of the equation \sum_{i=۱}{n}T_iX_i=U.

Keywords:

Operator equation , Hermitian solution , Common solution , Existence of solution , Moore penrose inverse