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

In this article we consider the system of operator equations TiX = Ui for i = 1, 2, 3, ..., n, between Hilbert spaces 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 × 1 block operator matrix and then give the general form of common Hermitian solutions to this system of equations. Cosequently, we give the necessary and sufficient conditions for the existence of common Hermitian solutions to the system of operator equations TiXVi = Ui , for i = 1, 2, 3, ..., n and also present the necessary conditions for solvability of the equation Pn i=1 TiXi = U.