Positive Solutions of Operator Equations AX = B, XC = D

Abstract

In this paper, using the technique of operator matrix, we consider the positive solution of the system of operator equations AX=B,XC=D in the framework of the Hilbert space; here, the ranges R(A) of A and R(C) of C are not necessarily closed. Firstly, we provide a new necessary and sufficient condition for the existence of positive solutions of AX=B and also provide a representation of positive solutions, which generalize previous conclusions. Furthermore, using the above result, a condition of equivalence for the existence of common positive solutions of AX=B,XC=D is given, as well as the general forms of positive solutions.