The Existence and Representation of the Solutions to the System of Operator Equations <i>A_iXB_i</i> + <i>C_iYD_i</i> + <i>E_iZF_i</i> = <i>G_i</i>(<i>i</i> = 1, 2)

Abstract

In this paper, we give the necessary and sufficient conditions for the existence of general solutions, self-adjoint solutions, and positive solutions to the system of AiXBi+CiYDi+EiZFi=Gi(i=1,2) under additional conditions. In addition, we derive the representation of general solutions to the system of AiXBi+CiYDi+EiZFi=Gi(i=1,2), and provide the matrix representation of the self-adjoint solutions and the positive solutions in the sense of the star order.

Keywords

• operator equation
• Moore–Penrose inverse
• general solution
• self-adjoint solution
• positive solution