Solutions to the System of Operator Equations A 1 X = C 1 , X B 2 = C 2 , and A 3 X B 3 = C 3 on Hilbert C * -Modules

Abstract

We study the solvability of the system of the adjointable operator equations $A_{1}X=C_1$, $X{B}_2=C_2$, and $A_{3}X{B}_3=C_3$ over Hilbert $C^{\mathrm{*}}$-modules. We give necessary and sufficient conditions for the existence of a solution and a positive solution of the system. We also derive representations for a general solution and a positive solution to this system. The above results generalize some recent results concerning the equations for operators with closed ranges.