Solutions to some systems of adjointable operator equations over Hilbert $C^*$-modules

Abstract

In this paper, by using operator matrix techniques, we present necessary and sufficient conditions for the existence of a solution to the system of equations $AXD+FX^{*}B=C,$ $GXF^{*}+FX^{*}G^{*}=H$ for adjointable operators between Hilbert $C^*$-modules, and derive an expression for the general solution to the system. We establish necessary and sufficient conditions for the existence of a solution to the system of adjointable operator equations $AXF=H_{1},$ $CXD=H_{2},$ $BXD=H_{3}$ over Hilbert $C^*$-modules. Some of the findings of this paper extend some known results in the literature.