Operator equations $AX+YB=C$ and $AXA^*+BYB^*=C$ in Hilbert $C^*$-modules

Abstract

Let $A,B$ and $C$ be adjointable operators on a Hilbert $C^*$-module $\mathscr{E}$. Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert $C^*$-modules, we present the general solution of the equation $AX+YB=C$ when the ranges of $A,B$ and $C$ are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert $C^*$-modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for $AXA^*+BYB^*=C$. Finally, we deduce that there exist nonzero operators $X, Y\geq 0$ and $Z$ such that $AXA^*+BYB^*=CZ$, when $A, B$ and $C$ are given subject to some conditions.