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Let $A,B$ and $C$ be adjointable operators on a Hilbert $C^*$-module$\mathscr{E}$. Giving a suitable version of the celebrated Douglas theorem inthe context of Hilbert $C^*$-modules, we present the general solution of theequation $AX+YB=C$ 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 forexistence of a solution for $AXA^*+BYB^*=C$. Finally, we deduce that thereexist nonzero operators $X, Y\geq 0$ and $Z$ such that $AXA^*+BYB^*=CZ$, when$A, B$ and $C$ are given subject to some conditions.

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

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