Operator equations $AX+YB=C$ and $AX{A}^{\ast }+BY{B}^{\ast }=C$ in Hilbert $C^{\ast }$-modules

Abstract

Let $A,B$ and $C$ be adjointable operators on a Hilbert $C^{\ast }$-module $\mathcal{E}$. Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert $C^{\ast }$-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 $C^{\ast }$-modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for $AX{A}^{\ast }+BY{B}^{\ast }=C$. Finally, we deduce that there exist nonzero operators $X,Y\ge 0$ and $Z$ such that $AX{A}^{\ast }+BY{B}^{\ast }=CZ$, when $A,B$ and $C$ are given subject to some conditions.

13 pages, to appear in Linear Algebra Appl. (LAA)