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

TL;DR

This paper extends the Douglas theorem to Hilbert $C^*$-modules, providing solutions for operator equations involving adjointable operators without the need for closed ranges, and explores conditions for solutions to related operator equations.

Contribution

It generalizes the Douglas theorem to Hilbert $C^*$-modules and derives conditions for solutions of operator equations without assuming closed ranges.

Findings

General solution for $AX+YB=C$ in Hilbert $C^*$-modules

Necessary and sufficient conditions for $AXA^*+BYB^*=C$

Existence of nonzero positive solutions under certain conditions

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.