Douglas factorization theorem revisited

TL;DR

This paper revisits the Douglas factorization theorem within Hilbert C*-modules, providing new solutions, conditions for positivity, counterexamples, and perturbation results for operator equations.

Contribution

It extends the Douglas lemma to Hilbert C*-modules, offering explicit solutions, conditions for positive solutions, and counterexamples for certain operator equations.

Findings

General solution for AX=C when A is semi-regular

Characterization of positive solutions via range inclusion and inequalities

Counterexample showing non-existence of solutions in some cases

Abstract

Inspired by the Douglas lemma, we investigate the solvability of the operator equation $AX=C$ in the framework of Hilbert C*-modules. Utilizing partial isometries, we present its general solution when $A$ is a semi-regular operator. For such an operator $A$, we show that the equation $AX=C$ has a positive solution if and only if the range inclusion ${\mathcal R}(C) \subseteq {\mathcal R}(A)$ holds and $CC^*\le t\, CA^*$ for some $t>0$. In addition, we deal with the solvability of the operator equation $(P+Q)^{1/2}X=P$, where $P$ and $Q$ are projections. We provide a counterexample to show that there exists a $C^*$-algebra $\mathfrak{A}$, a Hilbert $\mathfrak{A}$-module $\mathscr{H}$ and projections $P$ and $Q$ on $\mathscr{H}$ such that the operator equation $(P+Q)^{1/2}X=P$ has no solution. Moreover, we give a perturbation result related to the latter equation.