Douglas factorization theorem revisited

Abstract:

Inspired by the Douglas factorization theorem, 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 tricky counterexample to show that there exist 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.