Douglas factorization theorem revisited

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 $C{C}^{\ast }\le tC{A}^{\ast }$ 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 $C^{\ast }$-algebra $\mathfrak{A}$, a Hilbert $\mathfrak{A}$-module $\mathcal{H}$ and projections $P$ and $Q$ on $\mathcal{H}$ 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.

14 pages, title changed, final version

References in corpus (2)

• Positivity of $2\times 2$ block matrices of operators
• On majorization and range inclusion of operators on Hilbert $C^{\ast }$-modules

Cited by in corpus (3)

• Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators
• The polar decomposition of the product of three operators
• The Douglas lemma for von Neumann algebras and some applications