Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

Abstract

We extend a work of Pedersen and Takesaki by giving some equivalent conditions for the existence of a positive solution of the so-called Pedersen--Takesaki operator equation $XHX=K$ in the setting of Hilbert $C^*$-modules. It is known that the Douglas lemma does not hold in the setting of Hilbert $C^*$-modules in its general form. In fact, if $\mathscr{E}$ is a Hilbert $C^*$-module and $A, B \in \mathcal{L}(\mathscr E)$, then the operator inequality $B B^*\le \lambda AA^*$ with $\lambda>0$ does not ensure that the operator equation $AX=B$ has a solution, in general. We show that under a mild orthogonally complemented condition on the range of operators, $AX=B$ has a solution if and only if $BB^*\leq \lambda AA^*$ and $\mathscr R(A) \supseteq \mathscr R(BB^*)$. Furthermore, we prove that if $\mathcal{L}(\mathscr E)$ is a $W^*$-algebra, $A,B\in \mathcal{L}(\mathscr E)$, and $\overline{\mathscr R(A^*)}=\mathscr E$, then $BB^*\leq\lambda AA^*$ for some $\lambda>0$ if and only if $\mathscr R (B)\subseteq \mathscr R(A)$. Several examples are given to support the new findings.