Pedersen--Takesaki operator equation in Hilbert $C^{\ast }$-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 $C^{\ast }$-modules. It is known that the Douglas lemma does not hold in the setting of Hilbert $C^{\ast }$-modules in its general form. In fact, if $\mathcal{E}$ is a Hilbert $C^{\ast }$-module and $A,B\in \mathcal{L}(\mathcal{E})$, then the operator inequality $B{B}^{\ast }\le \hat{I}»A{A}^{\ast }$ with $\hat{I}»>0$ 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 $B{B}^{\ast }\le \hat{I}»A{A}^{\ast }$ and $\mathcal{R}(A)\supseteq \mathcal{R}(B{B}^{\ast })$. Furthermore, we prove that if $\mathcal{L}(\mathcal{E})$ is a $W^{\ast }$-algebra, $A,B\in \mathcal{L}(\mathcal{E})$, and $\overline{\mathcal{R}(A^{\ast })}=\mathcal{E}$, then $B{B}^{\ast }\le \hat{I}»A{A}^{\ast }$ for some $\hat{I}»>0$ if and only if $\mathcal{R}(B)\subseteq \mathcal{R}(A)$. Several examples are given to support the new findings.

16 pages