Fuglede-Putnam type theorems via the Moore-Penrose inverse and Aluthge transform

Abstract

Let $A, B\in B(H)$ and let $\mbox{Com}(A, B)=\{X \in B(H): AX = XB\}$.
A pair $(A, B)$ is said to have the $FP$-property if
$\mbox{Com}(A, B) \subseteq \mbox{Com}(A^{*} , B^{*})$.
Let $\widetilde{T}$ and $T^\dag$ denote the
Aluthge transform and the Moore-Penrose inverse of $T$,
respectively. We show that (i) if $A^{*}$ is quasinormal, then
$((\widetilde{A})^{\dag},(\widetilde{B})^{\dag})$ has the
$FP$-property; (ii) if $(A^{\dag},B^{\dag})$ has the $FP$-property, then so is
$((\widetilde{A})^{\dag},(\widetilde{B})^{\dag})$. In general,
$(\widetilde{T})^{\dag}\neq \widetilde{T^{\dag}}$.
Finally, we give some
applications  to the Lambert multiplication operator
$M_wEM_u$ on $L^2(\Sigma)$, where $E$ is the
conditional expectation operator