On intertwining and w-hyponormal operators

Abstract

Given $A, B\in B(H)$, the algebra of operators on a Hilbert Space $H$, define $\delta_{A,B}: B(H) \to B(H)$ and $\Delta_{A,B}: B(H) \to B(H)$ by $\delta_{A,B}(X)=AX-XB$ and $\Delta_{A,B}(X)=AXB-X$. In this note, our task is a twofold one. We show firstly that if $A$ and $B^{*}$ are contractions with $C_{.}o$ completely non unitary parts such that $X \in \ker \Delta_{A,B}$, then $X \in \ker \Delta_{A*,B*}$. Secondly, it is shown that if $A$ and $B^{*}$ are $w$-hyponormal operators such that $X \in \ker \delta_{A,B}$ and $Y \in \ker \delta_{B,A}$, where $X$ and $Y$ are quasi-affinities, then $A$ and $B$ are unitarily equivalent normal operators. A $w$-hyponormal operator compactly quasi-similar to an isometry is unitary is also proved