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.
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