| Summary: | 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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