On intertwining and w-hyponormal operators

Given A, B mem B(H), the algebra of operators on a Hilbert Spaces H, define deltaA,B : B(H) arr B(H) and DeltaA,B : B(H) arr B(H) by deltaA,B(X) = AX - XB and DeltaA,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 mem ker DeltaA,B, then X mem ker DeltaA*,B*. Secondly, it is shown that if A and B* are w-hyponormal operators such that X mem ker deltaA,B and Y mem ker deltaB,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.