Class \(wA(s,t)\) operators and quasisimilarity
The paper under review is partly motivated by the Fuglede-Putnam theorem, which can be rephrased as follows. For operators \(S, T \in B(\mathcal H),\) consider the operator \(\delta_{T, S}\) on \(B(\mathcal H)\) given by \[ \delta_{T, S}(X)=TX-XS. \] If \(S, T\) are normal operators, then \(\ker (\delta_{T, S})=\ker(\delta_{T^*, S^*}).\) A number of generalizations of this result to various classes (including subnormal and hyponormal operators) can be found in the literature. For instance, the inclusion \(\ker (\delta_{T, S}) \subseteq \ker(\delta_{T^*, S^*})\) can be ensured under hypotheses weaker than normality of \(S\) and \(T.\) Although it is difficult to reproduce the definition of the class of \(wA(s, t)\) operators, we note that this class contains the class of the so-called \(w\)-hyponormal operators. It is unfortunate that the paper does not contain a single concrete example of operators belonging to the class \(wA(s, t)\), other than known examples. Some results related to quasi-similar class \(wA(s, t)\) operators are also discussed.
scientific article; zbMATH DE number 6143506
scientific article