The Putnam-Fuglede property for paranormal and \(\ast\)-paranormal operators (Q2851293)

An operator \(T \in {\mathcal{B(H)}}\) has the Putnam-Fuglede commutativity property (PF property) if \(T^*X = XJ\) for any \(X \in {\mathcal{B(K,H)}}\) and any isometry \(J \in {\mathcal{B(K)}}\) such that \(TX = XJ^*\). In this paper, the author presents necessary and sufficient conditions for power bounded operators and completely nonunitary operators to have the PF property. Besides, it is shown that \(k^*\)- paranormal operators have the PF property. This is a generalization of the author's previously obtained result which states that \(k^*\)-paranormal contractions have the PF property [Linear Algebra Appl. 436, No. 9, 3065--3071 (2012; Zbl 1254.47021)]. Nevertheless, paranormal operators need not have the PF property. An example of such an operator is presented in the last section.