The Putnam-Fuglede property for paranormal and ∗-paranormal operators

Abstract. An operator \(T \in {\cal B}(H)\) is said to have the Putnam-Fuglede commutativity property (PF property for short) if \(T^*X = XJ\) for any \(X \in {\cal B}(K,H)\) and any isometry \(J \in {\cal B}(K)\) such that \(TX = XJ^*\). The main purpose of this paper is to examine if paranormal operators have the PF property. We prove that \(k*\)-paranormal operators have the PF property. Furthermore, we give an example of a paranormal without the PF property.