Opuscula Math. 33, no. 3 (), 565-574
http://dx.doi.org/10.7494/OpMath.2013.33.3.565
Opuscula Mathematica

# 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.
Keywords: power-bounded operators, paranormal operators, $$*$$-paranormal operators, $$k$$-paranormal operators, $$k*$$-paranormal operators, the Putnam-Fuglede theorem.
Mathematics Subject Classification: 47B20, 47A05, 47A62.