An asymmetric Putnam-Fuglede theorem for *-paranormal operators

Abstract

The well-known asymmetric form of Putnam-Fuglede theorem asserts that if $A$ and $B$ are bounded normal operators and $AX = XB^*$ for some bounded operator $X$, then $A^*X = XB$. In this paper we showed that the above theorem does not hold for paranormal operator $A$, even if we assume that $B$ has to be unitary and an operator $X$ is taken from Hilbert-Schmidt class. Additionally, we showed the similar resualt for *-paranormal operators.Comment: 5 page