The Berberian's transform and an asymmetric Putnam-Fuglede theorem

TL;DR

This paper applies Berberian's technique to asymmetric Putnam-Fuglede theorems, establishing new operator relations and providing a counterexample for paranormal operators.

Contribution

It introduces a novel application of Berberian's method to asymmetric theorems and extends the understanding of operator classes involved.

Findings

Proved operator relation $A^*X = XB$ under certain class conditions.

Applied Berberian's technique to asymmetric Putnam-Fuglede theorems.

Provided a new counterexample for paranormal operators.

Abstract

We present how to apply a Berberian's technique to asymmetric Putnam-Fuglede theorems. In particular, we proved that if $A, B \in B(H)$ belong to the union of classes of $*$-paranormal operators, p-hyponormal operators, dominant operators and operators of class Y and $AX = XB^*$ for some $X \in B(H)$, then $A^*X = XB$. Moreover, we gave a new counterexample for an asymmetric Putnam-Fuglede theorem for paranormal operators