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