  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
