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

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 $\ast$-paranormal operators, p-hyponormal operators, dominant operators and operators of class Y and $AX=X{B}^{\ast }$ for some $X\in B(H)$, then $A^{\ast }X=XB$. Moreover, we gave a new counterexample for an asymmetric Putnam-Fuglede theorem for paranormal operators