The Berberian’s Transform and an Asymmetric Putnam–Fuglede Theorem

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