We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.

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.