A closed densely defined operator T on a Hilbert space \({\mathcal {H}}\) is called M-hyponormal if \({\mathcal {D}}(T) \subset {\mathcal {D}}(T^{*})\) and there exists \(M>0\) for which \(\Vert (T-zI)^{*}x \Vert \le M \Vert (T-zI)x \Vert \) for all \(z \in {\mathbb {C}}\) and \(x\in {\mathcal {D}}(T)\). In this paper, we prove that if \(A:{\mathcal {H}}\rightarrow {\mathcal {K}}\) is a bounded linear operator, such that \(AB^*\subseteq TA\), where B is a closed subnormal (resp. a closed M-hyponormal) on \({\mathcal {H}}\), T is a closed M-hyponormal (resp. a closed subnormal) on a Hilbert space \({\mathcal {K}}\), then (i) \( AB\subseteq T^*A\) (ii) \({\overline{\text{ ran }\,A^{*}}}\) reduces B to the normal operator \( B\vert _{{\overline{\text{ ran }\,A^{*}}}}\) and (iii) \({\overline{\text{ ran }\,A}}\) reduces T to the normal operator \( T\vert _{{\overline{\text{ ran }\,A}}}\).