Asymmetric Fuglede-Putnam Theorem for Unbounded M-Hyponormal Operators

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