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 \( T\vert_{{\overline{ran(A)}}}.\)