Asymmetric Fuglede-Putnam Theorem for Unbounded M-Hyponormal Operators

Abstract

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

9 Pages