Asymmetric Fuglede-Putnam theorem for unbounded M-hyponormal operators

A closed densely defined operator T on a Hilbert space H is called M-hyponormal if D(T) subset of D(T*) and there exists M > 0 for which parallel to(T-zI)*x parallel to <= M parallel to(T-zI)x parallel to for all z is an element of C and x is an element of D(T). In this paper, we prove that if A:H -> K is a bounded linear operator, such that AB* subset of TA, where B is a closed subnormal (resp. a closed M-hyponormal) on H, T is a closed M-hyponormal (resp. a closed subnormal) on a Hilbert space K, then (i) AB subset of T*A (ii) (ranA*) over tilde reduces B to the normal operator B vertical bar((ranA*) over tilde) over bar and (iii) (ranA*) over tilde reduces T to the normal operator T vertical bar((ranA) over tilde).