Quasinormal extensions of subnormal operator-weighted composition operators in ℓ2-spaces

Abstract

We prove the subnormality of an operator-weighted composition operator whose symbol is a transformation of a discrete measure space and weights are multiplication operators in L 2 -spaces under the assumption of existence of a family of probability measures whose Radon-Nikodym derivatives behave regular along the trajectories of the symbol. We build the quasinormal extension which is a weighted composition operator induced by the same symbol. We give auxiliary results concerning commutativity of operator-weighted composition operators with multiplication operators.

Subnormal $n$th roots of quasinormal operators are quasinormal

“…By [23, Proposition 1], a closed densely defined operator A in H is quasinormal if and only if U |A| ⊂ |A|U , where A = U |A| is the polar decomposition of A (see [26,Theorem 7.20]). For more information on quasinormal operators we refer the reader to [2,4,24] for the bounded case, and to [14,23,16,11,3,24] for the unbounded one.…”