Quasinormal extensions of subnormal operator-weighted composition operators in $\ell^2$-spaces

TL;DR

This paper establishes conditions for subnormality of operator-weighted composition operators in discrete measure spaces and constructs their quasinormal extensions, advancing understanding of their spectral properties.

Contribution

It introduces a new criterion for subnormality based on probability measures and constructs explicit quasinormal extensions for these operators.

Findings

Subnormality characterized by probability measures with regular Radon-Nikodym derivatives.

Construction of quasinormal extensions as weighted composition operators.

Auxiliary results on commutativity with multiplication operators.

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.