Operators with absolute continuity properties: an application to quasinormality

TL;DR

This paper develops an absolute continuity framework to analyze quasinormal operators, providing algebraic characterizations and examples, including weighted shifts on directed trees, and extends results to q-quasinormal operators.

Contribution

It introduces a novel absolute continuity approach to quasinormality and offers new algebraic characterizations and generalizations for q-quasinormal operators.

Findings

Established an absolute continuity approach to quasinormality

Provided algebraic characterizations of operator classes

Constructed examples using weighted shifts on directed trees

Abstract

An absolute continuity approach to quasinormality which relates the operator in question to the spectral measure of its modulus is developed. Algebraic characterizations of some classes of operators that emerged in this context are invented. Various examples and counterexamples illustrating the concepts of the paper are constructed by means of weighted shifts on directed trees. Generalizations of these results that cover the case of q-quasinormal operators are established.