Aluthge transforms of unbounded weighted composition operators in $L^2$-spaces

TL;DR

This paper studies the Aluthge transform of unbounded weighted composition operators in $L^2$-spaces, revealing its properties, effects on hyponormality, and fixed points, with implications for operator theory.

Contribution

It characterizes the Aluthge transform of unbounded weighted composition operators, including its closure, dense definiteness, and fixed points, advancing understanding in operator theory.

Findings

The closure of the Aluthge transform is a weighted composition operator with the same symbol.

The paper characterizes $p$-hyponormality and its preservation under the Aluthge transform.

The only fixed points of the Aluthge transform are quasinormal operators.

Abstract

We describe the Aluthge transform of an unbounded weighted composition operator acting in an $L^2$-space. We show that its closure is also a weighted composition operator with the same symbol and a modified weight function. We investigate its dense definiteness. We characterize $p$-hyponormality of unbounded weighted composition operators and provide results on how it is affected by the Aluthge transformation. We show that the only fixed points of the Aluthge transformation on weighted composition operators are quasinormal ones.