A multiplicative property characterizes quasinormal composition operators in $L^2$-spaces

TL;DR

This paper characterizes quasinormal composition operators in $L^2$-spaces through a multiplicative property of their Radon-Nikodym derivatives, providing a clear criterion for quasinormality.

Contribution

It establishes a necessary and sufficient condition for quasinormality of composition operators based on the multiplicative behavior of Radon-Nikodym derivatives.

Findings

Quasinormality is equivalent to the multiplicative property of derivatives.

Provides a characterization criterion for composition operators.

Connects operator theory with measure-theoretic properties.

Abstract

A densely defined composition operator in an $L^2$-space induced by a measurable transformation $\phi$ is shown to be quasinormal if and only if the Radon-Nikodym derivatives $h_{\phi^n}$ attached to powers $\phi^n$ of $\phi$ have the multiplicative property: $h_{\phi^n} = h_{\phi}^n$ almost everywhere for n = 0, 1, 2, ....