On Unbounded Composition Operators in $L^2$-Spaces

TL;DR

This paper investigates the fundamental properties of unbounded composition operators in $L^2$-spaces, providing characterizations of normality, quasinormality, and generating Stieltjes moment sequences, along with counterexamples to bounded operator characterizations.

Contribution

It offers new characterizations for unbounded composition operators, including conditions for normality and quasinormality, and refutes bounded operator analogs of Lambert's subnormality criteria.

Findings

Formally normal operators are shown to be normal.

Composition operators generating Stieltjes moment sequences are fully characterized.

Counterexamples demonstrate the failure of bounded operator subnormality characterizations.

Abstract

Fundamental properties of unbounded composition operators in $L^2$-spaces are studied. Characterizations of normal and quasinormal composition operators are provided. Formally normal composition operators are shown to be normal. Composition operators generating Stieltjes moment sequences are completely characterized. The unbounded counterparts of the celebrated Lambert's characterizations of subnormality of bounded composition operators are shown to be false. Various illustrative examples are supplied.