Essentially Normal Composition Operators on $H^2$

TL;DR

This paper establishes a criterion for when composition operators on the Hardy space are essentially normal, and characterizes all rational self-maps inducing such operators using boundary interpolation theory.

Contribution

It introduces a simple criterion for essential normality and provides a complete parametrization of rational self-maps inducing essentially normal composition operators.

Findings

Established a criterion for essential normality of composition operators.

Provided a parametrization for rational self-maps inducing essentially normal operators.

Connected boundary interpolation theory with operator normality conditions.

Abstract

We prove a simple criterion for essential normality of composition operators on the Hardy space induced by maps in a reasonably large class of analytic self-maps of the unit disk. By combining this criterion with boundary Carath\'{e}odory-Fej\'{e}r interpolation theory, we exhibit a parametrization for all rational self-maps of the unit disk which induce essentially normal composition operators.