Essential Spectra of Quasi-parabolic Composition Operators on Hardy Spaces of Analytic Functions

TL;DR

This paper investigates the essential spectra of a class of quasi-parabolic composition operators on Hardy spaces, introducing a new method to analyze their spectral properties and identify their essential spectra.

Contribution

It develops a novel approach linking these composition operators to C*-algebras of Toeplitz operators and Fourier multipliers, enabling spectral calculations.

Findings

Identified the essential spectra of quasi-parabolic composition operators.

Provided new examples of essentially normal composition operators.

Developed a new method for spectral analysis of these operators.

Abstract

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as "quasi-parabolic". This is the class of composition operators on H^{2} with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form \phi(z) = z+\psi(z) where \psi\in H^{2}(\mathbb{H}) and \Im(\psi(z)) >\delta > 0. We especially examine the case where \psi is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.