Parabolic Non-Automorphism Induced Toeplitz-Composition C*-Algebras with Piece-wise Quasi-continuous Symbols

TL;DR

This paper studies a specific commutative C*-algebra generated by Toeplitz and composition operators with piece-wise quasi-continuous symbols, characterizing its maximal ideal space and analyzing the essential spectra of certain operator combinations.

Contribution

It provides a detailed characterization of the maximal ideal space of the algebra and applies this to determine the essential spectra of related operators.

Findings

The C*-algebra is commutative.

Maximal ideal space characterized explicitly.

Essential spectra of operator combinations determined.

Abstract

In this paper we consider the C*-algebra $C^{*}(\{C_{\varphi}\}\cup\mathcal{T}(PQC(\mathbb{T})))/K(H^{2})$ generated by Toeplitz operators with piece-wise quasi-continuous symbols and a composition operator induced by a parabolic linear fractional non-automorphism symbol modulo compact operators on the Hilbert-Hardy space $H^{2}$. This C*-algebra is commutative. We characterize its maximal ideal space. We apply our results to the question of determining the essential spectra of linear combinations of a class of composition operators and Toeplitz operators.