# An invertibility criterion in a C*-algebra acting on the Hardy space with applications to composition operators

## TL;DR

This paper establishes an invertibility criterion for a class of operators on the Hardy space, linking invertibility to Fredholm properties, and applies it to analyze spectra of certain composition operators.

## Contribution

It introduces a new invertibility criterion for operators combining Toeplitz operators and Fourier multipliers, extending the theory to composition operators on Hardy spaces.

## Key findings

- Operators are invertible iff they are Fredholm with zero index
- Spectra and essential spectra coincide for quasi-parabolic composition operators
- Provides a new criterion connecting invertibility and Fredholm properties

## Abstract

In this paper we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application we prove that for "quasi-parabolic" composition operators the spectra and the essential spectra are equal.