# On the essential spectrum of $\lambda$-Toeplitz operators over compact Abelian groups

## TL;DR

This paper generalizes the spectral properties of $ ext{lambda}$-Toeplitz operators from the circle to arbitrary compact Abelian groups with totally ordered duals, expanding understanding of their essential spectrum.

## Contribution

It extends the analysis of $ ext{lambda}$-Toeplitz operators' essential spectrum invariance and circularity from the circle to more general compact Abelian groups.

## Key findings

- Essential spectrum invariance under rotation for generalized groups
- Circularity of the essential spectrum when $ ext{lambda}$ is not of finite order
- Generalization from $ ext{T}$ to arbitrary compact Abelian groups

## Abstract

In the recent paper by Mark C. Ho (2014) the notion of a $\lambda$-Toeplitz operator on the Hardy space $H^2(\mathbb{T})$ over the one-dimensional torus $\mathbb{T}$ was introduced and it was shown (under the supplementary condition) that for $\lambda\in \mathbb{T}$ the essential spectrum of such an operator is invariant with respect to the rotation $z\mapsto \lambda z$; if in addition $\lambda$ is not of finite order the essential spectrum is circular. In this paper, we generalize these results to the case when $\mathbb{T}$ is replaced by an arbitrary compact Abelian group whose dual is totally ordered.