Limit Operators, Compactness and Essential Spectra on Bounded Symmetric Domains

TL;DR

This paper extends the analysis of Toeplitz operators' essential spectra from the unit ball to general bounded symmetric domains, characterizing Fredholmness, compactness, and the role of limit operators and Berezin transform.

Contribution

It generalizes previous results to all bounded symmetric domains, establishing criteria for Fredholmness and compactness of Toeplitz and band-dominated operators.

Findings

Toeplitz operator is Fredholm iff all limit operators are invertible

Compactness characterized by Berezin transform vanishing at boundary

Band-dominated operators include Toeplitz algebra for p=2

Abstract

This paper is a follow-up to a recent article about the essential spectrum of Toeplitz operators acting on the Bergman space over the unit ball. As mentioned in the said article, some of the arguments can be carried over to the case of bounded symmetric domains and some cannot. The aim of this paper is to close the gaps to obtain comparable results for general bounded symmetric domains. In particular, we show that a Toeplitz operator on the Bergman space $A^p_{\nu}$ is Fredholm if and only if all of its limit operators are invertible. Even more generally, we show that this is in fact true for all band-dominated operators, an algebra that contains the Toeplitz algebra. Moreover, we characterize compactness and explain how the Berezin transform comes into play. In particular, we show that a bounded linear operator is compact if and only if it is band-dominated and its Berezin transform vanishes at the boundary. For $p = 2$ "band-dominated" can be replaced by "contained in the Toeplitz algebra".