Extreme cases of limit operator theory on metric spaces

TL;DR

This paper extends the theory of limit operators on metric spaces to include the extreme cases of p=0, 1, and infinity, filling important gaps in the existing framework for analyzing operator Fredholmness.

Contribution

It generalizes the limit operator theory to the remaining extreme p-values in metric spaces, completing the existing theoretical framework.

Findings

Extended limit operator theory to p=0, 1, and infinity.

Provided new criteria for Fredholmness in these extreme cases.

Filled gaps in the mathematical understanding of operators on metric spaces.

Abstract

The theory of limit operators was developed by Rabinovich, Roch and Silbermann to study the Fredholmness of band-dominated operators on $\ell^p(\mathbb{Z}^N)$ for $p \in \{0\} \cup [1,\infty]$, and recently generalised to discrete metric spaces with property A by \v{S}pakula and Willett for $p \in (1,\infty)$. In this paper, we study the remained extreme cases of $p \in\{0,1,\infty\}$ (in the metric setting) to fill the gaps.