Fredholm theory for band-dominated and related operators: a survey

TL;DR

This survey reviews Fredholm theory for band-dominated operators on l^p-spaces, highlighting recent advances, especially for p=, using limit operator methods and algebraic frameworks.

Contribution

It fills gaps in existing Fredholm theory for p= and addresses open questions, providing a comprehensive overview of the limit operator approach.

Findings

Extended Fredholm theory to p= cases

Resolved open questions from previous literature

Unified algebraic framework for Fredholmness

Abstract

This paper presents the Fredholm theory on l^p-spaces for band-dominated operators and important subclasses, such as operators in the Wiener algebra. It particularly closes several gaps in the previously known results for the case p=\infty and addresses the open questions raised in a paper by Chandler-Wilde and Lindner. The main tools are provided by the limit operator method and an algebraic framework for the description and adaption of Fredholmness and convergence. A comprehensive overview of this approach is given.