Band-Dominated Fredholm Operators and a Question of Rabinovich, Roch and Silbermann

TL;DR

This paper proves that for band-dominated operators, the Fredholm property can be characterized without requiring uniform boundedness of the inverses of the operator spectrum, answering an open question.

Contribution

It establishes that the invertibility of all operators in the spectrum suffices for Fredholmness, removing the need for uniform boundedness conditions.

Findings

Fredholmness characterized without uniform boundedness

Answer to Rabinovich, Roch, and Silbermann's question

Advances understanding of band-dominated operators

Abstract

Withdrawn due to a likely error with the homeomorphism at line (4). Old abstract: In the monograph 'Limit Operators and their Applications in Operator Theory', the authors define the operator spectrum of a band-dominated operator T and prove that T is Fredholm if and only if all of the operators in its operator spectrum are invertible with uniformly bounded inverses. They also ask whether the uniform boundedness condition can in fact be dispensed with. In this note we answer this question affirmatively.