On Semi-Fredholm Band-Dominated Operators

TL;DR

This paper investigates the semi-Fredholm property of band-dominated operators and demonstrates that, under certain conditions, it implies the operators are Fredholm, with implications for the role of limit operators in determining Fredholmness.

Contribution

It proves that semi-Fredholm band-dominated operators are necessarily Fredholm unless obvious obstructions exist, and links Fredholmness to the surjectivity of all limit operators.

Findings

Semi-Fredholm property implies Fredholmness for band-dominated operators.

Fredholmness can be deduced from the surjectivity of all limit operators.

The result clarifies the relationship between semi-Fredholm and Fredholm properties.

Abstract

In this paper we study the semi-Fredholm property of band-dominated operators $A$ and prove that it already implies the Fredholmness of $A$ in all cases where this is not disqualified by obvious reasons. Moreover, this observation is applied to show that the Fredholmness of a band-dominated operator already follows from the surjectivity of all its limit operators.