Regular Fredholm Pairs

TL;DR

This paper introduces and classifies regular Fredholm pairs, exploring their properties, index, and relationships with other operator classes in Banach spaces.

Contribution

It defines regular Fredholm pairs, provides a complete classification, and links them to other operator concepts, expanding the understanding of Fredholm theory.

Findings

Regular Fredholm pairs are fully classified.

The index is a key tool in their analysis.

Characterizations involve symmetrical pairs and invertible operators.

Abstract

In this work it is introduced the notion of regular Fredholm pair, i.e. a Fredholm pair whose operators are regular. The main properties of these objects are studied, and what is more, they are entirely classified. Furthermore, the index of a Fredholm pair turns out to be an extremely useful tool in the description of the aforementioned objects. Finally, regular Fredholm pairs are characterized in terms of regular Fredholm symmetrical pairs, exact chains of multiplication operators, and invertible Banach space operators.