Index theorems for Fredholm, semi-Fredholm, and almost periodic operators: all in one example

TL;DR

This paper presents a unified approach to index theorems for various classes of operators, including Fredholm, semi-Fredholm, and almost periodic, using operators from scattering theory within C*-algebras.

Contribution

It introduces a single partial isometry framework that encompasses multiple operator classes and provides explicit index computations for each case.

Findings

Explicit index formulas for different operator classes

Unified framework using a single partial isometry

Concrete realizations in scattering theory context

Abstract

Based on operators borrowed from scattering theory, several concrete realizations of index theorems are proposed. The corresponding operators belong to some C*-algebras of pseudo-differential operators with coefficients which either have limits at plus and minus infinity, or which are periodic or asymptotically periodic, or which are uniformly almost periodic. These various situations can be deduced from a single partial isometry which depends on several parameters. All computations are explicitly performed.