On the spectral flow for paths of essentially hyperbolic bounded operators on Banach spaces

TL;DR

This paper defines a spectral flow for paths of essentially hyperbolic operators on Banach spaces, explores its topological properties, and relates it to Fredholm indices, revealing differences from the Hilbert space case.

Contribution

It introduces a new spectral flow concept for Banach space operators, analyzes its homotopy properties, and connects it to Fredholm indices, extending previous Hilbert space results.

Findings

Spectral flow induces a group homomorphism on the fundamental group.

Characterization of the kernel and image of the homomorphism.

Spectral flow coincides with negative Fredholm index for essentially splitting paths.

Abstract

We give a definition of the spectral flow for paths of bounded essentially hyperbolic operators on a Banach space. The spectral flow induces a group homomorphism on the fundamental group of every connected component of the space of essentially hyperbolic operators. We prove that this homomorphism completes the exact homotopy sequence of a Serre fibration. This allows us to characterise its kernel and image and to produce examples of spaces where it is not injective or not surjective, unlike what happens for Hilbert spaces. For a large class of paths, namely the essentially splitting, the spectral flow of $ A $ coincides with $ -\ind(F_A) $, the Fredholm index of the differential operator $ F_A (u) = u' - A u $.