Ordinary differential equations in Banach spaces and the spectral flow

TL;DR

This paper defines and characterizes the spectral flow for operator paths in Banach spaces, linking it to topological properties and demonstrating its behavior in infinite-dimensional settings.

Contribution

It introduces a homotopical characterization of spectral flow for Banach space operators and explores its properties and implications in infinite-dimensional analysis.

Findings

Spectral flow is not always injective or surjective in certain Banach spaces.

Existence of a path with a given spectral flow relates to the existence of specific projectors.

For hyperbolic and splitting paths, the associated differential operator is Fredholm with index related to spectral flow.

Abstract

We give a definition of the spectral flow for continuous paths in the space of bounded and essentially hyperbolic operators. We provide a homotopical characterization of the spectral flow in terms of a group homomorphism of the fundamental group of the projectors of the Calkin algebra with the infinite cyclic group Z. This characterization helps us to exhibit examples of infinite-dimensional Banach spaces where the spectral flow is not injective nor surjective. We prove that a path with spectral flow equal to an integer m exists if and only if there exists a projector P connected by an arc to a projector Q such that Range(Q) has co-dimension m in Range(P). We prove that if A is an asymptotically hyperbolic and essentially splitting path the differential operator F(u) = du/dt - Au is Fredholm. Moreover if A is also essentially hyperbolic the Fredholm index coincides with minus the spectral flow of A.