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

#### 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 $.

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 $.

#### Keywords

Spectral flow; projectors; hyperplanes

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