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 $.