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

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

2008-06-25

arxiv.org/abs/0806.4094v3

Mathematics

Functional Analysis

Final version to appear in Topological Methods in Nonlinear Analysis. 23 pages, LaTeX

Scientific paper

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 $.

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