On the spectral flow for paths of essentially hyperbolic bounded operators on Banach spaces (Q647568)

From the author's abstract: A definition of the spectral flow for paths of bounded essentially hyperbolic operators on a Banach space is given. The spectral flow induces a group homomorphism on the fundamental group of every connected component of the space of essentially hyperbolic operators. The author proves that this homomorphism completes the exact homotopy sequence of a Serre fibration. This allows to characterize 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 \(-{\mathrm{ind}}(F_A)\), the Fredholm index of the differential operator \(F_A (u) = u' - A u\).