Ordinary differential equations in Banach spaces and the spectral flow

Details Ordinary differential equations in Banach spaces and the spectral flow Ordinary differential equations in Banach spaces and the spectral flow

2008-03-11

arxiv.org/abs/0803.1685v2

Mathematics

Functional Analysis

Correction to the definition of spectral flow and some examples added

Scientific paper

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.

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