  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.
