On Fredholm properties of Lu = u′ − A(t)u for paths of sectorial operators

Abstract

We consider a path of sectorial operators t ↦ A (t) ∈ Cα (R, L (D, X)), 0 < α < 1, in general Banach space X, with common domain D (A (t)) = D and with hyperbolic limits at ±∞. We prove that there exist exponential dichotomies in the half-lines (−∞, −T] and [T, +∞) for large T, and we study the operator (Lu)(t) = u′(t) − A(t)u(t) in the space Cα (R, D) ∩ C1+α (R, X). In particular, we give sufficient conditions in order that L is a Fredholm operator. In this case, the index of L is given by an explicit formula, which coincides to the well-known spectral flow formula in finite dimension. Such sufficient conditions are satisfied, for instance, if the embedding D ↪ X is compact.