Fredholm Differential Operators and Dichotomy

Abstract

We prove that a first order linear differential operator G=-d/dt+A(t) with unbounded operator coefficients A(t) is Fredholm on spaces of vector functions on real line if and only if the corresponding differential equation has exponential dichotomies on both positive and negative semi-lines, and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. This provides a complete infinite dimensional generalization of well-known finite dimensional results by K. Palmer, and by A. Ben-Artzi and I. Gohberg, used in the study of stability of traveling waves for applied partial differential equations. It also gives new connections of dichotomy to infinite dimensional Morse theory, and the celebrated Atiyah-Patodi-Singer Index Theorem.