  We prove that a first order linear differential operator G with unbounded
operator coefficients is Fredholm on spaces of functions on the real line with
values in a reflexive Banach space if and only if the corresponding strongly
continuous evolution family has exponential dichotomies on both semiaxises 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. The operator G
is the generator of the evolution semigroup associated with the evolution
family. In the case when the evolution family is the propagator of a well-posed
differential equation u'(t)=A(t)u(t) with, generally, unbounded operators A(t),
the operator G is a closure of the operator -d/dt+A(t). Thus, this paper
provides a complete infinite dimensional generalization of well-known finite
dimensional results by K. Palmer, and by A. Ben-Artzi and I. Gohberg.
