The Dichotomy Theorem for evolution bi-families

We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L2(R,X), is Fredholm if and only if the not well-posed equation u′(t)=D(t)u(t), t∈R, has exponential dichotomies on R+ and R− and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.