Sufficient conditions for exponential stability and dichotomy of evolutionequations

We present several sufficient conditions for exponential stability and dich otomy of solutions of the evolution equation u'(t) = A(t)u(t) (*) on a Bana ch space X. Our main theorem says that if the operators A(t) generate analy tic semigroups on X having exponential dichotomy with uniform constants and A(.) has a sufficiently small Holder constant, then (*) has exponential di chotomy. We further study robustness of exponential dichotomy under time de pendent unbounded Miyadera-type perturbations. Our main tool is a character ization of exponential dichotomy of evolution families by means of the spec tra of the so-called evolution semigroup on C-0(R, X) or L-1(R, X).