Exponential dichotomy and admissibility for evolution families on the real line

We give necessary and sufficient conditions for uniform exponential dichotomy of discrete evolution families in terms of the admissibility of the pairs (l(infinity)(Z, X); c(0)(Z, X)), (l(infinity)(Z, X), l(infinity)(Z, X)) and (c(0)(Z, X), c(0)(Z, X)), respectively. We prove that the uniform exponential dichotomy of an evolution family is equivalent with the uniform exponential dichotomy of the discrete evolution family associated to it. Thus, we obtain that the uniform exponential dichotomy of an evolution family is equivalent with the admissibility of one of the pairs (l(infinity)(Z, X), c(0)(Z, X)), (l(infinity)(Z; X): l(infinity)(Z, X)) or (c(0)(Z, X), c(0)(Z, X)) and the uniform exponential dichotomy of a strongly continuous evolution family is equivalent with the admissibility of one of the pairs (C-b(R, X), C-0(R, X)), (C-b(R: X) C-b(R, X)) or (C-0(R, X), C-0(R, X)), respectively. Finally, we apply our results at the characterization of the exponential dichotomy of C-0-semigroups.