Pairs of Function Spaces and Exponential Dichotomy on the Real Line

We provide a complete diagram of the relation between the admissibility of pairs of Banach function spaces and the exponential dichotomy of evolution families on the real line. We prove that if W ∈ H R and V ∈ T R are two Banach function spaces with the property that either W ∈ W R or V ∈ V R , then the admissibility of the pair W R, X , V R, X implies the existence of the exponential dichotomy. We study when the converse implication holds and show that the hypotheses on the underlying function spaces cannot be dropped and that the obtained results are the most general in this topic. Finally, our results are applied to the study of exponential dichotomy of C0-semigroups.