Exponential dichotomy and $(L^p,L^q)$-admissibility

We consider the notion of an exponential dichotomy with respect to a family of norms for an evolutionary family in a Banach space, and we characterize it by the admissibility of the pair $(L^p,L^q)$ for $p,q\in [1,\mathrm{\infty }]$ with $p\ge q$. We then use this characterization to establish the robustness of an exponentially dichotomic evolutionary family with respect to a family of norms.