A characterization of $(\mu,\nu)$-dichotomies via admissibility

We present a characterization of $(\mu,\nu)$-dichotomies in terms of the admissibility of certain pairs of weighted spaces for nonautonomous discrete time dynamics acting on Banach spaces. Our general framework enables us to treat various settings in which no similar result has been previously obtained as well as to recover and refine several known results. We emphasize that our results hold without any bounded growth assumption and the statements make no use of Lyapunov norms. Moreover, as a consequence of our characterization, we study the robustness of $(\mu, \nu)$-dichotomies, i.e. we show that this notion persists under small but very general linear perturbations.