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

TL;DR

This paper characterizes $(, u)$-dichotomies using admissibility of weighted spaces, allowing analysis without bounded growth assumptions or Lyapunov norms, and examines their robustness under perturbations.

Contribution

It introduces a new characterization of $(, u)$-dichotomies via admissibility, applicable to broader settings and without traditional growth constraints.

Findings

Characterization of $(, u)$-dichotomies through admissibility.

Results hold without bounded growth assumptions or Lyapunov norms.

Dichotomies are shown to be robust under small linear perturbations.

Abstract

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.