A characterization of $(μ,ν)$-dichotomies via admissibility

TL;DR

The paper develops a Lyapunov-norm-free, admissibility-based characterization of $(\mu,\nu)$-dichotomies for nonautonomous discrete dynamics $x_{n+1}=A_n x_n$ on Banach spaces. It introduces and leverages weighted sequence spaces $\ell^{\infty}_{\beta}$, $\ell^{1}_{\beta}$ (and two-sided variants) to translate dichotomy properties into input–output admissibility, providing explicit reconstruction formulas and proving both one- and two-sided cases. A key advance is removing bounded-growth and Lyapunov-norm requirements, while establishing robustness under small linear perturbations. The results unify and extend known exponential, polynomial, and general-growth dichotomies, and include an alternative, compact-operator-based characterization in the one-sided case, with analogous persistence results in the two-sided setting. Overall, the work offers a flexible, general framework for verifying and preserving dichotomic behavior in infinite-dimensional, nonautonomous discrete systems, with potential applications to stability analysis and perturbation resilience.

Abstract

We present a characterization of $(μ,ν)$-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 $(μ, ν)$-dichotomies, i.e. we show that this notion persists under small but very general linear perturbations.

Theorems & Definitions (52)

• Definition 2.1
• Remark 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof : Proof of Lemma \ref{['aux']}
• Lemma 3.4
• proof : Proof of Lemma \ref{['lem: spliting']}