# Admissibility and polynomial dichotomies for evolution families

## TL;DR

This paper characterizes polynomial dichotomies for evolution families using admissibility, introduces Lyapunov norms to recover nonuniform cases, and proves robustness under small perturbations.

## Contribution

It provides a new characterization of polynomial dichotomies via admissibility and demonstrates their robustness with Lyapunov norms.

## Key findings

- Characterization of polynomial dichotomies through admissibility.
- Introduction of Lyapunov norms to recover nonuniform dichotomies.
- Proof of robustness of strong nonuniform polynomial dichotomies under small perturbations.

## Abstract

For an arbitrary evolution family, we consider the notion of a polynomial dichotomy with respect to a family of norms and characterize it in terms of the admissibility property, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms, we recover the notion of a (strong) nonuniform polynomial dichotomy. As a nontrivial application of the characterization, we establish the robustness of the notion of a strong nonuniform polynomial dichotomy under sufficiently small linear perturbations.