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

TL;DR

This paper explores exponential dichotomy in Banach spaces, characterizing it through $(L^p,L^q)$-admissibility, and demonstrates its robustness under certain norm variations.

Contribution

It introduces a new characterization of exponential dichotomy via $(L^p,L^q)$-admissibility and proves its robustness in Banach space evolutionary families.

Findings

Characterization of exponential dichotomy using $(L^p,L^q)$-admissibility

Proof of robustness of exponential dichotomy under norm perturbations

Extension of the concept to families of norms in Banach spaces

Abstract

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 $p,q \in [1,\infty]$ 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.