Exponential Dichotomy on the Real Line and Admissibility of Function Spaces

Abstract.

The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted $\mathcal{T}{\left( \mathbb{R} \right)}$ and we prove that if $B \in \mathcal{T}{\left( \mathbb{R} \right)}$ with $B\backslash L^{1} {\left( {\mathbb{R},\mathbb{R}} \right)} \ne \emptyset$ and the pair ${\left( {C_{b} {\left( {\mathbb{R},X} \right)},B{\left( {\mathbb{R},X} \right)}} \right)}$ is admissible for an evolution family $\mathcal{U} = {\left\{ {U{\left( {t,s} \right)}} \right\}}_{{t \geqslant s}} ,$ then $\mathcal{U}$ is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair ${\left( {C_{b} {\left( {\mathbb{R},X} \right)},L^{1} {\left( {\mathbb{R},X} \right)}} \right)}$ for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs ${\left( {C_{b} {\left( {\mathbb{R},X} \right)},L^{p} {\left( {\mathbb{R},X} \right)}} \right)},{\left( {C_{b} {\left( {\mathbb{R},X} \right)},C_{b} {\left( {\mathbb{R},X} \right)}} \right)},{\left( {C_{b} {\left( {\mathbb{R},X} \right)},C_{0} {\left( {\mathbb{R},X} \right)}} \right)}$ and ${\left( {C_{b} {\left( {\mathbb{R},X} \right)},C_{0} {\left( {\mathbb{R},X} \right)} \cap L^{p} {\left( {\mathbb{R},X} \right)}} \right)},$ with $p \in \left[ {1,\infty } \right).$