Exponential dichotomy on the real line and admissibility of function spaces (Q2494132)

The purpose of the paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. The authors prove that if \(\mathcal{U}\) is an evolution family on a Banach space \(X\) and if \(B(\mathbb R, X)\) is a special space of Bochner measurable functions from \(\mathbb R\) into \(X\), then the admissibility of the pair \((C_b (\mathbb R,X),B(\mathbb R, X))\) for \(\mathcal{U}\) implies the uniform exponential dichotomy of \(\mathcal{U}\). By an example they show that the admissibility of the pair \((C_b (\mathbb R,X),L^1(\mathbb R, X))\) for \(\mathcal{U}\) is not a sufficient condition for the exponential dichotomy of \(\mathcal{U}\). As applications, they deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families \(\mathcal{U}\) in terms of the admissibility of pairs \((C_b (\mathbb R,X),F)\), where \(F\) is one of the spaces \(L^p (\mathbb R, X)\), \(C_b (\mathbb R,X)\), \(C_0 (\mathbb R,X)\), and \(C_0 (\mathbb R,X)\cap L^p (\mathbb R,X)\) with \(p\in (1,\infty]\).