The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions

Abstract

(see e.g. [Da-K], [Fat], [Paz], [Tan]). In the following a family (U(t, s))(t,s)∈D in L(X) satisfying (E1)–(E3) is called an evolution family. It has been noticed by several authors (see [LM1], [LM2], [LRa], [Na2], [RaS], [Ra1], [Ra2], [Ra3], [Rha] and the references therein) that asymptotic properties of the evolution family (U(t, s))(t,s)∈D are strongly related to the asymptotic behaviour of an associated evolution semigroup (TE(t))t≥0 of operators on a Banach space E(X) of X –valued functions (see Section 1). For a large class of these function spaces this evolution semigroup is strongly continuous and hence has a generator GE . It has been shown by R. Rau [Ra1, Prop. 1.7] and Y. Latushkin and S. Montgomery–Smith [LM1, Thm. 3.1], [LM2, Thm. 4] that on the function spaces C0(IR, X) and L (IR, X), 1 ≤ p <∞ , these semigroups always satisfy the spectral mapping theorem