Perturbation and an Abstract Characterization of Evolution SemigroupsU

Evans and Howland used an abstract characterization of evolution semigroups and a perturbation result for C0-semigroups to treat non-autonomous Schr¨odinger equations and problems in scattering theory. Here we proceed in a similar way. At first we characterize evolution semigroups on spaces of vector-valued functions E X. which are induced by a strongly continuous evolution family U t, s..tGs defined on the Banach space X. From this and a perturbation result of Voigt we obtain a sufficient condition for an operator on L p R, X.to be the generator of an evolution semigroup. This will be applied to non-autonomous heat equations with absorption.