HYPERBOLIC EVOLUTION SEMIGROUPS ON VECTOR-VALUED FUNCTION-SPACES

In [13] we characterized exponentially dichotomic evolution operators (U(t, s))t,s is-an-element-of R on a Banach space E in terms of the spectrum of an associated C0-group on an E-valued function space. In this paper we investigate the more general case of hyperbolic evolution families (U(t, s))t greater-than-or-equal-to is-an-element-of R and derive a spectral characterization through an associated C0-semigroup. We then apply the results to periodic initial value problems and show that the semigroup can be interpreted as a generalized monodromy operator. Furthermore we briefly discuss the spectral properties of a C0-semigroup associated with an evolution family (U(t, s))t greater-than-or-equal-to s greater-than-or-equal-to 0.