Hyperbolic evolution groups and dichotomic evolution families

Recently, Ben-Artzi and Gohberg [2] used the concept ofC₀-semigroups in order to characterize the existence of dichotomies for nonautonomous differential equations on _n. A similar task was performed by Latushkin and Stepin [11] for dichotomies of linear skew-product flows. In this paper we will useC_o-semigroups to characterize existence of dichotomies for strongly continuous evolution families (U(t,s))_t.s on Hilbert and Banach spaces. Under an exponential growth condition we show that the concepts of hyperbolic evolution groups and exponentially dichotomic evolution families are equivalent.