Characterizations of Dichotomies of Evolution Equations on the Half-Line

In this paper we investigate the characterization of dichotomies of an evolution family = U t s t≥s≥0 of bounded linear operators on Banach space X. We introduce operators I0 and IX on subspaces of Lp R + X using the integral equation u t = U t s u s + t s U t ξ f ξ dξ. The exponential and ordinary dichotomies of are characterized by properties of I0 IX.