Generalized exponential behavior on the half-line via evolution semigroups

We modify the classical concept of an evolution semigroup associated to an evolution family on the half-line to fit to the general case when linear flows may not agree with the restricted hypothesis of uniform exponential growth. We study the connections between spectral properties of the corresponding generator and a wide class of behavior of the evolution family. As a consequence, we prove that the generalized exponential dichotomy of possible non-invertible evolution families persists under sufficiently small linear perturbations.