Evolutionary Semigroups and Dichotomy of Linear Skew-Product Flows on Locally Compact Spaces with Banach Fibers

We study evolutionary semigroups generated by a strongly continuous semi-cocycle over a locally compact metric space acting on Banach fibers. This setting simultaneously covers evolutionary semigroups arising from non-autonomous abstract Cauchy problems andC0-semigroups, and linear skew-product flows. The spectral mapping theorem for these semigroups is proved. The hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew-product flow. To this end a Banach algebra of weighted composition operators is studied. The results are applied in the study of: “roughness” of the dichotomy, dichotomy and solutions of nonhomogeneous equations, Greenʹs function for a linear skew-product flow, “pointwise” dichotomy versus “global” dichotomy, and evolutionary semigroups along trajectories of the flow.