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Evolutionary semigroups and dichotomy of linear skew-product flows on locally compact spaces with Banach fibers. (English) Zbl 0881.47020
Summary: 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 and C 0 -semigroups, and linear skew-product flows. The spectral mapping theorem for these semigroups is proved. The hyperbolicity of the semigroups 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 and solutions of nonhomogeneous equations, Green’s function for a linear skew-product flow, “pointwise” dichotomy versus “global” dichotomy, and evolutionary semigroups along trajectories of the flow.
MSC:
47D06One-parameter semigroups and linear evolution equations
34G10Linear ODE in abstract spaces
47B38Operators on function spaces (general)