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    Evolutionary semigroups and Lyapunov theorems in Banach spaces

    Latushkin, Yuri, 1956-
    Montgomery-Smith, Stephen, 1963-
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    [PDF] EvolutionarySemigroupsLyapunov.pdf (251.7Kb)
    Date
    2011
    Format
    Preprint
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    Abstract
    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 nonautonomuous abstract Cauchy problems and C0-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, Green's function for a linear skew-product flow, "pointwise" dichotomy versus "global" dichotomy, and evolutionary semigroups along trajectories of the flow.
    URI
    http://hdl.handle.net/10355/9760
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