Evolutionary semigroups and Lyapunov theorems in Banach spaces

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Evolutionary semigroups and Lyapunov theorems in Banach spaces

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Title: Evolutionary semigroups and Lyapunov theorems in Banach spaces
Author: Latushkin, Yuri, 1956-; Montgomery-Smith, Stephen, 1963-
Keywords: Exponential dichotomy
Hyperbolicity
Date: 2011-01-27
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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