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Zbl 0881.47020
Latushkin, Y.; Montgomery-Smith, S.; Randolph, T.
Evolutionary semigroups and dichotomy of linear skew-product flows on locally compact spaces with Banach fibers. (English)
[J] J. Differ. Equations 125, No.1, 73-116 (1996). ISSN 0022-0396

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 2000:: *47D06 One-parameter semigroups and linear evolution equations
34G10 Linear ODE in abstract spaces
47B38 Operators on function spaces

Keywords: roughness; evolutionary semigroups; strongly continuous semi-cocycle over a locally compact metric space acting on Banach fibers; non-autonomous abstract Cauchy problems; $C\sb 0$-semigroups; linear skew-product flows; spectral mapping theorem; hyperbolicity; exponential dichotomy; Banach algebra of weighted composition operators; Green's function

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