EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LINEAR SKEW-PRODUCT FLOWS ONLOCALLY COMPACT SPACES WITH BANACH FIBERS

We study evolutionary semigroups generated by a strongly continuous se mi-cocycle over a locally compact metric space acting on Banach fibers . This setting simultaneously covers evolutionary semigroups arising f rom non-autonomous abstract Cauchy problems and C-0-semigroups, and li near skew-product flows. The spectral mapping theorem for these semigr oups is proved. The hyperbolicity of the semigroup is related to the e xponential dichotomy of the corresponding linear skew-product flow. To this end a Banach algebra of weighted composition operators is studie d. The results are applied in the study of: ''roughness'' of the dicho tomy, dichotomy and solutions of nonhomogeneous equations, Green's fun ction for linear skew-product flow, ''pointwise'' dichotomy versus ''g lobal'' dichotomy, and evolutionary semigroups along trajectories of t he flow. (C) 1996 Academic Press, Inc.