Y. Latushkin et al., EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LINEAR SKEW-PRODUCT FLOWS ONLOCALLY COMPACT SPACES WITH BANACH FIBERS, Journal of differential equations, 125(1), 1996, pp. 73-116
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.