Y. Latushkin et S. Montgomerysmith, EVOLUTIONARY SEMIGROUPS AND LYAPUNOV THEOREMS IN BANACH-SPACES, Journal of functional analysis, 127(1), 1995, pp. 173-197
We present a spectral mapping theorem for continuous semigroups of ope
rators on any Banach space E. The condition for the hyperbolicity of a
semigroup on E is given in terms of the generator of an evolutionary
semigroup acting in the space of E-valued functions. The evolutionary
semigroup generated by the propagator of a nonautonomous differential
equation in E is also studied. A ''discrete'' technique for investigat
ing the evolutionary semigroup is developed and applied in describing
the hyperbolicity (exponential dichotomy) of the nonautonomous equatio
n. (C) 1995 Academic Press, Inc.