EVOLUTIONARY SEMIGROUPS AND LYAPUNOV THEOREMS IN BANACH-SPACES

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.