**Yuri Latushkin and Stephen Montgomery-Smith, Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces.**
*J. Func. Anal. ***127**, (1995), 173-197.
We present a spectral mapping theorem for continuous semigroups of operators 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 the investigating of the evolutionary semigroup is developed and applied to describe the hyperbolicity (exponential dichotomy) of the nonautonomuos equation.
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