  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.
  File Length: 68K
