Y. Latushkin et S. Montgomerysmith, LYAPUNOV THEOREMS FOR BANACH-SPACES, Bulletin, new series, of the American Mathematical Society, 31(1), 1994, pp. 44-49
We present a spectral mapping theorem for semigroups on any Banach spa
ce E. From this, we obtain a characterization of exponential dichotomy
for nonautonomous differential equations for E-valued functions. This
characterization is given in terms of the spectrum of the generator o
f the semigroup of evolutionary operators.