The present paper considers two concepts of nonuniform exponential dichotomy
(in the sense of Barreira-Valls) for evolution operators in Banach spaces. Some
examples clarify the relations between these concepts. A variant for the case
of nonuniform exponential dichotomy of a well-known theorem due to Datko is
obtained. We also prove a sufficient condition for the existence of exponential
dichotomy of evolution operators in terms of the existence of a Lyapunov
function in the general case of Banach spaces.