  We give a sufficient condition for existence of an exponential dichotomy for
a general linear dynamical system (not necessarily invertible) in a Banach
space, in discrete or continuous time. We provide applications to the backward
heat equation with a potential varying in time and to the heat equation with a
finite number of slowly moving potentials. We also consider the Klein-Gordon
equation with a finite number of potentials whose centres move at sub-light
speed with small accelerations.
