  In this article we revisit the perturbation of exponential trichotomy of
linear difference equation in Banach space by using a Perron-Lyapunov fixed
point formulation for the perturbed evolution operator. This approach allows us
to directly re-construct the perturbed semiflow without using shift spectrum
arguments. These arguments are presented to the case of linear autonomous
discrete time dynamical system. This result is then coupled to Howland
semigroup procedure to obtain the persistence of exponential trichotomy for
non-autonomous difference equations as well as for linear random difference
equations in Banach spaces.
