Persistence of exponential trichotomy for linear operators: A Lyapunov-Perron approach

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.