Shadowing for infinite dimensional dynamics and exponential trichotomies

Let (A(m))(m is an element of Z) be a sequence of bounded linear maps acting on an arbitrary Banach space X and admitting an exponential trichotomy and let f(m) : X -> X be a Lispchitz map for every m is an element of Z. We prove that whenever the Lipschitz constants of fm, m is an element of Z, are uniformly small, the nonautonomous dynamics given by x(m+1) = A(m)x(m) + f(m)(x(m)), m is an element of Z, has various types of shadowing. Moreover, if X is finite dimensional and each A(m) is invertible we prove that a converse result is also true. Furthermore, we get similar results for one-sided and continuous time dynamics. As applications of our results, we study the Hyers-Ulam stability for certain difference equations and we obtain a very general version of the Grobman-Hartman's theorem for nonautonomous dynamics.