Let $(A_m)_{m \in {\mathop Z}}$ be a sequence of bounded linear maps acting on an arbitrary Banach space X and admitting an exponential trichotomy and let $f_m:X \to X$ be a Lispchitz map for every $m\in {\mathop Z} $. We prove that whenever the Lipschitz constants of $f_m$, $m \in {\mathop Z} $, are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in {\mathop 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.

For a dynamics on the whole line, for both discrete and continuous time, we extend a result of Pliss that gives a characterization of the notion of a trichotomy in various directions. More precisely, the result gives a characterization in terms of an admissibility property in the whole line (namely, the existence of bounded solutions of a linear dynamics under any nonlinear bounded perturbation) of the existence of a trichotomy, i.e. of exponential dichotomies in the future and in the past, together with a certain transversality condition at time zero. In particular, we consider arbitrary linear operators acting on a Banach space as well as sequences of norms instead of a single norm, which allows us to consider the general case of non-uniform exponential behaviour.