Robustness and dichotomy radii for discrete dynamics on the half-line

Abstract We present a complete study on the dichotomy roughness and dichotomy radii for nonautonomous discrete dynamics on the half-line, in infinite-dimensional spaces. Our approach is based on new characterizations of the uniform exponential dichotomy that extend the results in [72]. We prove novel criteria for the robustness of the uniform exponential dichotomy, providing general bounds for the perturbations that preserve it. We introduce for the first time the notions of dichotomy radius on the half-line and of dichotomy radius relative to a subspace and obtain lower bounds for both of these radii. Furthermore, we determine an interval in which the dichotomy radius relative to an unstable subspace varies, expressed in terms of input–output operators. Using explicit examples, we show that the bounds and the dichotomy radii can be calculated, and in certain cases these radii may coincide. All the results are obtained in the most general framework, without imposing any additional hypotheses on the coefficients of the original or perturbed systems or on their propagators.