Cocycles for equations with infinite delay and hyperbolicity

We show that the hyperbolicity of a linear delay-difference equation with infinite delay, expressed in terms of the existence of an exponential dichotomy, can be completely characterized by the hyperbolicity of a linear cocycle obtained from the solutions of the equation. As an application of this characterization, we obtain several consequences: the extension of hyperbolicity to all equations in the invariant hull; the robustness of the existence of hyperbolicity for all equations in this hull under sufficiently small linear perturbations; the equality of all spectra in the invariant hull; and a characterization of hyperbolicity for all equations in the invariant hull in terms of an admissibility property taking bounded perturbations to bounded solutions.