Roughness of tempered exponential dichotomies for infinite-dimensional random difference equations

In this paper we study the roughness of tempered exponential dichotomies for linear random dynamical systems in Banach spaces. Such a dichotomy has a tempered bound and describes nonuniform hyperbolicity. We prove the roughness without assuming their invertibility and the integrability condition of the Multiplicative Ergodic Theorem. We give an explicit bound for the linear perturbation such that the dichotomy is persistent. We also obtain explicit forms for the exponent and the bound of tempered exponential dichotomy of the perturbed random system in terms of the original ones and the perturbations.