Input–output admissibility and exponential trichotomy of difference equations

We propose a new method for the study of the asymptotic behavior of difference equations in infinite-dimensional spaces, providing characterizations for the property of uniform exponential trichotomy. We deduce the structure of the stable, unstable and bounded subspace and prove the uniqueness of the projection families. We introduce a new admissibility concept with respect to a discrete input–output system and prove that this is a necessary and sufficient condition for the existence of uniform exponential trichotomy. Throughout the paper, there is no assumption on the coefficients and the obtained results are applicable to any class of difference equations.