On Uniform Dichotomies for the Growth Rates of Linear Discrete-time Dynamical Systems in Banach Spaces

The aim of the present paper is to give some characterizations for the growth rates of linear discrete-time dynamical systems in Banach spaces. More precisely, necessary and sufficient conditions of Datko–Zabczyk type as well as characterizations using Lyapunov functions are given using both invariant and strongly invariant projection sequences. Also, as consequences we obtain characterizations for the uniform exponential dichotomy behavior.