Asymptotic Expansive and Hyperbolic Linear Operators

Abstract

We study the topological dynamical notion of asymptotic expansivity for linear operators. We also introduce two new notions, namely strong asymptotic expansivity and super asymptotic expansivity. We show that all three notions are equivalent to hyperbolicity for linear operators on finite dimensional spaces. In case of infinite-dimension, we construct examples to show that asymptotic expansivity is weaker than strong asymptotic expansivity which is further weaker than super asymptotic expansivity. More interestingly, we prove that super asymptotic expansivity is equivalent to the notion of hyperbolicity. Then, we prove that the super asymptotically expansive linear operators is a dense class in strong operator topology. Finally, we show elegant ways to construct variants of asymptotically expansive linear operators.