Chain Recurrence and Positive Shadowing in Linear Dynamics

Abstract: We study shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even on normed vector spaces. We show that for linear operators there is only one chain recurrent set, and this set is actually a closed invariant subspace. We prove that every chain transitive linear dynamical system with the shadowing property is frequently hypercyclic, and as a corollary obtain that every shadowing hypercyclic linear dynamical system is frequently hypercyclic.