  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.
