  In the present work we study the concepts of shadowing and chain recurrence
in the setting of linear dynamics. We prove that shadowing and finite shadowing
always coincide for operators on Banach spaces, but we exhibit operators on the
Fr\'echet space $H(\mathbb{C})$ of entire functions that have the finite
shadowing property but do not have the shadowing property. We establish a
characterization of mixing for continuous maps with the finite shadowing
property in the setting of uniform spaces, which implies that chain recurrence
and mixing coincide for operators with the finite shadowing property on any
topological vector space. We establish a characterization of dense
distributional chaos for operators with the finite shadowing property on
Fr\'echet spaces. As a consequence, we prove that if a Devaney chaotic (resp. a
chain recurrent) operator on a Fr\'echet space (resp. on a Banach space) has
the finite shadowing property, then it is densely distributionally chaotic. We
obtain complete characterizations of chain recurrence for weighted shifts on
Fr\'echet sequence spaces. We prove that generalized hyperbolicity implies
periodic shadowing for operators on Banach spaces. Moreover, the concepts of
shadowing and periodic shadowing coincide for unilateral weighted backward
shifts, but these notions do not coincide in general, even for bilateral
weighted shifts.
