  We introduce and study the notions of (generalized) hyperbolicity,
topological stability and (uniform) topological expansivity for operators on
locally convex spaces. We prove that every generalized hyperbolic operator on a
locally convex space has the finite shadowing property. Contrary to what
happens in the Banach space setting, hyperbolic operators on Fr\'echet spaces
may fail to have the shadowing property, but we find additional conditions that
ensure the validity of the shadowing property. Assuming that the space is
sequentially complete, we prove that generalized hyperbolicity implies the
strict periodic shadowing property, but we also show that the hypothesis of
sequential completeness is essential. We show that operators with the periodic
shadowing property on topological vector spaces have other interesting
dynamical behaviors, including the fact that the restriction of such an
operator to its chain recurrent set is topologically mixing and Devaney
chaotic. We prove that topologically stable operators on locally convex spaces
have the finite shadowing property and the strict periodic shadowing property.
As a consequence, topologically stable operators on Banach spaces have the
shadowing property. Moreover, we prove that generalized hyperbolicity implies
topological stability for operators on Banach spaces. We prove that uniformly
topologically expansive operators on locally convex spaces are neither Li-Yorke
chaotic nor topologically transitive. Finally, we characterize the notion of
topological expansivity for weighted shifts on Fr\'echet sequence spaces.
Several examples are provided.
