  In the early 1970's Eisenberg and Hedlund investigated relationships between
expansivity and spectrum of operators on Banach spaces. In this paper we
establish relationships between notions of expansivity and hypercyclicity,
supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space
is $c_0$ or $\ell_p$ ($1 \leq p < \infty$), we give complete characterizations
of weighted shifts which satisfy various notions of expansivity. We also
establish new relationships between notions of expansivity and spectrum.
Moreover, we study various notions of shadowing for operators on Banach spaces.
In particular, we solve a basic problem in linear dynamics by proving the
existence of nonhyperbolic invertible operators with the shadowing property.
This also contrasts with the expected results for nonlinear dynamics on compact
manifolds, illuminating the richness of dynamics of infinite dimensional linear
operators.
