Dynamical properties of weighted shifts on sequence spaces

Motivated by three recent open questions in the study of linear dynamics, we study weighted shifts on sequence spaces. First, we provide an example of a weighted shift on a locally convex space whose topology is generated by a sequence of complete seminorms which is generalized hyperbolic, but does not have the shadowing property. Next, we characterise uniform topological expansivity on Fréchet spaces satisfying some very natural conditions. Finally, we study the periodic shadowing property on normed spaces leading to a condition formulated purely in terms of weights which we show is necessary for the periodic shadowing property on $\ell_p$ and equivalent on $c_0$.