Dynamics of shift operators on non-metrizable sequence spaces

Abstract

We investigate dynamical properties such as topological transitivity, (sequential) hypercyclicity, and chaos for backward shift operators associated to a Schauder basis on LF-spaces. As an application, we characterize these dynamical properties for weighted generalized backward shifts on Köthe coechelon sequence spaces $k_p((v^{(m)}{)}_{m\in \mathbb{N}})$ in terms of the defining sequence of weights $(v^{(m)}{)}_{m\in \mathbb{N}}$. We further discuss several examples and show that the annihilation operator from quantum mechanics is mixing, sequentially hypercyclic, chaotic, and topologically ergodic on ${\mathcal{S}}^{\prime }(\mathbb{R})$.