Dynamics of shift operators on non-metrizable sequence spaces

TL;DR

This paper studies the dynamical behavior of shift operators on complex non-metrizable sequence spaces, providing characterizations and examples including an operator from quantum mechanics.

Contribution

It offers new characterizations of dynamical properties for weighted backward shifts on LF-spaces and analyzes the quantum mechanical annihilation operator.

Findings

Backward shift operators can be topologically transitive, hypercyclic, and chaotic on LF-spaces.

Weighted generalized backward shifts' properties are characterized by their weight sequences.

The quantum mechanical annihilation operator exhibits mixing, hypercyclicity, chaos, and ergodicity.

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 K\"othe 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 $\mathscr{S}'(\mathbb{R})$.