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Dynamics of shift operators on non-metrizable sequence spaces

José Bonet, Thomas Kalmes, Alfred Peris

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

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 in terms of the defining sequence of weights . We further discuss several examples and show that the annihilation operator from quantum mechanics is mixing, sequentially hypercyclic, chaotic, and topologically ergodic on .

Paper Structure

This paper contains 9 sections, 18 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. The backward shift on certain LF-spaces
  3. The backward shift on Köthe coechelon spaces
  4. Examples and open problems
  5. Weighted generalized backward shifts on duals of power series spaces of infinite type
  6. The annihilation operator on $\mathscr{S}'(\mathbb{R})$.
  7. Separating examples.
  8. Snake shift operators
  9. Open problems

Key Result

Proposition 2.3

Let $E=\text{ind}_m E_m$ be an LF-space with stepwise Schauder basis $(e_j)_{j\in\mathbb{N}}$. Then, for every $m\in\mathbb{N}$, on the Fréchet space $E_m$ there is an increasing fundamental sequence of seminorms $(p_k)_{k\in\mathbb{N}}$ satisfying

Theorems & Definitions (39)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • Proposition 2.7
  • ...and 29 more