Hypercyclic Toeplitz operators

TL;DR

This paper investigates the conditions under which Toeplitz operators with specific symbols exhibit hypercyclicity in the Hardy space, providing nearly complete characterizations especially for linear polynomial symbols.

Contribution

It offers new necessary and sufficient conditions for hypercyclicity of Toeplitz operators with symbols combining polynomials and bounded analytic functions, advancing understanding in operator theory.

Findings

Characterization of hypercyclicity for Toeplitz operators with polynomial plus bounded analytic symbols

Nearly complete criteria for hypercyclicity when the polynomial degree is one

Conditions that unify necessary and sufficient criteria in specific cases

Abstract

We study hypercyclicity of the Toeplitz operators in the Hardy space $H^2(\mathbb{D})$ with symbols of the form $p(\bar{z}) +\phi(z)$, where $p$ is a polynomial and $\phi \in H^\infty(\mathbb{D})$. We find both necessary and sufficient conditions for hypercyclicity which almost coincide in the case when ${\rm deg}\, p =1$.