Hypercyclicity of Toeplitz operators with smooth symbols

TL;DR

This paper investigates the hypercyclicity of Toeplitz operators with smooth symbols on Hardy spaces, providing new conditions and extending previous results on their dynamical properties.

Contribution

It offers new necessary and sufficient conditions for hypercyclicity of Toeplitz operators with smooth symbols, extending prior work in the field.

Findings

Established criteria for hypercyclicity of Toeplitz operators with smooth symbols.

Extended previous results by Baranov-Lishanskii and Abakumov-Baranov-Charpentier.

Analyzed additional dynamical properties of these operators.

Abstract

This paper is devoted to the study of the dynamics of Toeplitz operators $T_F$ with smooth symbols $F$ on the Hardy spaces of the unit disk $H^p$, $p>1$. Building on a model theory for Toeplitz operators on $H^2$ developed by Yakubovich in the 90's, we carry out an in-depth study of hypercyclicity properties of such operators. Under some rather general smoothness assumptions on the symbol, we provide some necessary/sufficient/necessary and sufficient conditions for $T_F$ to be hypercyclic on $H^p$. In particular, we extend previous results on the subject by Baranov-Lishanskii and Abakumov-Baranov-Charpentier-Lishanskii. We also study some other dynamical properties for this class of operators.