Disjoint hypercyclic Toeplitz operators

Abstract

The aim of this work is to describe new classes of disjoint hypercyclic Toeplitz operators on the Hardy space \(H^2({\mathbb {D}})\) in the unit disc \({\mathbb {D}}\). We examine the disjoint hypercyclicity of the coanalytic Toeplitz operators, the Toeplitz operators with the symbols \(a{\bar{z}}+b+cz\), where \(a,b,c\in {\mathbb {C}}\), and the Toeplitz operators with the symbols \(p(\bar{z})+\varphi (z)\), where p is a polynomial and \(\varphi \in H^\infty (\mathbb {D})\). The hypercyclicity of these classes of Toeplitz operators has been characterized by G. Godefroy and J. Shapiro (J. Funct. Anal., 98, 1991), S. Shkarin (arXiv:1210.3191v1, 2012), and A. Baranov and L. Lishanskii (Results Math., 70, 2016), respectively. Based on their results, we first provide a criterion for the bounded linear operators to be disjoint hypercyclic. Using this criterion, we then establish certain conditions under which the aforementioned classes of Toeplitz operators are disjoint hypercyclic in terms of their symbols.