Frequent hypercyclicity and chaoticity of Toeplitz operators and their tensor products

Abstract

In this paper, we investigate the frequent hypercyclicity of the Toeplitz operator $T_{\Phi }$ and their tensor products on the Hardy space with the symbol of the form $\Phi (z)=p\big (\frac{1}{z}\big )+\varphi (z)$, where p is a polynomial and $\varphi \in H^{\infty }$. We also give some sufficient conditions for $T_{\Phi _{1}}\otimes T_{\Phi _{2}}$ to be hypercyclic and construct an example such that neither $T_{\Phi _{1}}$ nor $T_{\Phi _{2}}$ is hypercyclic, but $T_{\Phi _{1}}\otimes T_{\Phi _{2}}$ is frequently hypercyclic. Moreover, we also characterize the mixing property and chaoticity of $T_{\Phi }$ and $T_{\Phi _{1}}\otimes T_{\Phi _{2}}$.