Hypercyclic Toeplitz operators

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 ${\rm deg}\, p =1$.