Point spectrum and hypercyclicity problem for a class of truncated Toeplitz operators

In this note we discuss an open problem whether a truncated Toeplitz operator on a model space can be hypercyclic. We compute point spectrum and eigenfunctions for a class of truncated Toeplitz operators with polynomial analytic and antianalytic parts. We show that, for a class of model spaces, truncated Toeplitz operators with symbols of the form $\Phi(z) =a \bar{z} +b + cz$, $|a| \ne |c|$, have complete sets of eigenvectors, and, in particular, are not hypercyclic.