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In this note we discuss an open problem whether a truncated Toeplitz operatoron a model space can be hypercyclic. We compute point spectrum andeigenfunctions for a class of truncated Toeplitz operators with polynomialanalytic 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, arenot hypercyclic.

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