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

Abstract

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.

13 pages. Section 3 is substantially expanded