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

Abstract

Truncated Toeplitz operators are restrictions of usual Toeplitz operators onto model subspaces $K_\theta =H^2 \ominus \theta H^2$ of the Hardy space $H^2$, where $\theta$ is an inner function. In this note we study the structure of eigenvectors for a class of truncated Toeplitz operators and discuss an open problem whether a truncated Toeplitz operator on a model space can be hypercyclic, that is, whether there exists a vector with a dense orbit. For the classical Toeplitz operators on $H^2$ with antianalytic symbols a hypercyclicity criterion was given by G. Godefroy and J. Shapiro, while for Toeplitz operators with polynomial or rational antianalytic part some partial answers were obtained by the authors jointly with E. Abakumov and S. Charpentier. We find point spectrum and eigenfunctions for a class of truncated Toeplitz operators with polynomial analytic and antianalytic parts. It is shown that the eigenvectors are linear combinations of reproducing kernels at some points such that the values of the inner function $\theta$ at these points have a polynomial dependence. Next we show that, for a class of model spaces, truncated Toeplitz operators with symbols of the form $\Phi(z) =a \bar{z} +b + cz$, where $|a| \ne |c|$, have complete sets of eigenvectors and, in particular, are not hypercyclic. Our main tool here is the factorization of functions in an associated Hardy space in an annulus. We also formulate several open problems.