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

TL;DR

This paper investigates the spectral properties of certain truncated Toeplitz operators on model spaces, showing that some classes are not hypercyclic due to their complete eigenvector sets.

Contribution

It computes eigenfunctions and spectra for a class of truncated Toeplitz operators and establishes conditions under which these operators are not hypercyclic.

Findings

Operators have complete eigenvector sets

Operators with specific symbols are not hypercyclic

Spectral properties depend on symbol parameters

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.