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Ufa Mathematical Journal, 2023, Volume 15, Issue 3, Pages 106–117
DOI: https://doi.org/10.13108/2023-15-3-106
(Mi ufa668)
 

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

A. D. Baranova, A. A. Lishanskiib

a Department of Mathematics and Mechanics, St. Petersburg State University, 28, Universitetskii prosp., St. Petersburg, 198504, Russia
b St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg, 199034, Russia
References:
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.
Keywords: Hypercyclic operator, Toeplitz operator, model space, truncated Toeplitz operator.
Funding agency Grant number
Russian Science Foundation 19-11-00058P
Ministry of Science and Higher Education of the Russian Federation 075-15-2022-287
The work of A. Lishanskii in Section 2 was performed at the Saint Petersburg Leonhard Euler International Mathematical Institute and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2022-287). The results of Section 3 were obtained with the support of the Russian Science Foundation project 19-11-00058P.
Received: 03.11.2022
Document Type: Article
UDC: 517.958
MSC: 47A16, 47B35, 30H10
Language: English
Original paper language: English
Citation: A. D. Baranov, A. A. Lishanskii, “Point spectrum and hypercyclicity problem for a class of truncated Toeplitz operators”, Ufa Math. J., 15:3 (2023), 106–117
Citation in format AMSBIB
\Bibitem{BarLis23}
\by A.~D.~Baranov, A.~A.~Lishanskii
\paper Point spectrum and hypercyclicity problem for a class of truncated Toeplitz operators
\jour Ufa Math. J.
\yr 2023
\vol 15
\issue 3
\pages 106--117
\mathnet{http://mi.mathnet.ru//eng/ufa668}
\crossref{https://doi.org/10.13108/2023-15-3-106}
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