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Linear dynamical systems on Hilbert spaces: typical properties and explicit examples
About this Title
S. Grivaux, CNRS, Laboratoire Paul Painlevé, UMR 8524, Université de Lille, Cité Scientifique, Bâtiment M2, 59655 Villeneuve d’Ascq Cedex, France, É. Matheron, Laboratoire de Mathématiques de Lens, Université d’Artois, Rue Jean Souvraz SP 18, 62307 Lens, France and Q. Menet, Service de Probabilité et Statistique, Département de Mathématique, Université de Mons, Place du Parc 20, 7000 Mons, Belgium
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 269, Number 1315
ISBNs: 978-1-4704-4663-5 (print); 978-1-4704-6468-4 (online)
DOI: https://doi.org/10.1090/memo/1315
Published electronically: March 31, 2021
Keywords: Linear dynamical systems; Hilbert spaces; Baire category; frequent and $\mathcal U$-frequent hypercyclicity; ergodicity; chaos; topological mixing.
MSC: Primary 47A16, 37A05, 54E52, 54H05
Table of Contents
Chapters
- 1. Introduction
- 2. Typical properties of hypercyclic operators
- 3. Descriptive set-theoretic issues
- 4. Ergodicity for upper-triangular operators
- 5. Periodic points at the service of hypercyclicity
- 6. Operators of C-type and of C$_{+}$-type
- 7. Explicit counterexamples
- 8. A few questions
- Short list of abbreviations
Abstract
We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results.
-
A typical hypercyclic operator is not topologically mixing, has no eigenvalues and admits no non-trivial invariant measure, but is densely distributionally chaotic.
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A typical upper-triangular operator with coefficients of modulus $1$ on the diagonal is ergodic in the Gaussian sense, whereas a typical operator of the form “diagonal with coefficients of modulus $1$ on the diagonal plus backward unilateral weighted shift” is ergodic but has only countably many unimodular eigenvalues; in particular, it is ergodic but not ergodic in the Gaussian sense.
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There exist Hilbert space operators which are chaotic and $\mathcal U$-frequently hypercyclic but not frequently hypercyclic, Hilbert space operators which are chaotic and frequently hypercyclic but not ergodic, and Hilbert space operators which are chaotic and topologically mixing but not $\mathcal U$-frequently hypercyclic.
We complement our results by investigating the descriptive complexity of some natural classes of operators defined by dynamical properties.
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