Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-12-01T00:27:41.666Z Has data issue: false hasContentIssue false
  • English
  • Français

INVERSE OF FREQUENTLY HYPERCYCLIC OPERATORS

Part of: General theory of linear operators

Published online by Cambridge University Press:  04 February 2021

Quentin Menet Open the ORCID record for Quentin Menet [Opens in a new window]
Quentin Menet*
Affiliation:
Quentin Menet, Département de Mathématique, Université de Mons, 20 Place du Parc, 7000Mons, Belgique ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that there exists an invertible frequently hypercyclic operator on $\ell ^1(\mathbb {N})$ whose inverse is not frequently hypercyclic.

Keywords

frequent hypercyclicityoperator of C-typeinvertible operator

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 1867 - 1886
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ansari, S. I., Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), 384–390.CrossRefGoogle Scholar
Bayart, F. and Grivaux, S., Hypercyclicité: le r $\hat{\mathrm{o}}$ le du spectre ponctuel unimodulaire [Hypercyclicity: the role of the unimodular point spectrum], C. R. Math. Acad. Sci. Paris 338 (2004), 703–708.CrossRefGoogle Scholar
Bayart, F. and Grivaux, S., Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358 (2006), 5083–5117.CrossRefGoogle Scholar
Bayart, F. and Grivaux, S., Invariant Gaussian measures for operators on Banach spaces and linear dynamics, Proc. Lond. Math. Soc. (3) 94 (2007), 181–210.CrossRefGoogle Scholar
Bayart, F. and Matheron, É., Dynamics of Linear Operators, Cambridge Tracts in Mathematics No. 179 (Cambridge University Press, New York, USA, 2009).CrossRefGoogle Scholar
Bayart, F. and Ruzsa, I., Difference sets and frequently hypercyclic weighted shifts, Ergod. Theor. Dyn. Syst. 35 (2015), 691–709.CrossRefGoogle Scholar
Bernal-gonzález, L., On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), 1003–1010.CrossRefGoogle Scholar
Bonilla, A. and Grosse-erdmann, K.-G., Frequently hypercyclic operators and vectors, Ergod. Theor. Dyn. Syst. 27 (2007), 383–404. Erratum: Ergod. Theor. Dyn. Syst. 29 (2009), 1993–1994.CrossRefGoogle Scholar
Grivaux, S., Matheron, E. and Menet, Q., Linear dynamical systems on Hilbert spaces: typical properties and explicit examples, Mem. Amer. Math. Soc., to appear.Google Scholar
Grosse-erdmann, K.-G., Frequently hypercyclic bilateral shifts, Glas. Math. J. 61 (2019), 271–286. doi: https://doi.org/10.1016/j.jfa.2020.108543 CrossRefGoogle Scholar
Grosse-erdmann, K.-G. and Peris, A., Linear Chaos (Universitext) (Springer, London, 2011).CrossRefGoogle Scholar
Guirao, A. J., Montesinos, V. and Zizler, V., Open Problems in the Geometry and Analysis of Banach Spaces (Springer, Springer: New York, USA, 2016).CrossRefGoogle Scholar
Menet, Q., Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc. 369 (2017), 4977–4994.CrossRefGoogle Scholar
Menet, Q., Inverse of $\mathbf{\mathcal{U}}$ -frequently hypercyclic operators, J. Funct. Anal. 279 (2020).CrossRefGoogle Scholar
Shkarin, S., On the spectrum of frequently hypercyclic operators, Proc. Amer. Math. Soc. 137 (2009), 123–134.CrossRefGoogle Scholar