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Existence and non-existence of frequently hypercyclic subspaces for weighted shifts


Author: Quentin Menet
Journal: Proc. Amer. Math. Soc. 143 (2015), 2469-2477
MSC (2010): Primary 47A16
DOI: https://doi.org/10.1090/S0002-9939-2015-12444-6
Published electronically: January 16, 2015
MathSciNet review: 3326029
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the existence and the non-existence of frequently hypercyclic subspaces of frequently hypercyclic operators living on Banach spaces. In particular, we give an example of a weighted shift on $ l^p$ possessing a frequently hypercyclic subspace and an example of a frequently hypercyclic weighted shift on $ l^p$ possessing a hypercyclic subspace but no frequently hypercyclic subspace. The latter example allows us to answer positively Problem 1 posed by Bonilla and Grosse-Erdmann in [Monatsh. Math. 168 (2012)].


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  • [1] Frédéric Bayart and Sophie Grivaux, Hypercyclicité: le rôle du spectre ponctuel unimodulaire, C. R. Math. Acad. Sci. Paris 338 (2004), no. 9, 703-708 (French, with English and French summaries). MR 2065378 (2005c:47009), https://doi.org/10.1016/j.crma.2004.02.012
  • [2] Frédéric Bayart and Sophie Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5083-5117 (electronic). MR 2231886 (2007e:47013), https://doi.org/10.1090/S0002-9947-06-04019-0
  • [3] Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318 (2010m:47001)
  • [4] F. Bayart and I. Z. Ruzsa, Difference sets and frequently hypercyclic weighted shifts, Ergod. Th. Dynam. Sys. (2013), available on CJO2013. DOI 10.107/edts.2013.77.
  • [5] A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 383-404. MR 2308137 (2008c:47016), https://doi.org/10.1017/S014338570600085X
  • [6] A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic subspaces, Monatsh. Math. 168 (2012), no. 3-4, 305-320. MR 2993952, https://doi.org/10.1007/s00605-011-0369-2
  • [7] Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812
  • [8] Manuel González, Fernando León-Saavedra, and Alfonso Montes-Rodríguez, Semi-Fredholm theory: hypercyclic and supercyclic subspaces, Proc. London Math. Soc. (3) 81 (2000), no. 1, 169-189. MR 1757050 (2001g:47013), https://doi.org/10.1112/S0024611500012454
  • [9] Fernando León-Saavedra and Vladimír Müller, Hypercyclic sequences of operators, Studia Math. 175 (2006), no. 1, 1-18. MR 2261697 (2007g:47012), https://doi.org/10.4064/sm175-1-1
  • [10] Quentin Menet, Hypercyclic subspaces and weighted shifts, Adv. Math. 255 (2014), 305-337. MR 3167484, https://doi.org/10.1016/j.aim.2014.01.012

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Additional Information

Quentin Menet
Affiliation: Institut de Mathématique, Université de Mons, 20 Place du Parc, 7000 Mons, Belgique
Email: Quentin.Menet@umons.ac.be

DOI: https://doi.org/10.1090/S0002-9939-2015-12444-6
Keywords: Hypercyclicity, frequent hypercyclicity, hypercyclic subspaces, weighted shifts
Received by editor(s): September 30, 2013
Received by editor(s) in revised form: January 7, 2014
Published electronically: January 16, 2015
Additional Notes: The author was supported by a grant of FRIA
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2015 American Mathematical Society

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