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Chaos and frequent hypercyclicity for weighted shifts

Part of: General theory of linear operators Special classes of linear operators Topological linear spaces and related structures

Published online by Cambridge University Press:  28 December 2020

STÉPHANE CHARPENTIER ,
KARL GROSSE-ERDMANN Open the ORCID record for KARL GROSSE-ERDMANN [Opens in a new window]  and
QUENTIN MENET
STÉPHANE CHARPENTIER
Affiliation:
Institut de Mathématiques de Marseille, UMR 7373, Aix-Marseille Université, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France (e-mail: stephane.charpentier.1@univ-amu.fr)
KARL GROSSE-ERDMANN*
Affiliation:
Département de Mathématique, Université de Mons, 20 Place du Parc, 7000Mons, Belgium (e-mail: quentin.menet@umons.ac.be)
QUENTIN MENET
Affiliation:
Département de Mathématique, Université de Mons, 20 Place du Parc, 7000Mons, Belgium (e-mail: quentin.menet@umons.ac.be)
*
e-mail: kg.grosse-erdmann@umons.ac.be
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Abstract

Bayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys.35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc.358 (2006), 5083–5117], who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . We first generalize the Bayart–Ruzsa theorem to all Banach sequence spaces in which the unit sequences form a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular, on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on $H(\mathbb {D})$ is chaotic, while $H(\mathbb {C})$ admits a non-chaotic frequently hypercyclic weighted shift.

Keywords

chaotic operatorfrequently hypercyclic operatorKöthe sequence spacepower series spaceweighted shift operator

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators
Secondary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 46A45: Sequence spaces (including Köthe sequence spaces)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3634 - 3670
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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