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Hereditarily frequently hypercyclic operators and disjoint frequent hypercyclicity

Part of: General theory of linear operators Topological dynamics Ergodic theory

Published online by Cambridge University Press:  16 April 2025

FRÉDÉRIC BAYART Open the ORCID record for FRÉDÉRIC BAYART [Opens in a new window] ,
SOPHIE GRIVAUX Open the ORCID record for SOPHIE GRIVAUX [Opens in a new window] ,
ÉTIENNE MATHERON Open the ORCID record for ÉTIENNE MATHERON [Opens in a new window]  and
QUENTIN MENET Open the ORCID record for QUENTIN MENET [Opens in a new window]
FRÉDÉRIC BAYART
Affiliation:
Laboratoire de Mathématiques Blaise Pascal UMR 6620 CNRS, Université Clermont Auvergne, Campus universitaire des Cézeaux, 3 place Vasarely, 63178 Aubière Cedex, France (e-mail: [email protected])
SOPHIE GRIVAUX
Affiliation:
Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France (e-mail: [email protected])
ÉTIENNE MATHERON
Affiliation:
Univ. Artois, UR 2462 - Laboratoire de Mathématiques de Lens (LML) F-62300 Lens, France (e-mail: [email protected])
QUENTIN MENET*
Affiliation:
Service d’Analyse fonctionnelle, Département de Mathématique, Université de Mons, Place du Parc 20, 7000 Mons, Belgium
*
e-mail: [email protected]
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Abstract

We introduce and study the notion of hereditary frequent hypercyclicity, which is a reinforcement of the well-known concept of frequent hypercyclicity. This notion is useful for the study of the dynamical properties of direct sums of operators; in particular, a basic observation is that the direct sum of a hereditarily frequently hypercyclic operator with any frequently hypercyclic operator is frequently hypercyclic. Among other results, we show that operators satisfying the frequent hypercyclicity criterion are hereditarily frequently hypercyclic, as well as a large class of operators whose unimodular eigenvectors are spanning with respect to the Lebesgue measure. However, we exhibit two frequently hypercyclic weighted shifts $B_w,B_{w'}$ on $c_0(\mathbb {Z}_+)$ whose direct sum ${B_w\oplus B_{w'}}$ is not $\mathcal {U}$-frequently hypercyclic (so that neither of them is hereditarily frequently hypercyclic), and we construct a C-type operator on $\ell _p(\mathbb {Z}_+)$, $1\le p<\infty $, which is frequently hypercyclic but not hereditarily frequently hypercyclic. We also solve several problems concerning disjoint frequent hypercyclicity: we show that for every $N\in \mathbb {N}$, any disjoint frequently hypercyclic N-tuple of operators $(T_1,\ldots ,T_N)$ can be extended to a disjoint frequently hypercyclic $(N+1)$-tuple $(T_1,\ldots ,T_N, T_{N+1})$ as soon as the underlying space supports a hereditarily frequently hypercyclic operator; we construct a disjoint frequently hypercyclic pair which is not densely disjoint hypercyclic; and we show that the pair $(D,\tau _a)$ is disjoint frequently hypercyclic, where D is the derivation operator acting on the space of entire functions and $\tau _a$ is the operator of translation by $a\in \mathbb {C}\setminus \{ 0\}$. Part of our results are in fact obtained in the general setting of Furstenberg families.

Keywords

frequent hypercyclicityFurstenberg familiescountable Lebesgue spectrumdisjointness

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators 37A30: Ergodic theorems, spectral theory, Markov operators
Secondary: 37B20: Notions of recurrence
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 52
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© The Author(s), 2025. Published by Cambridge University Press

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