scholarly journals $q$-Frequently hypercyclic operators

2015 ◽  
Vol 9 (2) ◽  
pp. 114-126 ◽  
Author(s):  
Manjul Gupta ◽  
Aneesh Mundayadan
Keyword(s):  
Hypercyclic Operators ◽  
Frequently Hypercyclic Operators

scholarly journals Frequently hypercyclic operators

2006 ◽  
Vol 358 (11) ◽  
pp. 5083-5117 ◽  
Author(s):  
Frédéric Bayart ◽  
Sophie Grivaux
Keyword(s):  
Hypercyclic Operators ◽  
Frequently Hypercyclic Operators

scholarly journals Difference sets and frequently hypercyclic weighted shifts

2013 ◽  
Vol 35 (3) ◽  
pp. 691-709 ◽  
Author(s):  
FRÉDÉRIC BAYART ◽  
IMRE Z. RUZSA
Keyword(s):  
Difference Sets ◽  
Difference Set ◽  
Hypercyclic Operators ◽  
Weighted Shifts ◽  
Frequently Hypercyclic Operators ◽  
Upper Density ◽  
The Difference ◽  
Positive Integers

AbstractWe solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on${\ell }^{p} ( \mathbb{Z} )$,$p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is$ \mathcal{U} $-frequently hypercyclic but not frequently hypercyclic, and that there exists an operator which is frequently hypercyclic but not distributionally chaotic. These (surprising) counterexamples are given by weighted shifts on${c}_{0} $. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.

scholarly journals INVERSE OF FREQUENTLY HYPERCYCLIC OPERATORS

2021 ◽  
pp. 1-20
Author(s):  
Quentin Menet
Keyword(s):  
Hypercyclic Operator ◽  
Hypercyclic Operators ◽  
Frequently Hypercyclic Operator ◽  
Frequently Hypercyclic Operators

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

scholarly journals Chaos and frequent hypercyclicity for weighted shifts

2020 ◽  
pp. 1-37
Author(s):  
STÉPHANE CHARPENTIER ◽  
KARL GROSSE-ERDMANN ◽  
QUENTIN MENET
Keyword(s):  
Difference Sets ◽  
Weighted Shift ◽  
Sequence Spaces ◽  
Hypercyclic Operators ◽  
Weighted Shifts ◽  
Banach Sequence Spaces ◽  
Frequently Hypercyclic Operators ◽  
Köthe Sequence Spaces ◽  
The Relationship ◽  
Banach Sequence

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.

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