scholarly journals Maximal distributional chaos of weighted shift operators on Köthe sequence spaces

2014 ◽  
Vol 64 (1) ◽  
pp. 105-114 ◽  
Author(s):  
Xinxing Wu
Keyword(s):  
Weighted Shift ◽  
Sequence Spaces ◽  
Distributional Chaos ◽  
Shift Operators ◽  
Köthe Sequence Spaces

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.

scholarly journals Uniform distributional chaos for weighted shift operators

2013 ◽  
Vol 26 (1) ◽  
pp. 130-133 ◽  
Author(s):  
Xinxing Wu ◽  
Peiyong Zhu ◽  
Tianxiu Lu
Keyword(s):  
Weighted Shift ◽  
Distributional Chaos ◽  
Shift Operators

CHAOS IN A CLASS OF NONCONSTANT WEIGHTED SHIFT OPERATORS

2013 ◽  
Vol 23 (01) ◽  
pp. 1350010 ◽  
Author(s):  
XINXING WU ◽  
PEIYONG ZHU
Keyword(s):  
Positive Integer ◽  
Linear Space ◽  
Normed Linear Space ◽  
Shift Operator ◽  
Weighted Shift ◽  
Constructive Proof ◽  
Sufficient Condition ◽  
Weighted Shift Operator ◽  
Shift Operators ◽  
Devaney Chaos

In this paper, chaos generated by a class of nonconstant weighted shift operators is studied. First, we prove that for the weighted shift operator Bμ : Σ(X) → Σ(X) defined by Bμ(x0, x1, …) = (μ(0)x1, μ(1)x2, …), where X is a normed linear space (not necessarily complete), weak mix, transitivity (hypercyclity) and Devaney chaos are all equivalent to separability of X and this property is preserved under iterations. Then we get that [Formula: see text] is distributionally chaotic and Li–Yorke sensitive for each positive integer N. Meanwhile, a sufficient condition ensuring that a point is k-scrambled for all integers k > 0 is obtained. By using these results, a simple example is given to show that Corollary 3.3 in [Fu & You, 2009] does not hold. Besides, it is proved that the constructive proof of Theorem 4.3 in [Fu & You, 2009] is not correct.

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