scholarly journals FREQUENTLY HYPERCYCLIC BILATERAL SHIFTS

2018 ◽  
Vol 61 (2) ◽  
pp. 271-286 ◽  
Author(s):  
KARL-G. GROSSE-ERDMANN
Keyword(s):  
Positive Answer ◽  
Weighted Shift ◽  
Sequence Spaces ◽  
Weighted Shifts ◽  
Banach Sequence Spaces ◽  
Bilateral Weighted Shift ◽  
Banach Sequence

AbstractIt is not known whether the inverse of a frequently hypercyclic bilateral weighted shift on c0(ℤ) is again frequently hypercyclic. We show that the corresponding problem for upper frequent hypercyclicity has a positive answer. We characterise, more generally, when bilateral weighted shifts on Banach sequence spaces are (upper) 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.

Weak orthogonality and weak property (β) in some banach sequence spaces

1999 ◽  
Vol 49 (2) ◽  
pp. 303-316 ◽  
Author(s):  
Yunan Cui ◽  
Henryk Hudzik ◽  
Ryszard Płuciennik
Keyword(s):  
Sequence Spaces ◽  
Banach Sequence Spaces ◽  
Weak Property ◽  
Banach Sequence

scholarly journals On the Dvoretzky-Rogers theorem

1984 ◽  
Vol 27 (2) ◽  
pp. 105-113
Author(s):  
Fuensanta Andreu
Keyword(s):  
Sequence Space ◽  
Normed Space ◽  
Sequence Spaces ◽  
Finite Dimensional ◽  
Banach Sequence Spaces ◽  
Perfect Sequence ◽  
Normal Topology ◽  
Banach Sequence

The classical Dvoretzky-Rogers theorem states that if E is a normed space for which l1(E)=l1{E} (or equivalently , then E is finite dimensional (see [12] p. 67). This property still holds for any lp (l<p<∞) in place of l1 (see [7]p. 104 and [2] Corollary 5.5). Recently it has been shown that this result remains true when one replaces l1 by any non nuclear perfect sequence space having the normal topology (see [14]). In this context, De Grande-De Kimpe [4] gives an extension of the Devoretzky-Rogers theorem for perfect Banach sequence spaces.

scholarly journals Banach-Saks Type and Gurariǐ Modulus of Convexity of Some Banach Sequence Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Henryk Hudzik ◽  
Vatan Karakaya ◽  
Mohammad Mursaleen ◽  
Necip Simsek
Keyword(s):  
Sequence Spaces ◽  
Modulus Of Convexity ◽  
Banach Sequence Spaces ◽  
Banach Sequence

Banach-Saks type is calculated for two types of Banach sequence spaces and Gurariǐ modulus of convexity is estimated from above for the spaces of one type among them.

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