Topologically mixing hypercyclic operators

By George Costakis, Mart'in Sambarino. 2003

96 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/ffb20d12e8e2ff14f4400c38a6fee8a28a70525b

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Distributional chaos for operators with full scrambled sets (2013, background): “The fact that B is mixing can be deduced from [6] (see also Chapter 4 in [10] for more details), since limj vj = 0.”
Frequently hypercyclic operators and vectors (2007, background): “Thus, every topologically mixing operator is hypercyclic, but the converse is not true (see [8]).”
Linear chaos and frequent hypercyclicity (2014, background): “ws from Theorem 1.1 and each of the other implications is obvious by deﬁnition. Moreover, there are no other implications since there exist: • a weakly mixing weighted shift on ℓp which is not mixing [18], • a chaotic operator which is not mixing [2], • a frequently hypercyclic weighted shift on c0 ”
Disjoint mixing operators (2012, background): “The point of such operators is that they are all mixing [15].”
Recurrence properties of hypercyclic operators (2014, background): “Indeed, we know that a weighted shift Bw on (Z+) is mixing if and only if ∏∞ k=1 |wk | tends to infinity [13].”
Szemeredi's theorem, frequent hypercyclicity and multiple recurrence (2010, background): “recurrent but not frequently hypercyclic. Proof. Consider the unilateral weighted shift T on l2(N) with weight sequence w q n = n+1 n , n = 1,2,.... Then Qn i=1 wi = √ n+1 → +∞. By the main result of [13] it follows that T is topologically mixing and thus by Corollary 4.8 the operator T is topologic”
Expansivity and Shadowing in Linear Dynamics (2016, background): “For instance, Salas [30] characterized hypercyclicity and weak mixing whereas Costakis and Sambarino [12] characterized mixing for unilateral and bilateral weighted shifts on ℓ2(N) and ℓ2(Z), respectively.”