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(Non-)weakly mixing operators and hypercyclicity sets. (English) Zbl 1178.47003

Let \(X\) be an infinite-dimensional separable Banach space. For any \(x\in X\) and any nonempty open set \(V\subset X\), define the set \(N(x,V):=\{n\in\mathbb{N}: T^n(x)\in V\}\). An operator \(T\in L(X)\) is called hypercyclic if there exists \(x\in X\) such that \(N(x,V)\neq\emptyset\) for any nonempty open subset \(V\) of \(X\). Computing the frequency in which every nonempty open subset is visited by a hypercyclic vector \(x\), F. Bayart and S. Grivaux introduced the notion of frequent hypercyclicity [Trans. Am. Math. Soc. 358, No. 11, 5083–5117 (2006; Zbl 1115.47005)]. A hypercyclic operator \(T\in L(X)\) is said to be frequently hypercyclic if there is a hypercyclic vector \(x\in X\) such that for every nonempty open set \(V\) of \(X\), the set \(N(x,V)\) can be enumerated as an increasing sequence \((n_k)\) with \(n_k=\mathcal{O}(k)\) as \(k\rightarrow\infty\).
An operator \(T\) is said to be weakly mixing if \(T\oplus T\) is hypercyclic on \(X\oplus X\). K.-G. Grosse-Erdmann and A. Peris proved that every frequently hypercyclic operator is weakly mixing [C. R., Math., Acad. Sci. Paris 341, No. 2, 123–128 (2005; Zbl 1068.47012)]. On the other hand, De la Rosa and Read have solved a long-standing question posed by D. A. Herrero [J. Funct. Anal. 99, No. 1, 179–190 (1991; Zbl 0758.47016)], proving that there are hypercyclic operators which are not weakly mixing [M. De La Rosa and C. Read, J. Oper. Theory 61, No. 2, 369–380 (2009; Zbl 1193.47014); see also J. Funct. Anal. 250, No. 2, 426–441 (2007; Zbl 1131.47006)]. In fact, this question results to be equivalent to see wheter every hypercyclic operator verifies the Hypercyclicity Criterion [J. Bès and A. Peris, J. Funct. Anal. 167, No. 1, 94–112 (1999; Zbl 0941.47002)].
In the paper under review, the authors make a deep analysis of the constructions shown in [Zbl 1193.47014] and in [Zbl 1131.47006] in order to quantify how big the sets \(N(x,V)\) can be for a weakly mixing operator, and also for a hypercyclic operator which is not weakly mixing. In these constructions, the arithmetic properties of the sets \(N(x,V)\) play an important role, such as Sidon-type properties. The authors also present examples in the Fréchet space setting.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
37B99 Topological dynamics
11B99 Sequences and sets
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References:

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