## (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

### Keywords:

hypercyclic operators; weak mixing; Sidon sequences
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### References:

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