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Frequently hypercyclic operators and vectors. (English) Zbl 1119.47011

Let $X$ be an $ℱ$ space, i.e., a topological vector space whose topology is defined by an invariant metric. A continuous linear operator $T$ on $X$ is said to be frequently hypercyclic if there exists $x\in X$ so that for every nonempty open set $U$, the set $\left\{n\in ℕ:{T}^{n}x\in U\right\}$ has positive lower density. Thus the operator $T$ is not only hypercyclic, but the orbit of $x$ under powers of $T$ visits each open set quite often. This fruitful concept was introduced by F. Bayart and S. Grivaux [Trans. Am. Math. Soc. 358, No. 11, 5083–5117 (2006; Zbl 1115.47005)]. They gave a Frequently Hypercyclicity Criterion (an adaptation of the well-known Hypercyclicity Criterion to the new situation).

The present authors give a strengthened version, actually, a Frequently Universality Criterion. (A sequence of operators $\left\{{T}_{n}:n\in ℕ\right\}$ is considered instead of the powers ${T}^{n}$.) Among other things, they study under which conditions every vector in $X$ can be written as the sum of two frequently hypercyclic vectors. One important tool, for the case when $X$ is a Fréchet space but not a Banach space, is their “Runge transitivity” notion.

There are a few open questions in the paper under review. The following is their Problem 5.11: Is there a frequently hypercyclic operator on a Banach space for which every every vector can be written as the sum of two frequently hypercyclic vectors?

##### MSC:
 47A16 Cyclic vectors, hypercyclic and chaotic operators