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Frequently hypercyclic operators and vectors. (English) Zbl 1119.47011
Let $X$ be an $\mathcal F$ 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 $\{n\in\mathbb N:T^nx\in U\}$ 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 {\it F. Bayart} and {\it 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 $\{T_n: n \in \mathbb N\}$ 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?

47A16Cyclic vectors, hypercyclic and chaotic operators
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