Frequently hypercyclic operators and vectors. (English) Zbl 1119.47011

Bonilla, A.; Grosse-Erdmann, K.-G.

Ergodic Theory Dyn. Syst. 27, No. 2, 383-404 (2007); erratum ibid. 29, No. 6, 1993-1994 (2009).

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 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 \(\{T_n: n \in \mathbb N\}\) 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?

Reviewer: Héctor N. Salas (Mayagüez)

MSC:

47A16

Cyclic vectors, hypercyclic and chaotic operators

Keywords:

hypercyclic operators; frequently hypercyclic operators; frequently universality criterion; Runge transitivity

Citations:

Zbl 1115.47005