Let be an space, i.e., a topological vector space whose topology is defined by an invariant metric. A continuous linear operator on is said to be frequently hypercyclic if there exists so that for every nonempty open set , the set has positive lower density. Thus the operator is not only hypercyclic, but the orbit of under powers of 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 is considered instead of the powers .) Among other things, they study under which conditions every vector in can be written as the sum of two frequently hypercyclic vectors. One important tool, for the case when 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?