zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Frequently hypercyclic operators and vectors. (English) Zbl 1119.47011

Let X be an 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 xX so that for every nonempty open set U, the set {n:T n xU} 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} 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:
47A16Cyclic vectors, hypercyclic and chaotic operators