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Frequently hypercyclic semigroups. (English) Zbl 1232.47007
If X is a separable infinite-dimensional Banach space, a C 0 -semigroup (T t ) t0 of bounded linear operators on X is said to be hypercyclic if there exists a vector xX such that {T t xt0} is dense in X, and frequently hypercyclic if there exists a vector xX such that for any non-empty open subset U of X, the set {t0T t xU} has positive lower density. In this paper, the authors prove a version for C 0 -semigroups of the so-called Frequent Hypercyclicity Criterion. Applications are given to semigroups generated by Orstein-Uhlenbeck operators, in particular to translation semigroups on weighted spaces of L p -functions or continuous functions which, when multiplied by the weight, vanish at infinity.
MSC:
47A16Cyclic vectors, hypercyclic and chaotic operators
47D06One-parameter semigroups and linear evolution equations