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Frequently hypercyclic semigroups. (English) Zbl 1232.47007
If $$X$$ is a separable infinite-dimensional Banach space, a $$C_{0}$$-semigroup $$(T_{t})_{t\geq 0}$$ of bounded linear operators on $$X$$ is said to be hypercyclic if there exists a vector $$x\in X$$ such that $$\{T_{t} x \mid t\geq 0\}$$ is dense in $$X$$, and frequently hypercyclic if there exists a vector $$x\in X$$ such that for any non-empty open subset $$U$$ of $$X$$, the set $$\{t\geq 0 \mid T_{t}x\in U\}$$ 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 Ornstein-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:
 47A16 Cyclic vectors, hypercyclic and chaotic operators 47D06 One-parameter semigroups and linear evolution equations
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