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Frequently hypercyclic semigroups. (English) Zbl 1232.47007
If $X$ is a separable infinite-dimensional Banach space, a $C_{0}$-semigroup $(T_{t})_{t\ge 0}$ of bounded linear operators on $X$ is said to be {\it hypercyclic} if there exists a vector $x\in X$ such that $\{T_{t} x \mid t\ge 0\}$ is dense in $X$, and {\it frequently hypercyclic} if there exists a vector $x\in X$ such that for any non-empty open subset $U$ of $X$, the set $\{t\ge 0 \mid T_{t}x\in U\}$ 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 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.

47A16Cyclic vectors, hypercyclic and chaotic operators
47D06One-parameter semigroups and linear evolution equations
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