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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.

47A16 Cyclic vectors, hypercyclic and chaotic operators
47D06 One-parameter semigroups and linear evolution equations
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