On products of hypercyclic semigroups
If \(X\) is a separable infinite-dimensional Banach space and \(J=\mathbb{N}\) or \(\mathbb{R}_{+}\), a semigroup \((S(t))_{t\in J}\) with values in \(X\) is hypercyclic if there exists a vector \(x\in X\) such that the orbit \(\{S(t)x ; t\in J\}\) is dense in \(X\). If such a semigroup is hypercyclic, it is known that \((S(t)\times S(t))_{t\in J}\) is hypercyclic on \(X\times X\) if and only if it satisfies the so-called Hypercyclicity Criterion. The authors give a condition, which they call the Recurrent Hypercyclicity Criterion, which is necessary and sufficient for \((S(t))_{t\in J}\) to have the property that \((S(t)\times T(t))_{t\in J}\) is hypercyclic for every semigroup \((T(t))_{t\in J}\) satisfying the Hypercyclicity Criterion.
scientific article; zbMATH DE number 5014186