Periods of strongly continuous semigroups (Q2893265)

In this paper, the authors study the set of periods of (chaotic) strongly continuous semigroups of operators. Recall that a strongly continuous semigroup of operators \(\{ T(t)\}_{t\geq 0}\) on a Banach space \(X\) is called chaotic if it has a dense orbit and a dense set of periodic points. A positive real number \(t\) is called a (prime) period of \(T(t)\) if there is some \(x\in X\) such that \(T(t)x=x\) and \(T(s)x\neq x\) for all \(0<s<t\). Furthermore, we call \(t=0\) a period if there is some non-zero vector \(x\in X\) such that \(T(s)x=x\) for all \(s\geq 0\).NEWLINENEWLINEFirstly, the authors establish a relationship between eigenvalues on the imaginary axis of the generator of a strongly continuous semigroup of operators and the set of periods of the semigroup itself. Namely, they show that a non-zero vector \(x\in X\) is a periodic point with period \(t=0\) if and only if \(x\) is an eigenvector with eigenvalue \(0\) of the generator \(A\) of \(\{ T(t)\}_{t\geq 0}\).NEWLINENEWLINEFor the case of a positive period \(t\), they show that, if \(x\in X\) is a periodic point with period \(t\), then \(x=\sum_k h_{\alpha_k}\), where the sum runs over all \(k\in\mathbb{Z}\) such that \(\alpha_k=\alpha_k (t)=2\pi ik/t\) is an eigenvalue of the generator \(A\) of \(\{ T(t)\}_{t\geq 0}\) and \(h_{\alpha_k}\) are corresponding eigenvectors.NEWLINENEWLINEHence, \(t>0\) is a period if and only if there exist \(\alpha_k\in i\mathbb{R}\cap \sigma_p (A)\), \(k=1,\dots,l\) (where \(l=\infty\) is allowed) such that \(\sum_{k=1}^{l}h_{\alpha_k}\) is convergent and \(t\) is the smallest positive number with \(t\alpha_k\in 2\pi i\mathbb{Z}\) for all \(k\), where \(\sigma_p (A)\) denotes the point spectrum of \(A\). Then, they use this relationship to obtain some information about the structure of the set of periods and to construct (chaotic) semigroups with prescribed periods.