The set of periods of chaotic operators and semigroups
This expository paper reviews recent, still unpublished results about the set of possible periods of a chaotic operator \(T\) or a chaotic semigroup \((T(t) : t \geq 0)\) of operators acting on an infinite dimensional complex Banach space \(X\). The results about a single operator are due to Conejero, Martínez-Giménez and Peris and those about semigroups to the authors of the article under review (cf. [Bull. Lond. Math. Soc. 44, No.~3, 480--488 (2012; Zbl 1251.47009)]). In the case of one single operator, the picture is rather complete: An infinite subset \(P\) of the natural numbers is the set of periods of a chaotic operator on a Banach (even a Hilbert) space if and only if \(P\) is closed under least common multiple. The similar question for semigroups seems to be more delicate and only partial results are obtained. The paper concludes with a list of open questions.
periods of strongly continuous semigroups
scientific article; zbMATH DE number 6112717