Oberseminar "Dynamische Systeme und Kontrolltheorie" - Prof. Dr. Jochen Glück, Bergische Universität Wuppertal

Operator semigroups are, both in discrete and continuous time, a powerfool tool in the analysis of linear autonomous systems. Stability analysis of such systems is closely related to the question whether a semigroup converges (to zero or to an equilibrium) as time tends to infinity.

In this talk we consider the particular case of state spaces that are endowed with an order structure (such as, for instance, function spaces or C^*-algebras) and of semigroups which preserve this order structure - so-called positive semigroups. Positivity is prevalent in various applications, and often it considerably facilitates the analysis of the long-term behaviour of the semigroup.

We give a brief walk-through of the topic, starting with some classical results, stopping at several examples to demonstrate how positivity occurs and can be used in concrete applications, and finishing with some recent insights and research questions.