Asymptotics of Operator Semigroups via the Semigroup at Infinity

Abstract: We systematize and generalize recent results of Gerlach and Gl\"uck on the strong convergence and spectral theory of bounded (positive) operator semigroups $(T_s)_{s\in S}$ on Banach spaces (lattices). (Here, $S$ can be an arbitrary commutative semigroup, and no topological assumptions neither on $S$ nor on its representation are required.) To this aim, we introduce the "semigroup at infinity" and give useful criteria ensuring that the well-known Jacobs--de Leeuw--Glicksberg splitting theory can be applied to it. Next, we confine these abstract results to positive semigroups on Banach lattices with a quasi-interior point. In that situation, the said criteria are intimately linked to so-called AM-compact operators (which entail kernel operators and compact operators); and they imply that the original semigroup asymptotically embeds into a compact group of positive invertible operators on an atomic Banach lattice. By means of a structure theorem for such group representations (reminiscent of the Peter--Weyl theorem and its consequences for Banach space representations of compact groups) we are able to establish quite general conditions implying the strong convergence of the original semigroup. Finally, we show how some classical results of Greiner (1982), Davies (2005), Keicher (2006) and Arendt (2008) and more recent ones by Gerlach and Gl\"uck (2017) are covered and extended through our approach.