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We systematize and generalize recent results of Gerlach and Gluck on the strong convergence and spectral theory of bounded (positive) operator semigroups (Ts)s ∈ 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.

Asymptotics of Operator Semigroups via the Semigroup at Infinity