Asymptotics of Operator Semigroups via the Semigroup at Infinity

Abstract

Dedicated to Ben de Pagter on the occasion of his 65th birthday.Abstract. We systematize and generalize recent results of Gerlach and Glück 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.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 Glück (2017) are covered and extended through our approach.

Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups

“…Define T ∞ := s∈J {T t : s ≤ t ∈ J}, where the closure is taken in the strong operator topology. According to [26,Proposition 2.5(i) and (vi)] the assumptions on our semigroup imply that T ∞ is non-empty and strongly compact. Hence we can apply [26,Theorem 2.2] which yields a projection P ∈ T ∞ (denoted by P ∞ in [26,Theorem 2.2]) and a group G, given as the strong closure of {T t | P X , : t ∈ J} in L(P X), with the desired properties.…”

“…According to [26,Proposition 2.5(i) and (vi)] the assumptions on our semigroup imply that T ∞ is non-empty and strongly compact. Hence we can apply [26,Theorem 2.2] which yields a projection P ∈ T ∞ (denoted by P ∞ in [26,Theorem 2.2]) and a group G, given as the strong closure of {T t | P X , : t ∈ J} in L(P X), with the desired properties. Note that the positivity of P follows from P ∈ T ∞ , and the positivity of the elements of G follows from the positivity of the operators T t | P X .…”

“…In the above theorem we employed the concept of the semigroup at infinity T ∞ which was recently studied in [26]. More classical approaches apply the Jacobsde Leeuw-Glicksberg theory directly to the strong closure of {T t : t ∈ J}, which requires the stronger assumption that all orbits of the semigroup be relatively compact.…”

Ergodic properties of Markov semigroups in von Neumann algebras

“…The notion 'lim t∈G ', which shall be crucial in our further considerations, was (in the language of filters) exploited in [7] for an abelian semigroup of operators acting on a Banach space. Quite recently, this notion has also been exploited in the same setup in [11].…”

“…It was shown there that the following assertions hold: Theorem 11]). However, even without the assumption Fix T = {0} a map E may still be defined by formula (11), and it is seen that also in this case assertions (i) and (ii) hold. Indeed, observe first that for every ϕ ∈ M * and every t ∈ G, we have, by the definition of m T ,…”