No abstract

We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a purely algebraic result about positive group representations. Thus we obtain convergence theorems not only for one-parameter semigroups but for a much larger class of semigroup representations.Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive C 0 -semigroup containing or dominating a kernel operator converges strongly as t → ∞. We gain new insights into the structure theoretical background of those theorems and generalise them in several respects; especially we drop any kind of continuity or regularity assumption with respect to the time parameter.

Abstract. Given a positive, irreducible and bounded C 0 -semigroup on a Banach lattice with order continuous norm, we prove that the peripheral point spectrum of its generator is trivial whenever one of its operators dominates a non-trivial compact or kernel operator. For a discrete semigroup, i.e. for powers of a single operator T , we show that the point spectrum of some power T k intersects the unit circle at most in 1.As a consequence, we obtain a sufficient condition for strong convergence of the C 0 -semigroup and for a subsequence of the powers of T , respectively.

If (Tt) is a semigroup of Markov operators on an L 1 -space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as t → ∞. In this article we generalise and improve this result in several respects.First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalise a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results.Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.