Lower Bounds and the Asymptotic Behaviour of Positive Operator\n Semigroups

2016/04/06 by Moritz Gerlach, Gerlach, Moritz, Jochen Glück +1
Mathematics · #47B65 #47D06 #47D07 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.1604.01743

openalex publication_date 2016/04/06 · openalex created_date 2022/09/18 · openalex updated_date 2026/07/28

Abstract

If (T_t) is a semigroup of Markov operators on an L¹-space that admits a\nnon-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts\nthat the semigroup is strongly convergent as t \→ \∞. In this article we\ngeneralise and improve this result in several respects.\n First, we give a new and very simple proof for the fact that the same\nconclusion also holds if the semigroup is merely assumed to be bounded instead\nof Markov. As a main result we then prove a version of this theorem for\nsemigroups which only admit certain individual lower bounds. Moreover, we\ngeneralise a theorem of Ding on semigroups of Frobenius-Perron operators. We\nalso demonstrate how our results can be adapted to the setting of general\nBanach lattices and we give some counterexamples to show optimality of our\nresults.\n Our methods combine some rather concrete estimates and approximation\narguments with abstract functional analytical tools. One of these tools is a\ntheorem which relates the convergence of a time-continuous operator semigroup\nto the convergence of embedded discrete semigroups.\n

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