Convergence of positive operator semigroups

Gerlach, Moritz; Glück, Jochen

doi:10.1090/tran/7836

Cited by 21 publications

(48 citation statements)

References 48 publications

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“…We note that irreducibility of the semigroup can be replaced by other assumptions ensuring that the AM-compact operator K is "sufficiently large" when compared with the semigroup. A very general result of this type was proved by Gerlach and Glück in [GG17b,Theorem 3.11].…”

Section: Conclusion: Some Classical Theorems Revisitedmentioning

confidence: 81%

“…2) As a consequence of 1), the two main results from [GG17b], Theorem 3.7 and Theorem 3.11, are now unified. Moreover, our results hold without requiring the semigroup to have a quasi-interior fixed point.…”

Section: Introductionmentioning

confidence: 82%

“…To understand the relevance of the results obtained in this paper, one best looks into the pre-history of its predecessor [GG17b]. We decided to place such a historical narrative after our systematic considerations, in Section 7.…”

Section: )mentioning

confidence: 99%

“…1) A general Banach space principle (Theorem 3.1) is established and identified as the theoretical core which underlies the results of [GG17b]. This principle, which is based on the new notion of "quasi-compactness relative to a subspace" and on our systematic study of the "semigroup at infinity" (Theorem 2.2), has potential applications in the asymptotic theory of semigroups without any positivity assumptions.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotics of Operator Semigroups via the Semigroup at Infinity

Glück

Haase

2019

Trends in Mathematics

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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.

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Section: Conclusion: Some Classical Theorems Revisitedmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 82%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Asymptotics of Operator Semigroups via the Semigroup at Infinity

Glück

Haase

2019

Trends in Mathematics

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“…We note that Corollary 6.7 provides a sufficient condition for this to happen. If ker A µ = span{1}, then in particular T µ is Markovian and enjoys the strong Feller property and we can used recent results ( [18,20]) on the asymptotic behavior of such semigroups. Of particular importance are invariant probability measures of the semigroup.…”

Section: Asymptotic Behaviormentioning

confidence: 99%

Diffusion with nonlocal Dirichlet boundary conditions on domains

Kunze

2020

Studia Math.

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We consider a second order differential operator A on an open and Dirichlet regular set Ω ⊂ R d (which typically is unbounded) and subject to nonlocal Dirichlet boundary conditions of the formHere, µ : ∂Ω → M (Ω) takes values in the probability measures on Ω and is continuous in the weak topoly σ(M (Ω), C b (Ω)). Under suitable assumptions on the coefficients in A , which may be unbounded, we prove that a realization Aµ of A subject to the nonlocal boundary condition, generates a (not strongly continuous) semigroup on L ∞ (Ω). We establish a sufficient condition for this semigroup to be Markovian and prove that in this case, it enjoys the strong Feller property. We also study the asymptotic behavior of the semigroup.2010 Mathematics Subject Classification. 47D07, 60J35, 35B40.

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Cell cycle length and long‐time behavior of an age‐size model

Pichór

Rudnicki

2022

Math Methods in App Sciences

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We consider an age-size structured cell population model based on the cell cycle length. The model is described by a first order partial differential equation with initial-boundary conditions. Using the theory of semigroups of positive operators, we establish new criteria for an asynchronous exponential growth of solutions to such equations. We discuss the question of exponential size growth of cells. We study in detail a constant size growth model and a model with target size division. We also present versions of the model when the population is heterogeneous.

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Convergence of positive operator semigroups

Cited by 21 publications

References 48 publications

Asymptotics of Operator Semigroups via the Semigroup at Infinity

Asymptotics of Operator Semigroups via the Semigroup at Infinity

Diffusion with nonlocal Dirichlet boundary conditions on domains

Cell cycle length and long‐time behavior of an age‐size model

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