Asymptotic behavior of operator sequences on KB-spaces

Özcan, Nazife Erkurşun

doi:10.1007/s11117-017-0545-2

Cited by 5 publications

(5 citation statements)

References 16 publications

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“…Here, we use another approach and consider lower bounds with respect to a given strictly positive functional. A related idea can also be found in [17]. We make this precise in the following definition: (i) T is strongly convergent and the limit operator P fulfils P ′ ψ ≥ εψ for some ε > 0.…”

Section: Lower Bounds On Banach Latticesmentioning

confidence: 99%

Lower bounds and the asymptotic behaviour of positive operator semigroups

Gerlach¹,

Glück²

2017

Ergod. Th. Dynam. Sys.

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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.

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Section: Lower Bounds On Banach Latticesmentioning

confidence: 99%

Lower bounds and the asymptotic behaviour of positive operator semigroups

Gerlach¹,

Glück²

2017

Ergod. Th. Dynam. Sys.

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“…An operator net T = (T λ ) λ∈Λ of positive operators is said to be a c-Markov net if there exists a weak unit x 0 ∈ X + and a strictly positive c-continuous functional x * 0 ∈ X * c such that T λ (x 0 ) = x 0 and T * λ (x * 0 ) = x * 0 for all λ ∈ Λ. The n-Markov net version is defined and studied in [12]. Two nets T, S ∈ N Λ (L(X, c)) are said to be d-asymptotically equivalent if…”

Section: Denote By X *mentioning

confidence: 99%

“…The notions of d-martingale and d-L R-nets are well motivated, see [6,7,9,10,12,13,18,20] for their applications. Since operator nets may not be uniformly bounded in the settings of convergences, uniformly bounded n-martingale and n-L R-nets are martingale and L R-nets in the sense of [7,10], respectively.…”

Section: Denote By X *mentioning

confidence: 99%

Unbounded asymptotic equivalences of operator nets with applications

Erkurşun-Özcan

Gezer

2019

Positivity

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Present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two convergences c and d on a vector lattice, we study d-asymptotic properties of operator nets formed by c-continuous operators. Asymptotic equivalences are known to be useful and extremely important tools to study infinite behaviors of strongly convergent operator nets and continuous semigroups. After giving a general theory, paper focuses on d-martingale and d-Lotz-Räbiger nets.) is said to be a convergence vector lattice. Let J be an order dense ideal, see [1,2] for more on this notion, in the vector lattice (X, c). Following [5], we write x λwe write uc instead of u J c. Prime examples of c are order convergence and norm convergence which are respectively denoted by o and B Niyazi Anıl Gezer

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“…In the following theorem, we will establish the asymptotic properties of C 0 Markov semigroups on KB-spaces. For the proof, we refer to [1] and [3] Theorem 2.2. Let E be a KB-space with a quasi-interior point e, T = (T t ) t2R be a C 0 Markov semigroup on E and A T t be the Cesaro averages of T .…”

Section: Preliminariesmentioning

confidence: 99%

“…Actually, all of these de…nitions of Markov operators describe a subclass of the positive contraction operator class. In this paper, the main object is also a Markov operator which is de…ned on a Banach lattice and the de…nition is given by [5] and then the de…nition of Markov operator net is given in [3].…”

Section: Introductionmentioning

confidence: 99%