Partially ordered linear spaces and locally convex linear topological spaces

DeMarr, Ralph

doi:10.1215/ijm/1256059459

Cited by 36 publications

(23 citation statements)

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“…Following [42,18,46,34,26], a net (x α ) in a vector lattice X is said to converge in unbounded order (uo-converge for short) to x ∈ X, written as x α uo − → x, if |x α − x| ∧ y o − → 0 for any y ∈ X + ; (x α ) is said to be uo-Cauchy if the "double" net (x α − x β ) (α,β) uoconverges to zero. It is easily seen that uo-convergence (respectively, uo-Cauchy) coincides with order convergence (respectively, o-Cauchy) for order bounded nets.…”

Section: Unbounded Order Convergence and Regular Sublatticesmentioning

confidence: 99%

Uo-convergence and its applications to Cesàro means in Banach lattices

2017

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Abstract. A net (x α ) in a vector lattice X is said to uo-converge to x if |x α −x|∧u o − → 0 for every u ≥ 0. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [26,27]. In the second part, we use uo-convergence to study convergence of Cesàro means in Banach lattices. In particular, we establish an intrinsic version of Komlós' Theorem, which extends the main results of [35,16,31] in a uniform way. We also develop a new and unified approach to Banach-Saks properties and Banach-Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [21,24,25].

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Section: Unbounded Order Convergence and Regular Sublatticesmentioning

confidence: 99%

Uo-convergence and its applications to Cesàro means in Banach lattices

2017

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“…It follows that order intervals are uo-and un-closed. For sequences in L p (µ), where 1 p < ∞ and µ is a finite measure, it is easy to see that uo-convergence agrees with convergence almost everywhere, see, e.g., [DeM64,Example 2]. Under the same assumptions, un-convergence agrees with convergence in measure, see [Tro04,Example 23].…”

Section: Introductionmentioning

confidence: 99%

Unbounded norm topology in Banach lattices

Kandić

Marabeh

Troitsky

2017

Journal of Mathematical Analysis and Applications

165

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A net (x α ) in a Banach lattice X is said to un-converge to a vector x if |x α − x| ∧ u → 0 for every u ∈ X + . In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff X has a strong unit. Un-topology is metrizable iff X has a quasi-interior point. Suppose that X is order continuous, then un-topology is locally convex iff X is atomic. An order continuous Banach lattice X is a KB-space iff its closed unit ball B X is un-complete. For a Banach lattice X, B X is un-compact iff X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.

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“…The concept of unbounded order convergence under the name of individual convergence was first considered in [13] and " uo -convergence" was initially proposed in [6]. Recently, several papers about uo -convergence in Banach lattices have been published; see [3-5, 8-10, 16] for more details on these results.…”

Section: Introductionmentioning

confidence: 99%

Unbounded absolutely weak Dunford–Pettis operators

Özcan¹,

Gezer²,

Zabeti³

2019

Turk J Math

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In the present article, we expose various properties of unbounded absolutely weak Dunford-Pettis and unbounded absolutely weak compact operators on a Banach lattice E . In addition to their topological and lattice properties, we investigate relationships between M -weakly compact operators, L -weakly compact operators, and order weakly compact operators with unbounded absolutely weak Dunford-Pettis operators. We show that the square of any positive uaw -Dunford-Pettis ( M -weakly compact) operator on an order continuous Banach lattice is compact. Many examples are given to illustrate the essential conditions.

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Partially ordered linear spaces and locally convex linear topological spaces

Cited by 36 publications

References 1 publication

Uo-convergence and its applications to Cesàro means in Banach lattices

Uo-convergence and its applications to Cesàro means in Banach lattices

Unbounded norm topology in Banach lattices

Unbounded absolutely weak Dunford–Pettis operators

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