Locally Solid Riesz Spaces with Applications to Economics

Aliprantis, Charalambos D.; Burkinshaw, Owen

doi:10.1090/surv/105

Cited by 282 publications

(455 citation statements)

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“…But the order interval [0, x] is weakly closed and thus [0, x] is compact for the topology σ (E, E ). Finally, Theorem 22.1 of [1] implies that the norm of E is order continuous.…”

Section: ) If the Norm Of F Is Order Continuous Then Each Order Boumentioning

confidence: 96%

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The Duality Problem for the Class of Order Weakly Compact Operators

Aqzzouz¹,

H’michane

2009

Glasgow Math. J.

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Abstract. We study the duality problem for order weakly compact operators by giving sufficient and necessary conditions under which the order weak compactness of an operator implies the order weak compactness of its adjoint and conversely.2000 AMS Classification. 46A40; 46B40; 46B42. considered order weakly compact operators from a vector-valued function space into a Banach space and gave a characterization of an order weakly compact operator T in terms of the continuity of its adjoint relatively to some weak topologies. Introduction and notation.Let us recall that an operator T from a Banach lattice E into a Banach space F is said to be order weakly compact if for each x ∈ E + , the subset T ([0, x]) is relatively weakly compact in F, where E + = {x ∈ E : 0 ≤ x}.Contrarily to weakly compact operators [2, 13], the class of order weakly compact operators satisfies the domination problem. Indeed, if S and T are two operators from a Banach lattice E into another F such that 0 ≤ S ≤ T and T is order weakly compact, then S is order weakly compact [3].Also, the class of order weakly compact operators does not satisfy the duality property, that is, there exist order weakly compact operators whose adjoints are not order weakly compact. In fact, the identity operator of the Banach lattice l 1 is order weakly compact, but its adjoint, which is the identity operator of the Banach lattice l ∞ , is not order weakly compact. And conversely, there exist operators that are not order weakly compact but their adjoints are order weakly compact. In fact, the identity operator of the Banach lattice l ∞ is not order weakly compact but its adjoint, which is the identity operator of the topological dual (l ∞ ) , is order weakly compact.In [16], Zaanen investigated the duality problem for semi-compact operators. Also, in [5] and [16], the duality problem of AM-compact operators on Banach lattices was studied. They gave sufficient and necessary conditions for which the AM-compactness

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“…But the order interval [0, x] is weakly closed and thus [0, x] is compact for the topology σ (E, E ). Finally, Theorem 22.1 of [1] implies that the norm of E is order continuous.…”

Section: ) If the Norm Of F Is Order Continuous Then Each Order Boumentioning

confidence: 96%

“…Since the norm of F is order continuous, it follows from Theorem 22.1 of [1] that T[0, x] is relatively weakly compact, and then T is order weakly compact.…”

Section: ) If the Norm Of F Is Order Continuous Then Each Order Boumentioning

confidence: 99%

The Duality Problem for the Class of Order Weakly Compact Operators

Aqzzouz¹,

H’michane

2009

Glasgow Math. J.

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“…The lattice operations are discontinuous in the weak and weak * topologies in infinite dimensional spaces (see [2,Theorem 2.36]) which is a reason that "positive" properties are weaker than their "non-positive" prototypes. Sometimes the modulus is weakly sequentially continuous or sequentially weak * continuous.…”

Section: In the Sequel We Will Use Notations E ∈ (Sp) E ∈ (Psp) E ∈mentioning

confidence: 99%

On the dual positive Schur property in Banach lattices

Wnuk

2012

Positivity

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“…(See e.g. [10] and [1], and also [2], Theorems 2.42, 2.46, 4.28.) In this paper, we consider a still weaker concept of completeness for a tvl, the nested completeness along with its σ -variant, the σ -nested completeness.…”

Section: Introduction and Main Conceptsmentioning

confidence: 99%

“…Our terminology is rather standard, though slightly more neutral than that in [2,3] or [11], for instance. Thus we deal with topological vector lattices (or Hausdorff locally solid Riesz spaces), F-lattices (complete metrizable topological vector lattices), and Banach lattices.…”

Section: Introduction and Main Conceptsmentioning

confidence: 99%

On nestedly complete topological vector lattices

Drewnowski

2010

Positivity

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A topological vector lattice E is called (σ -)nestedly complete if every downward directed net (resp., decreasing sequence) of order intervals in E whose 'diameters' tend to zero has a nonempty intersection. Some characterizations of the (σ -)nested completeness are given, and it is shown that if E is metrizable and nestedly complete, so is each of its quotients E/I , where I is a closed ideal in E. Conversely, if a closed ideal I in E is (sequentially) complete and E/I is (σ -)nestedly complete, so is E. However, the nested completeness is not a three-space property: an example is given where both I and E/I are nestedly complete while E is not. It is also shown that the nested completeness and the related notion of nested density come up quite naturally when extending some positive linear operators. Finally, the nested and other completeness type properties of vector lattices C(S) are investigated. Keywords

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Locally Solid Riesz Spaces with Applications to Economics

Cited by 282 publications

References 0 publications

The Duality Problem for the Class of Order Weakly Compact Operators

The Duality Problem for the Class of Order Weakly Compact Operators

On the dual positive Schur property in Banach lattices

On nestedly complete topological vector lattices

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