The Duality Problem for the Class of Order Weakly Compact Operators (original) (raw)

Some results on order weakly compact operators

Mathematica Bohemica, 2009

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The Relationship between M-Weakly Compact Operator and Order Weaky Compact Operator

2013

In this note, we will show that the class of order weakly compact operators bigger than the class of M-weakly compact operators. Under a new condition, we will show that each M-weakly compact operator is an order weakly compact operator. We will show that, if Banach lattice E be an AM-space with unit and has the property (b), then the class of M-weakly compact operators from E into Banach space Y coincides with that of order weakly compact operators from E into Y. Also we establish some relationship between M-weakly compact operators and weakly compact operators and b-weakly compact operators and order weakly compact operators.

The duality problem for the Class of b-weakly compact operators

Positivity, 2009

We establish necessary and sufficient conditions under which b-weakly compact operators between Banach lattices have b-weakly compact adjoints or operators with b-weakly compact adjoints are themselves b-weakly compact. Also, we give some consequences.

ABSTRACTS On Operators whose Compactness Properties are defined by Order

2014

Abstract. Let E be a Banach lattice. A subset B of E is called order bounded if there exist a, b in E such that a ≤ x ≤ b for each x ∈ B. Considering E in E ′ ′ , the bidual of E, a subset B of E is called b-order bounded in E if it is order bounded in the Banach lattice E ′ ′. A bounded linear operator T: E → X is called o-weakly compact if T (B) is relatively weakly compact for each order bounded set B in E. T is called b-weakly compact if T (B) is relatively weakly compact for each b-order bounded subset B in E. T is called an operator of strong type B if T (B(E)) ⊂ X where B(E) is the band generated by E

Weakly compact operators onH ?

Integral Equations and Operator Theory, 1999

We prove that a weakly compact operator from H ~ or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (i) every operator defined on H ~ or any of its even duals either fixes a copy of g~ or factors through a Banach space having the Banach-Saks property; (2) every quotient of H cc or any of its even duals either contains a copy of s or is super-reflexive; (3) eve N subspace of LI/H~ or any of its even duals either contains a complemented copy of ~i or is super-reflexive.

$\Tilde{O}$Rder-Norm Continuous Operators and TildeO\Tilde{O}TildeORder Weakly Compact Operators

arXiv (Cornell University), 2022

Fo −→ x is that there exists another net (y α) in F with the same index set satisfying y α ↓ 0 in F and |x α − x| ≤ y α for all indexes α. In this paper, we will study some properties of this new class of operators and its relationships with some known classifications of operators. We also define the new class of operators that named order weakly compact operators. A continuous operator T : E → X is said to beõrder weakly compact, if T (A) in X is a relatively weakly compact set for each Fo-bounded A ⊆ E. In this manuscript, we study some properties of this class of operators and its relationships withõrder-norm continuous operators.

Some notes on bbb-weakly compact operators

arXiv (Cornell University), 2019

In this paper, we will study some properties of b-weakly compact operators and we will investigate their relationships to some variety of operators on the normed vector lattices. With some new conditions, we show that the modulus of an operator T from Banach lattice E into Dedekind complete Banach lattice F exists and is b-weakly operator whenever T is a b-weakly compact operator. We show that every Dunford-Pettis operator from a Banach lattice E into a Banach space X is b-weakly compact, and the converse holds whenever E is an AM-space or the norm of E ′ is order continuous and E has the Dunford-Pettis property. We also show that each order bounded operator from a Banach lattice into a KB-space admits a b-weakly compact modulus.

A note on topologically b-order bounded sets and generalized b-weakly compact operators

Hacettepe Journal of Mathematics and Statistics, 2021

We study the class of order weakly compact operators on locally solid Riesz spaces and the introduced class of generalized b-weakly compact operators on locally convex-solid Riesz spaces. We prove as a consequence that some well-known results on b-weakly compact operators on a Banach lattice extend in an improved version to generalized b-weakly compact operators on a Fréchet lattice.

The relation between b-weakly compact operator and KB-operator

TURKISH JOURNAL OF MATHEMATICS, 2019

Our aim is to solve the problem asked by Bahramnezhad and Azar in "KB-operators on Banach lattices and their relationships with Dunford-Pettis and order weakly compact operators". We show that a continuous operator R from a Banach lattice N into a Banach space M is a b-weakly compact operator if and only if R is a KB-operator.

Topological characterization of weakly compact operators revisited

2007

In this note we revise and survey some recent results established in (8). We shall show that for each Banach space X, there exists a locally convex topology for X, termed the "Right Topology", such that a linear map T, from X into a Banach space Y, is weakly compact, precisely when T is a continuous map from X, equipped with the "Right" topology, into Y equipped with the norm topology. We provide here a new and shorter proof of this result. We shall also survey the results concerning sequentially Right-to-norm continuous operators.