scholarly journals On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Barış Akay ◽  
Ömer Gök
Keyword(s):  
Linear Span ◽  
Compact Operators ◽  
Approximation Properties ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Weakly Compact ◽  
Domination Property ◽  
Weakly Compact Operators ◽  
Lattice Approximation ◽  
Order Continuous

We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

Sur Les M-Ideaux Dans Certains Espaces D′Operateurs Et L′Approximation Par Des Operateurs Compacts

1980 ◽  
Vol 23 (4) ◽  
pp. 401-411 ◽  
Author(s):  
H. Fakhoury
Keyword(s):  
Best Approximation ◽  
Compact Operators ◽  
Approximation Properties ◽  
Bounded Operators ◽  
Weakly Compact ◽  
The Best Approximation ◽  
Weakly Compact Operators

SommaireIt is shown that if V=C(X) or V = L1(μ) then the subspace of compact (resp. weakly compact) operators from V into itself is not an M-ideal in the space of bounded operators. This is the contrary to what happens when V= Co(ℕ) or lp(ℕ). The main result is proved via the best approximation properties of M-ideals and some results concerning norm one projections in C(X) and L1(μ) are deduced from this fact.

scholarly journals Weak Compactness of Almost Limited Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Aziz Elbour ◽  
Nabil Machrafi ◽  
Mohammed Moussa
Keyword(s):  
Banach Lattice ◽  
Compact Operators ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Weakly Compact Operators ◽  
Limited Operator ◽  
Almost Limited Operator ◽  
The Relationship ◽  
Order Continuous

This paper is devoted to the relationship between almost limited operators and weakly compact operators. We show that ifFis aσ-Dedekind complete Banach lattice, then every almost limited operatorT:E→Fis weakly compact if and only ifEis reflexive or the norm ofFis order continuous. Also, we show that ifEis aσ-Dedekind complete Banach lattice, then the square of every positive almost limited operatorT:E→Eis weakly compact if and only if the norm ofEis order continuous.

M-embedded symmetric operator spaces and the derivation problem

2019 ◽  
Vol 169 (3) ◽  
pp. 607-622
Author(s):  
JINGHAO HUANG ◽  
GALINA LEVITINA ◽  
FEDOR SUKOCHEV
Keyword(s):  
Symmetric Spaces ◽  
Symmetric Operator ◽  
Von Neumann Algebra ◽  
Bounded Operators ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Symmetric Operator Spaces ◽  
Order Continuous

AbstractLet ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

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