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 Uniformly factoring weakly compact operators and parametrised dualisation

2021 ◽  
Vol 9 ◽  
Author(s):  
L. Antunes ◽  
K. Beanland ◽  
B. M. Braga
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Borel Subset ◽  
Compact Operators ◽  
Schauder Basis ◽  
Bounded Operators ◽  
Weakly Compact ◽  
Weakly Compact Operator ◽  
Borel Map ◽  
Weakly Compact Operators

Abstract This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z. We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$ , the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$ .

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.

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