Uniformly factoring weakly compact operators

Beanland, Kevin; Freeman, Daniel

doi:10.1016/j.jfa.2013.12.015

Cited by 10 publications

(24 citation statements)

References 29 publications

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“…By [9,Proposition 22], W is coanalytic. Finally, we denote the coding for the weakly compact operators between spaces with separable dual by W SD ; that is,…”

Section: Remark 22mentioning

confidence: 99%

“…Given ∈ SB, write F 0 ( ) = { ∈ F( ) : is bounded, convex and symmetric}. Then ( ( , ) ) is a basis for ( , ) for all ( , ) ∈ SB( ) × F 0 ( ) (see [15] or [9,Theorem 9] Proof. This proof consists of noticing that the methods in [9] (mainly Proposition 14) together with Lemma 4.6 give us the desired result.…”

Section: Factoring Weakly Compact Operators Through a Single Spacementioning

confidence: 99%

“…As is often common for this kind of result, descriptive settheoretic tools provide us the appropriate framework to study which subclasses have this property (cf. [7,9,18]).…”

Section: Introductionmentioning

confidence: 99%

“…In [9] the second named author and D. Freeman initiated the program of studying uniform factorisation problems using descriptive set theory. For that, the authors introduced a natural coding for the class of all operators between arbitrary Banach spaces, denoted by L; that is, L is a standard Borel space that naturally codes this class of operators (we refer the reader to Section 2 for precise definitions).…”

Section: Introductionmentioning

confidence: 99%

“…The main result of [9] (Theorem 25) states that if A is an analytic subset of weakly compact operators between separable Banach spaces so that either then there exists a separable reflexive Banach space admitting Schauder basis such that every operator in A factors through . In Section 4 (Theorem 4.3), we extend this result to several other classes of weakly compact operators.…”

Section: Introductionmentioning

confidence: 99%

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Uniformly factoring weakly compact operators and parametrised dualisation

Antunes

Beanland

Braga

2021

Forum of Mathematics, Sigma

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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 $\mathcal {B}\to \mathcal {L}$ .

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“…By [9,Proposition 22], W is coanalytic. Finally, we denote the coding for the weakly compact operators between spaces with separable dual by W SD ; that is,…”

Section: Remark 22mentioning

confidence: 99%

Section: Factoring Weakly Compact Operators Through a Single Spacementioning

confidence: 99%

“…As is often common for this kind of result, descriptive settheoretic tools provide us the appropriate framework to study which subclasses have this property (cf. [7,9,18]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Uniformly factoring weakly compact operators and parametrised dualisation

Antunes

Beanland

Braga

2021

Forum of Mathematics, Sigma

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Quantifying properties (K) and (μs$\mu ^{s}$)

Chen

Kania

Ruan

2022

Mathematische Nachrichten

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A Banach space 𝑋 has property (𝐾), whenever every weak* null sequence in the dual space admits a convex block subsequence (𝑓 𝑛 ) ∞ 𝑛=1 so that ⟨𝑓 𝑛 , 𝑥 𝑛 ⟩ → 0 as 𝑛 → ∞ for every weakly null sequence (𝑥 𝑛 ) ∞ 𝑛=1 in 𝑋; 𝑋 has property (𝜇 𝑠 ) if every weak * null sequence in 𝑋 * admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property (𝜇 𝑠 ) and reflexivity (or even the Grothendieck property) imply property (𝐾). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.

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Classes of operators determined by ordinal indices

Beanland

Causey

Freeman

et al. 2016

Journal of Functional Analysis

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Abstract. We introduce and study the Bourgain index of an operator between two Banach spaces. In particular, we study the Bourgain ℓp and c0 indices of an operator. Several estimates for finite and infinite direct sums are established. We define classes determined by these indices and show that some of these classes form operator ideals. We characterize the ordinals which occur as the index of an operator and establish exactly when the defined classes are closed. We study associated indices for non-preservation of ℓ ξ p and c ξ 0 spreading models and indices characterizing weak compactness of operators between separable Banach spaces. We also show that some of these classes are operator ideals and discuss closedness and distinctness of these classes.

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Uniformly factoring weakly compact operators

Cited by 10 publications

References 29 publications

Uniformly factoring weakly compact operators and parametrised dualisation

Uniformly factoring weakly compact operators and parametrised dualisation

Quantifying properties (K) and (μs$\mu ^{s}$)

Classes of operators determined by ordinal indices

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