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Uniformly factoring weakly compact operators and parametrised dualisation
Published online by Cambridge University Press: 08 March 2021
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
$\mathcal {B}\to \mathcal {L}$.
- Type
- Analysis
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
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