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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统一分解弱紧算子和参数化二元化

本文处理给定集合时的问题 $\数学{C}$ 可分离 Banach 空间之间的弱紧算子，存在可分离自反 Banach 空间Z以 Schauder 为基础，使得每个元素 $\数学{C}$ 通过因素Z（或通过一个子空间Z）。特别是，我们证明存在一个自反空间Z具有 Schauder 基，因此对于每个可分离的 Banach 空间X, 每个弱紧算子来自X到 $L_1[0,1]$ 通过因素Z. 我们还证明了以下描述集理论结果： $\数学{L}$ 是可分离 Banach 空间之间有界算子的标准 Borel 空间。我们证明如果 $\数学{B}$ 是具有可分离对偶的 Banach 空间之间的弱紧致算子的 Borel 子集，那么对于 $A \in \mathcal {B}$ ， 那作业 $A \to A^*$ 可以通过Borel map来实现 $\mathcal {B}\to \mathcal {L}$ .