Abstract
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 $\mathcal {B}\to \mathcal {L}$ .
Highlights
Pełczyński factorisation theorem states that every weakly compact bounded operator factors through a reflexive space [15, Corollary 1]: Recall that a bounded operator : → between Banach spaces and is called weakly compact if the closure of the image of the unit ball of under is compact with respect to the weak topology of ; the operator is said to factor through a Banach space if there are bounded operators
A natural generalisation, considered by several authors, asks when a uniform factorisation is possible in the following sense: For which collections C of weakly compact operators between separable Banach spaces does there exist a separable reflexive space so that every operator in C factors through ? As we explain later, a uniform factorisation of this type is not possible for all weakly compact operators between separable Banach spaces
In addition to Theorem A, we obtain many other uniform factorisation-type theorems for certain collections of weakly compact operators between separable Banach spaces; to state these results, we will require some terminology from descriptive set theory
Summary
In addition to Theorem A, we obtain many other uniform factorisation-type theorems for certain collections of weakly compact operators between separable Banach spaces; to state these results, we will require some terminology from descriptive set theory. 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. Causey proved in [7, Theorem 5.8] that the class of weakly compact operators is strongly bounded over the class of separable reflexive Banach spaces. In Theorem 5.1 we prove that for an analytic subset A of the class of weakly compact operators , with domain separable and codomain with separable dual, one can find a reflexive so that each ∈ A factors through a subspace of ; the choice of this subspace can be done in a (
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