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Uniformly factoring weakly compact operators. (English) Zbl 1329.46020
Let \(X\) and \(Y\) be Banach spaces and let \({\mathcal{L}}(X,Y)\) denote the space of bounded linear operators from \(X\) to \(Y\). Let \({\mathcal{A}}\subset{\mathcal{L}}(X,Y)\) be a set of weakly compact operators or, respectively, a set of operators whose adjoints have separable range.
The authors study the uniform factorization problem: When does there exist a reflexive Banach space \(Z\) or, respectively, a Banach space \(Z\) with separable dual such that every \(T\in{\mathcal{A}}\) factors through \(Z\)? It is well known that if \({\mathcal{A}}\) consists of a single operator, then the factorizations hold.
Let now \(X\) and \(Y\) be separable Banach spaces. The Borel \(\sigma\)-algebra generated by the strong operator topology is considered on \({\mathcal{L}}(X,Y)\). Then \({\mathcal{L}}(X,Y)\) is Borel isomorphic to a Polish space (i.e., to a separable topological space, metrizable by a complete metric). Recall that \({\mathcal{A}}\) is analytic if there exists a Polish space \(P\) and a Borel map \(f : P \to {\mathcal{L}}(X,Y)\) with \(f(P)={\mathcal{A}}\). The main results of the paper are as follows.
Theorem 1. Suppose that either \(Y\) has a shrinking basis or \(Y=C(2^{\mathbb{N}})\). Let \({\mathcal{A}}\) be an analytic set of weakly compact operators. Then there exists a reflexive Banach space \(Z\) with a basis such that every \(T\in {\mathcal{A}}\) factors through \(Z\).
Theorem 2. Suppose that \(Y=C(2^{\mathbb{N}})\). Let \({\mathcal{A}}\) be an analytic set of operators whose adjoints have separable range. Then there exists a Banach space \(Z\) with a shrinking basis such that every \(T\in{\mathcal{A}}\) factors through \(Z\).
A major part of the paper is devoted to proving Theorems 1 and 2. The main tool is Proposition 14 that establishes the claims in the case when \({\mathcal{A}}\) is a Borel set (the above analytic sets can be embedded in Borel sets). The cases when \(Y=C(2^{\mathbb{N}})\) uses the method of slicing and selection developed by N. Ghoussoub et al. [Can. J. Math. 44, No. 3, 483–504 (1992; Zbl 0780.46008)] in its parametrized version due to [P. Dodos and V. Ferenczi, Fundam. Math. 193, No. 2, 171–179 (2007; Zbl 1115.03061)].
An application (Theorem 28) is, e.g., as follows. Let \(X\) be a Banach space with a shrinking basis such that \(X^{**}\) is separable. Then there exists a reflexive Banach space \(Z\) such that every weakly compact operator on \(X\) factors through \(Z\).
Reviewer: Eve Oja (Tartu)

MSC:
46B28 Spaces of operators; tensor products; approximation properties
46B03 Isomorphic theory (including renorming) of Banach spaces
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