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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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