Uniformly factoring weakly compact operators

Original · EN

Let X and Y be separable Banach spaces. Suppose Y either has a shrinking basis or Y is isomorphic to C(2ⁿ) and A is a subset of weakly compact operators from X to Y which is analytic in the strong operator topology. We prove that there is a reflexive space with a basis Z such that every T ∈ A factors through Z. Likewise, we prove that if A ⊂ L(X, C(2ⁿ)) is a set of operators whose adjoints have separable range and is analytic in the strong operator topology then there is a Banach space Z with separable dual such that every T ∈ A factors through Z. Finally we prove a uniformly version of this result in which we allow the domain and range spaces to vary.