Factorization of weakly compact operators through reflexive spaces with bases
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If T:X→Y is weakly compact and Y is separable with the bounded approximation property, must T factor through a reflexive Banach space with a Schauder basis?
Why it matters and what counts as progress
Why it matters: A basis strengthens general reflexive factorization and the target weakens earlier shrinking-basis assumptions on the range.
What counts as progress: A factorization theorem throughout the stated range-space class, or a counterexample retaining bounded approximation; arbitrary separable ranges form a different problem.
How the problem developed
- 1974 · Factorization theorem [1]
William J. Davis, Tadeusz Figiel, William B. Johnson and Aleksander Pełczyński Introduced the interpolation construction factoring every weakly compact operator through a reflexive space; their basis refinement used a shrinking basis in the target space.
- 2016 · Open question [2]
Kevin Beanland Asked whether the DFJP interpolation factorization can be arranged through a reflexive space with a Schauder basis beyond the shrinking-basis case.
- 19 Jul 2026 · Basis factorization [3]
Antonio and Pablo Acuaviva; model-generated proof ideas were verified and refined by the authors Proved that every weakly compact T:X→Y factors through a reflexive space with a Schauder basis when Y is separable with the bounded approximation property; in particular when Y has a basis.
AI contribution
Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory
If T:X→Y is weakly compact and Y is separable with the bounded approximation property, then T factors through a reflexive Banach space with a Schauder basis.
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