Title

On operators from separable reflexive spaces with asymptotic structure

Abstract

Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower-ℓq-tree estimate and let T be a bounded linear operator from X which satisfies an upper-ℓp-tree estimate. Then T factors through a subspace of (Σ Fn) ℓr, where (Fn) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an (ℓp, ℓq) FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic lower-ℓq-tree estimate. Let T be a bounded linear operator from X which satisfies an asymptotic upper-ℓp-tree estimate. Then T factors through a subspace of (Σ Gn) ℓr, where (Gn) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an asymptotic (ℓp, ℓq) FDD. © Instytut Matematyczny PAN, 2008.

Publication Title

Studia Mathematica

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