On operators from separable reflexive spaces with asymptotic structure

Bentuo Zheng

Studia Mathematica, Tome 187 (2008), p. 87-98 / Harvested from The Polish Digital Mathematics Library

Résumé

Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower-$ℓ_{q}$-tree estimate and let T be a bounded linear operator from X which satisfies an upper-$ℓ_{p}$-tree estimate. Then T factors through a subspace of $(∑ Fₙ)_{ℓ_{r}}$, where (Fₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an $(ℓ_{p}, ℓ_{q})$ FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic lower-$ℓ_{q}$-tree estimate. Let T be a bounded linear operator from X which satisfies an asymptotic upper-$ℓ_{p}$-tree estimate. Then T factors through a subspace of $(∑ Gₙ)_{ℓ_{r}}$, where (Gₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an asymptotic $(ℓ_{p},ℓ_{q})$ FDD.

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Publié le : 2008-01-01

Zbl 1148.46010

EUDML-ID : urn:eudml:doc:286583