The impact of the Radon-Nikodym property on the weak bounded approximation property.

Oja, Eve

RACSAM, Tome 100 (2006), p. 325-331 / Harvested from Biblioteca Digital de Matemáticas

Résumé

A Banach space X is said to have the weak λ-bounded approximation property if for every separable reflexive Banach space Y and for every compact operator T : X → Y, there exists a net (Sα) of finite-rank operators on X such that supα ||TSα|| ≤ λ||T|| and Sα → IX uniformly on compact subsets of X.

We prove the following theorem. Let X** or Y* have the Radon-Nikodym property; if X has the weak λ-bounded approximation property, then for every bounded linear operator T: X → Y, there exists a net (Sα) as in the above definition. It follows that the weak λ-bounded and λ-bounded approximation properties are equivalent for X whenever X* or X** has the Radon-Nikodym property. Relying on Johnson?s theorem on lifting of the metric approximation property from Banach spaces to their dual spaces, this yields a new proof of the classical result: if X* has the approximation property and X* or X** has the Radon-Nikodym property, then X* has the metric approximation property.

Détail

Publié le : 2006-01-01

MR MR2267414 | Zbl 1112.46017

DMLE-ID : 4157