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

Oja, Eve. "The impact of the Radon-Nikodym property on the weak bounded approximation property.." RACSAM 100.1-2 (2006): 325-331. <http://eudml.org/doc/41657>.

@article{Oja2006,
abstract = {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.},
author = {Oja, Eve},
journal = {RACSAM},
keywords = {bounded approximation property; bounded approximation property with conjugate operators},
language = {eng},
number = {1-2},
pages = {325-331},
title = {The impact of the Radon-Nikodym property on the weak bounded approximation property.},
url = {http://eudml.org/doc/41657},
volume = {100},
year = {2006},
}

TY - JOUR
AU - Oja, Eve
TI - The impact of the Radon-Nikodym property on the weak bounded approximation property.
JO - RACSAM
PY - 2006
VL - 100
IS - 1-2
SP - 325
EP - 331
AB - 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.
LA - eng
KW - bounded approximation property; bounded approximation property with conjugate operators
UR - http://eudml.org/doc/41657
ER -