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

RACSAM (2006)

- Volume: 100, Issue: 1-2, page 325-331
- ISSN: 1578-7303

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topOja, 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 -