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The approximation property in terms of the approximability of weak∗-weak continuous operators

✍ Scribed by Eve Oja; Anders Pelander

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
206 KB
Volume
286
Category
Article
ISSN
0022-247X

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✦ Synopsis

By a well-known result of Grothendieck, a Banach space X has the approximation property if and only if, for every Banach space Y , every weak*-weak continuous compact operator T : X * → Y can be uniformly approximated by finite rank operators from X ⊗ Y . We prove the following "metric" version of this criterion: X has the approximation property if and only if, for every Banach space Y , every weak*-weak continuous weakly compact operator T : X * → Y can be approximated in the strong operator topology by operators of norm T from X ⊗ Y . As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja.

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