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

[摘要] 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 circle times 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 less than or equal to parallel toTparallel to from X circle times Y. As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja. (C) 2003 Elsevier Inc. All rights reserved.

[发布日期] 2003-10-15