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

Abstract

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.