Ideals of operators and the metric approximation property

Abstract

We prove that a Banach space X has the metric approximation property if and only if F(Y,X) , the space of all finite rank operators, is an ideal in L(Y,X) , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if F(X,Y) is an ideal in L(X,Y) for every Banach space Y. Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.