Bounded approximation properties via integral and nuclear operators

Abstract

Let X be a Banach space and let A be a Banach operator ideal. We say that X has the lambda-bounded approximation property for A (lambda-BAP for A) if for every Banach space Y and every operator T is an element of A(X, Y), there exists a net (S-alpha) of finite rank operators on X such that S-alpha -> I-X uniformly on compact subsets of X and lim(alpha) sup parallel to TS alpha parallel to(A)<=lambda parallel to T parallel to(A). We prove that the (classical) lambda-BAP is precisely the lambda-BAP for the ideal I of integral operators, or equivalently, for the ideal SI of strictly integral operators. We also prove that the weak lambda-BAP is precisely the lambda-BAP for the ideal N of nuclear operators.