Inner and Outer Inequalities with Applications to Approximation Properties

Let X be a closed subspace of a Banach space W and let F be the operator ideal of finite-rank operators. If α is a tensor norm, A is a Banach operator ideal, and λ > 0, then we call the condition “‖S‖α ≤ λ‖S‖A(X,W ) for all S ∈ F(X,X)” an inner inequality and the condition “‖T‖α ≤ λ‖T‖A(Y,W ) for all Banach spaces Y and for all T ∈ F(Y,X)” an outer inequality. We describe cases when outer inequalities are determined by inner inequalities or by some subclasses of Banach spaces. This provides, among others, a unified approach to the study of approximation properties. We present various applications to Grothendieck’s classical approximation properties, to the weak bounded approximation property, and to approximation properties of order p.