Inner and outer inequalities with applications to approximation properties

Abstract:

Let $X$ be a closed subspace of a Banach space $W$ and let ${\mathcal {F}}$ be the operator ideal of finite-rank operators. If $\alpha$ is a tensor norm, $\mathcal {A}$ is a Banach operator ideal, and $\lambda >0$, then we call the condition “$\|S\|_\alpha \leq \lambda \| S\|_{\mathcal {A}(X,W)} \ \textrm {for\ all}\ S\in \mathcal {F}(X,X)$” an inner inequality and the condition “$\| T\|_\alpha \leq \lambda \| T\|_{\mathcal {A}(Y,W)}$ for all Banach spaces $Y$ and for all $T\in \mathcal {F}(Y,X)$” 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 $p$.