Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

We define the notion of factorization of a family of metric spaces through a bounded, linear operator between Banach spaces. This notion serves as the analogue of uniform bi-Lipschitz embeddings of this family of metric spaces into a given Banach space. We prove operator versions of well-known non-linear characterizations of superreflexivity due to Bourgain, Johnson–Schechtman, and Baudier. More precisely, we give a non-linear characterization of non-super weakly compact operators as those through which the binary tree, diamond, and Laakso graphs may be factored with uniform distortion.