Finite representability of operators.
While finite representability is a well stated term for Banach spaces revealing deep results regarding their geometric structure (such as super-reflexivity, local reflexivity or Dvoretzky's theorem), there exist several approaches to define finite representability in the context of linear operators including the notions of Heinrich and Bellenot. The paper under review is devoted to a detailed investigation of the relationship between these different existing notions. To this end, embedded in the framework of operator ideals and operator semigroups, the authors concentrate on a careful analysis of the concept of local representability in the sense of Pietsch (generalizing Heinrich's finite representability), introduce the notion of local supportability (generalizing Bellenot's finite representability) and prove that these local concepts can be characterized by factorization criteria involving certain ultrapower operators as factors, leading to a useful translation of these local concepts to the fruitful language of ultraproducts. In particular, given an arbitrary ultrafilter \(\mathfrak{U}\), the authors reveal that the dual of the ultrapower operator \(T_{\mathfrak{U}}\) is both Heinrich finitely representable and Bellenot finitely representable in the ultrapower operator \({T^{\ast}}_{\mathfrak{U}}\). Despite an appearing similarity of these local concepts at first glance, the authors construct counterexamples showing that neither Heinrich's finite representability implies local supportability nor Bellenot's finite representability implies local representability, so that both notions are mutually independent.
scientific article; zbMATH DE number 1915725