Finite representability of operators.
type: Item
instance of: scholarly article
title: Finite representability of operators.
zbMATH Open document ID: 1052.46005
DOI: 10.1016/S0022-247X(03)00008-8
author: J. Pello
author: Antonio Martínez-Abejón
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: 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.
reviewed by: Frank Oertel
Mathematics Subject Classification ID: 46B08
Mathematics Subject Classification ID: 47L20
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 1915725
zbMATH Keywords: operator finite representability
zbMATH Keywords: operator local supportability
zbMATH Keywords: ultraproduct techniques
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915725