Hyperreflexivity and operator ideals
type: Item
instance of: scholarly article
title: Hyperreflexivity and operator ideals
zbMATH Open document ID: 1137.47060
author: T. Oikhberg
published in: Journal of Functional Analysis
publication date: 2007-06-26T00:00:00Z
review text: A norm-closed subspace \(\mathcal A\subseteq B(X,Y)\) is hyperreflexive if there is a constant \(C\) such that for all \(T\in B(X,Y)\) \[ \inf_{a\in{\mathcal A}}\| T-a\|\leq C\sup\| uTv\|, \] where the supremum is taken over all finite rank contractions \(u\) and \(v\) such that \(u\mathcal A v=0\); see \textit{V.\,Müller} and \textit{M.\,Ptak} [J.~Funct.\ Anal.\ 218, No.\,2, 395--408 (2005; Zbl 1074.47032)]. In the present paper, using normed operator ideals \((\mathcal B,\beta)\), a more general definition of \textit{\(\mathcal B\)-hyperreflexivity} is given by replacing all occurrences of \(\|\cdot\|\) in the definition above by the ideal norm \(\beta\). Also, considering only subsets \({\mathcal A}\) instead of subspaces, a modification leads to the definition of \textit{\(\mathcal B\)-Azoff--Shehada hyperreflexivity}. Examples of spaces having and failing \((\mathcal B,\beta)\)-hyperreflexivity are given. In the last section, this type of hyperreflexivity is used to construct operator spaces with prescribed sets of completely bounded maps.
Mathematics Subject Classification ID: 47L20
Mathematics Subject Classification ID: 46L07
Mathematics Subject Classification ID: 47L25
Mathematics Subject Classification ID: 46B28
zbMATH DE Number: 5167473
zbMATH Keywords: hyperreflexivity
zbMATH Keywords: Azoff-Shehada-hyperreflexivity
zbMATH Keywords: operator spaces
zbMATH Keywords: operator ideals
MaRDI profile type: Publication
full work available at URL: j.jfa.2007.02.002
OpenAlex ID: W2009683298
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DOI: 10.1016/J.JFA.2007.02.002
description: scientific article; zbMATH DE number 5167473