Asymptotic reflexivity
type: Item
instance of: scholarly article
title: Asymptotic reflexivity
zbMATH Open document ID: 1123.47051
author: Hassan Yousefi
published in: Linear Algebra and its Applications
publication date: 2007-04-10T00:00:00Z
review text: Let \(\mathcal{S}\) be a subspace of the space \(B(H)\) of all bounded linear operators on a Hilbert space~\(H\). A new reflexive hull \(\text{AsymRef}(\mathcal{S})\) of \(\mathcal{S}\) is introduced in the following manner: an operator \(T\in B(H)\) belongs to \(\text{AsymRef}(\mathcal{S})\) precisely when \(\lim(Te_{\lambda},f_{\lambda})\rightarrow 0\) whenever \(\{e_{\lambda}\}\) and \(\{f_{\lambda}\}\) are nets of vectors of \(H\) such that \((Se_{\lambda},f_{\lambda})\rightarrow 0\) for every \(S\in\mathcal{S}\). The subspace \(\mathcal{S}\) is called asymptotically reflexive if \(\mathcal{S}=\text{AsymRef}(\mathcal{S})\). An operator \(T\in B(H)\) is called asymptotically reflexive if \(\text{AsymRef}(\mathbb{C}T)\) equals the algebra \(\mathcal{P}(T)\) of all polynomials on \(T\). The notion of asymptotic reflexivity is related to, but distinct from, the notion of approximate reflexivity where boundedness of the nets \(\{e_{\lambda}\}\) and \(\{f_{\lambda}\}\) is required. In particular, asymptotically reflexive spaces need not be norm closed. The author establishes a number of properties of the asymptotic reflexive hull showing, in particular, that \(\text{AsymRef}(\bigcup_{\alpha}\mathcal{S_{\alpha}}) = \bigcup_{\alpha}\text{AsymRef}(\mathcal{S_{\alpha}})\) for any upward directed family \(\{\mathcal{S_{\alpha}}\}\) of subspaces of \(B(H)\). He shows that every operator is asymptotically reflexive and completely characterises the asymptotic reflexive hull of \(\mathcal{S}\) in terms of the subspace of finite rank operators in \(\mathcal{S}\). In particular, he establishes the formula \(\text{AsymRef}(\mathcal{S}) = \mathcal{S} + \text{AsymRef}(\mathcal{S}\cap\mathcal{F})\), where \(\mathcal{F}\) is the set of all operators on \(H\) of finite rank. A beautiful analogue of the characterisation of hereditarily reflexive subspaces due to \textit{A.\,I.\thinspace Lo\-gi\-nov} and \textit{V.\,S.\thinspace Sul'man} [Math.\ USSR, Izv.\ 9 (1975), 1189--1201 (1977; Z…
Mathematics Subject Classification ID: 47L05
Mathematics Subject Classification ID: 46A32
zbMATH DE Number: 5141638
zbMATH Keywords: reflexivity
zbMATH Keywords: asymptotic reflexivity
zbMATH Keywords: algebraic reflexivity
zbMATH Keywords: approximate reflexivity
zbMATH Keywords: subspaces of operators
reviewed by: Ivan G. Todorov
MaRDI profile type: Publication
full work available at URL: j.laa.2006.11.019
OpenAlex ID: W4206713934
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DOI: 10.1016/J.LAA.2006.11.019
description: scientific article; zbMATH DE number 5141638