On reflexivity and hyperreflexivity for linear subspace of operators
type: Item
instance of: scholarly article
title: On reflexivity and hyperreflexivity for linear subspace of operators
zbMATH Open document ID: 0753.47018
published in: Chinese Annals of Mathematics. Series B
publication date: 1992-06-25T00:00:00Z
review text: Let \(H_ 1\) and \(H_ 2\) be Hilbert spaces. For \({\mathcal P}_ 1\) and \({\mathcal P}_ 2\) denote the set of all closed subspaces of \(H_ 1\) and \(H_ 2\) respectively. For a subset \({\mathcal A}\) of \({\mathcal L}(H_ 1,H_ 2)\), the map of \({\mathcal A}\) which is denoted by Map \({\mathcal A}\) is defined as a map \(\varphi:{\mathcal P}_ 1\to{\mathcal P}_ 2\) given by \(\varphi(P)=[{\mathcal A}P]\), where \([{\mathcal A}P]\) denotes the closure of span\(\{Ax:\;A\in{\mathcal A},\;x\in P\}\). Given a map \(\varphi:{\mathcal P}_ 1\to{\mathcal P}_ 2\), we denote by \(op\varphi\) the set \(\{A\in{\mathcal L}(H_ 1,H_ 2):\;AP\subseteq\varphi(P),\;\forall P\in{\mathcal P}_ 1\}\). A subset \({\mathcal A}\) of \({\mathcal L}(H_ 1,H_ 2)\) is called to be reflexive if \({\mathcal A}=op\text{ Map }{\mathcal A}\). A weakly closed subspace \(M\) of \({\mathcal L}(H_ 1,H_ 2)\) is said to be a Von Neumann type subspace if \({\mathcal {MM}}^*{\mathcal M}\subseteq{\mathcal M}\). The author obtains some results on reflexivity for linear subspaces of \({\mathcal L}(H_ 1,H_ 2)\), and in particular he shows that any Von Neumann type subspace is reflexive.
reviewed by: Jafar Zafarani
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 11933
zbMATH Keywords: hyperreflexivity
zbMATH Keywords: reflexivity
zbMATH Keywords: Von Neumann type subspace
author: Deguang Han
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 11933