Infima of Hilbert space effects
type: Item
instance of: scholarly article
title: Infima of Hilbert space effects
zbMATH Open document ID: 0964.46010
DOI: 10.1016/S0024-3795(98)10119-2
author: T. Moreland
author: Stan Gudder
published in: Linear Algebra and its Applications
publication date: 2001-02-14T00:00:00Z
review text: If \(H\) is a Hilbert space let \({\mathcal E}(H)\) denote the set of positive operators on \(H\) that are bounded above by the unit operator. \({\mathcal E}(H)\) is a partially ordered set which fails to be a lattice even if \(H\) is finite-dimensional, i.e., \(H= \mathbb{C}^n\). The authors investigate conditions under which the infimum \(A\wedge B\) of two elements \(A,B\in{\mathcal E}(H)\) exists. If \(A\in{\mathcal E}(H)\) let \(R(A)\) be the closure of the range of \(A\) and for \(A,B\in{\mathcal E}(H)\) let \(P_{A,B}\) denote the projection to \(R(A)\cap R(B)\). Using methods of linear algebra the authors show that for \(A,B\in{\mathcal E}(\mathbb{C}^n)\), \(n\in\mathbb{N}\), \(A\wedge B\) exists if and only if \(A\wedge P_{A,B}\) and \(B\wedge P_{A,B}\) are comparable in which case \(A\wedge B\) is the smaller of them.
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 1331722
zbMATH Keywords: positive operators
zbMATH Keywords: partially ordered set
zbMATH Keywords: infimum
reviewed by: Ludvík Janoš
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1331722