An operator bound related to regular operators
type: Item
instance of: scholarly article
title: An operator bound related to regular operators
zbMATH Open document ID: 0871.47031
DOI: 10.1007/BF01195708
published in: Archiv der Mathematik
publication date: 1997-10-06T00:00:00Z
review text: The main theorem in this paper asserts, essentially, that if \(T\) is a bounded linear operator on \(L^p(\mu)\), \(f_j, g_j \in L^\infty(\mu)\) for \(1\leq j\leq k\) and for \(h\in L^\infty(\mu)\) we let \(Q(h)\) denote the corresponding multiplication operator on \(L^p(\mu)\) defined by \(Q(h)\varphi=h\varphi\) then there is a constant \(C\) such that \[ |\sum^k_{j=1} Q(g_j)TQ(f_j)|\leq C|\sum^k_{j=1} f_j\otimes g_j|_\infty\tag{*} \] if and only if \(T\) is regular. Proposition 1, which asserts the automatic validity of (*) when \(p=1\) or \(p=\infty\) merely reflects the fact that on \(L^1(\mu)\) or on \(L^\infty(\mu)\) all bounded linear operators are regular. Applications to various concrete classes of operators are given. Reviewer's remark. The reviewer has been recently informed by A. Wickstead that in fact the constant \(C\) may be taken to be the regular norm of \(T\), which in this setting is simply \(||T||\). The main result above is valid in a much broader setting. A simple version is to consider a Dedekind complete Banach lattice \(E\), let \(Z(E)\) denote its centre (i.e. all linear operators on \(E\) for which there is \(\lambda\in\mathbb{R}\) with \(|Tx|\leq\lambda|x|\) for all \(x\in E)\) on which the regular and operator norms coincide. The space of regular operators on \(E\), \({\mathcal L}^r(E)\) is again a Dedekind complete Banach lattice under the regular norm and has its own centre \(Z({\mathcal L}^r(E))\). The mapping from the algebraic tensor product of \(Z(E)\) with itself, equipped with the norm \(|\cdot|_\lambda\), into \(Z({\mathcal L}^r(E))\) is an isometry. Furthermore the inequality analogous to (*) holds for some \(C\) if and only if \(T\) is regular.
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 938529
zbMATH Keywords: bounded linear operator
zbMATH Keywords: multiplication operator
zbMATH Keywords: regular
zbMATH Keywords: regular norm
zbMATH Keywords: Dedekind complete Banach lattice
zbMATH Keywords: centre
author: Brian Jefferies
author: S. Okada
reviewed by: Y. A. Abramovich
MaRDI profile type: Publication
cites work: Q3906660
cites work: An Operator Bound Related to Feynman-Kac Formulae
cites work: Q4049667
cites work: Q3669960
full work available at URL: https://doi.org/10.1007/bf01195708
OpenAlex ID: W2071958414
description: scientific article; zbMATH DE number 938529