The ultimate estimate of the upper norm bound for the summation of operators
type: Item
instance of: scholarly article
title: The ultimate estimate of the upper norm bound for the summation of operators
zbMATH Open document ID: 1116.47011
author: Chi-Kwong Li
author: Man-Duen Choi
published in: Journal of Functional Analysis
publication date: 2006-04-06T00:00:00Z
review text: Let \(A\) and \(B\) be bounded linear operators acting on a Hilbert space \(H\). It is shown that \[ \sup\{\| U^*AU+V^*BV\| : U,V \text{ unitaries}\}= \min\{\| A+\mu I\| +\| B-\mu I\|:\mu\in\mathbb{C}\}.\tag{1} \] Furthermore, it is shown that the above quantity is the same as \[ \sup\{\| AX+XB\|:X\in B(H),\;\| X\| \leq1\}, \] which leads to a proof of Stampfli's well-known result on the norm of generalised derivation. The equality in (1) may not hold if the Hilbert-space operator norm \(\| \cdot\| \) is replaced by other norms. In fact, condition (1) can be used to characterise those unitarily invariant norms on \(M_n\) which are multiples of the operator norm. As a consequence of (1), some results related to spectral sets, the von Neumann inequality, and normal dilations are discussed.
reviewed by: Aleksej Turnšek
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A25
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 5017418
zbMATH Keywords: triangle inequality
zbMATH Keywords: operator norm
zbMATH Keywords: unitarily invariant norm
zbMATH Keywords: normal dilations
zbMATH Keywords: spectral circles
MaRDI profile type: Publication
full work available at URL: j.jfa.2005.07.001
OpenAlex ID: W1967673811
cites work: Orthogonality of matrices and some distance problems
cites work: Inequalities and equalities for the Cartesian decomposition of complex matrices
cites work: Norm bounds for summation of two normal matrices.
cites work: Q5548609
cites work: Some Metric Inequalities in the Space of Matrices
cites work: Q4040931
cites work: The norm of a derivation
cites work: Q5600360
DOI: 10.1016/J.JFA.2005.07.001
description: scientific article; zbMATH DE number 5017418