Norm equalities in pre-Hilbert \(C^{*}\)-modules
type: Item
instance of: scholarly article
title: Norm equalities in pre-Hilbert \(C^{*}\)-modules
zbMATH Open document ID: 1231.47008
author: Dan Popovici
published in: Linear Algebra and its Applications
publication date: 2011-11-29T00:00:00Z
review text: Let \((E,\langle\cdot,\cdot\rangle)\) be a pre-Hilbert module over a \(C^*\)-algebra \(A\) and \(\theta_{x,y}(z):=x\langle y,z\rangle\) for \(x, y, z \in E\). The author proves that the equality case in the triangle inequality \(\|\theta_{x,x}+\theta_{y,y}\| \leq\|\theta_{x,x}\|+\|\theta_{y,y}\|\) holds if and only if \(\|\langle x,y\rangle\| =\|x\|\,\|y\|\). \textit{Lj. Arambašić} and \textit{R. Rajić} [Linear Algebra Appl. 414, No.~1, 19--28 (2006; Zbl 1100.47006)] proved that, for two elements \(x, y \in E\) such that \(\langle x,y\rangle\) is a nonnull idempotent, \(\|x + y\| = \|x\|+\|y\|\) if and only if \(\|x\|\,\|y\|=1\). The author extends this result to the case when, for some \(\alpha \in\mathbb{C}\), \(\langle x,y\rangle -\alpha \notin \{0,1\}\) is an idempotent in the unital \(C^*\)-algebra \(A\) by showing that \(\|x + y\| = \|x\|+\|y\|\) if and only if \(\alpha=\|x\|\,\|y\|-1 \geq -1/2\). \textit{Z. Sebestyén} and \textit{D. Popovici} [J. Oper. Theory 56, No. 1, 3--15 (2006; Zbl 1122.47037)] proved that the equality case is attained in the triangle inequality for positive nonnull elements in a \(C^*\)-algebra if and only if the norm of their product equals the product of their norms. Using this fact, the author characterizes Pythagoras' equality in pre-Hilbert \(C^*\)-modules.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 46L08
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 5982422
zbMATH Keywords: Pythagoras' equality
zbMATH Keywords: Cauchy-Schwarz inequality
zbMATH Keywords: triangle inequality
zbMATH Keywords: pre-Hilbert \(C^*\)-module
MaRDI profile type: Publication
full work available at URL: j.laa.2011.06.015
OpenAlex ID: W2004004656
cites work: On some norm equalities in pre-Hilbert \(C^{\ast}\)-modules
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cites work: Characterization of the norm triangle equality in pre-Hilbert C✻-modules and applications
DOI: 10.1016/J.LAA.2011.06.015
description: scientific article; zbMATH DE number 5982422