On some norm equalities in pre-Hilbert \(C^{\ast}\)-modules
type: Item
instance of: scholarly article
title: On some norm equalities in pre-Hilbert \(C^{\ast}\)-modules
zbMATH Open document ID: 1100.47006
author: Ljiljana Arambašić
author: Rajna Rajić
published in: Linear Algebra and its Applications
publication date: 2006-03-29T00:00:00Z
review text: Hilbert \(C^{*}\)-modules are generalizations of Hilbert spaces by allowing the inner product to take values in a \(C^{*}\)-algebra rather than in the field of complex numbers, but there are some differences between these two classes. For example, a closed submodule of a Hilbert \(C^{*}\)-module is not necessary complemented. In this paper, the authors investigate some norm equalities in Hilbert \(C^{*}\) -modules. The triangle inequality \(\left\| x+y\right\| \leq \left\| x\right\| +\left\| y\right\| \) in normed linear spaces is well-known, as is the fact that for two elements \(x\) and \(y\) in a Hilbert space, \(\left\| x+y\right\| =\left\| x\right\| +\left\| y\right\| \) if and only if \(\left\langle x,y\right\rangle =\left\| x\right\| \left\| y\right\| .\) The authors extend this result in the context of Hilbert \(C^{*}\)-modules and show that, given a Hilbert \(C^{*}\)-module \(X\) and two elements \(x\) and \(y\) in \(X\), \(\left\| x+y\right\| =\left\| x\right\| +\left\| y\right\| \) if and only if \(\left\| x\right\| \left\| y\right\| \) is an element in the numerical range of \(\left\langle x,y\right\rangle \). Using this result, the authors obtain a characterization of Pythagoras' equality in a Hilbert \(C^{*}\)-module.
reviewed by: Maria Joiţa
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 46L08
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 5016218
zbMATH Keywords: Hilbert \(C^{*}\)-module
zbMATH Keywords: \(C^{*}\)-algebra
zbMATH Keywords: numerical range
zbMATH Keywords: triangle inequality
zbMATH Keywords: Pythagoras' equality
MaRDI profile type: Publication
full work available at URL: j.laa.2005.09.006
OpenAlex ID: W2083378146
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DOI: 10.1016/J.LAA.2005.09.006
description: scientific article; zbMATH DE number 5016218