Linear mappings approximately preserving orthogonality
type: Item
instance of: scholarly article
title: Linear mappings approximately preserving orthogonality
zbMATH Open document ID: 1090.46017
author: Jacek Chmieliński
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-18T00:00:00Z
review text: The author gives several results for linear mappings between real or complex inner product spaces such that for each two orthogonal vectors in the domain, their values are \(\varepsilon\)-orthogonal in the target space. \(\varepsilon\)-orthogonality \((0\leq \varepsilon <1)\) means the relation \[ u\perp^\varepsilon v\Longleftrightarrow | (u| v)| \leq \varepsilon \| u\| \cdot \| v\| . \] The main result of the paper is Theorem 2, where it is shown that if \(X\) and \(Y\) are real or complex inner product spaces and if \(f:X\rightarrow Y\) is a nonzero linear mapping satisfying (AOP) with some \(\varepsilon\in (0,1)\), then \(f\) is injective, continuous and there exists \(\gamma > 0\) such that \[ \left| (\left( f(x)| f(y)\right) -\gamma^2(x| y) \right| \leq \delta \cdot \min \left\{\gamma^2 \| x\| \cdot \| y\| , \| f(x)\| \cdot \| f(y)\| \right\}, \quad x,y\in X, \tag{1} \] with \(\delta =4\varepsilon \left( {1\over 1-\varepsilon} +\sqrt {{1+\varepsilon\over 1-\varepsilon}}\right)\). Conversely, if \(f:X\rightarrow Y\) satisfies \((1)\) with some \(\delta\geq 0\) and \(\gamma > 0\), then \(f\) is a quasi-linear, approximately orthogonality preserving mapping.
reviewed by: Muhib Lohaj
Mathematics Subject Classification ID: 46C99
zbMATH DE Number: 2157560
zbMATH Keywords: orthogonality preserving mappings
zbMATH Keywords: approximate orthogonality
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.09.011
OpenAlex ID: W2043791734
cites work: The singular case in the stability of additive functions
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DOI: 10.1016/J.JMAA.2004.09.011
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description: scientific article; zbMATH DE number 2157560