Mappings preserving \(B\)-orthogonality
type: Item
instance of: scholarly article
title: Mappings preserving \(B\)-orthogonality
zbMATH Open document ID: 1411.46014
author: Paweł Wójcik
published in: Indagationes Mathematicae. New Series
publication date: 2018-12-11T00:00:00Z
review text: Recall that two vectors \(x\) and \(y\) in a real or complex normed space \(X\) are said to be Birkhoff orthogonal if \(\| x\|\leq \| x+\alpha y\|\) for every scalar \(\alpha\). In the papers [\textit{A. Koldobsky}, Proc. R. Soc. Edinb., Sect. A, Math. 123, No. 5, 835--837 (1993; Zbl 0806.46013)] and [\textit{A. Blanco} and \textit{A. Turnšek}, ibid. 136, No. 4, 709--716 (2006; Zbl 1115.46016)], it was proved that a nonzero linear map \(T\) from a normed space \(X\) into another normed space \(Y\) preserves Birkhoff orthogonality if, and only if, there is some \(c>0\) such that \(\| Tx\|=c\| x\|\) for all \(x\in X\). The author gives a simpler proof of the above result. Also, he proves that the above result is also true, in the real case, assuming only that the map \(T\) is additive, the linearity of \(T\) being a consequence.
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 6991914
zbMATH Keywords: Birkhoff orthogonality
zbMATH Keywords: orthogonality preserving linear mappings
zbMATH Keywords: additive mappings
zbMATH Keywords: geometry of Banach spaces
reviewed by: Ginés López-Pérez
MaRDI profile type: Publication
full work available at URL: j.indag.2018.09.007
OpenAlex ID: W2892919565
cites work: On maps that preserve orthogonality in normed spaces
cites work: Orthogonally additive and orthogonally increasing functions on vector spaces
cites work: Operators preserving orthogonality are isometries
cites work: On orthogonally additive mappings
DOI: 10.1016/J.INDAG.2018.09.007
description: scientific article; zbMATH DE number 6991914