Operators preserving orthogonality are isometries

Proceedings of the Royal Society of Edinburgh Section A Mathematics (Impact Factor: 0.64). 01/1993; DOI: 10.1017/S0308210500029528
Source: arXiv

ABSTRACT Let $E$ be a real Banach space. For $x,y \in E,$ we follow R.James in saying that $x$ is orthogonal to $y$ if $\|x+\alpha y\|\geq \|x\|$ for every $\alpha \in R$. We prove that every operator from $E$ into itself preserving orthogonality is an isometry multiplied by a constant.

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    ABSTRACT: We consider the Birkhoff-James orthogonality in normed spaces and classes of linear mappings exactly and approximately preserving this relation. Some related stability problems are posed.
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