Article

Operators preserving orthogonality are isometries

Proceedings of the Royal Society of Edinburgh Section A Mathematics (Impact Factor: 1.01). 01/1993; 123(5). 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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    • "Orthogonality preservers of C * -modules have been studied by many authors , e.g., [1], [3], [6], [10], [22]. In the case when A is a standard C * -algebra, the equivalence of (1.1) and (1.2) was established by D. Iliševi´c and A. Turnšek [8]. "
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    • "However in general normed spaces Birkhoff–James orthogonality is neither symmetric nor additive but it is homogeneous. It is known that a linear mapping T between normed spaces X and Y is OP if and only if it is a scalar multiple of an isometry; see [5] for the case of real normed spaces and [6] for the complex case. Therefore OP mappings between normed spaces are of the same form as OP mappings between inner product spaces. "
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