Operators preserving orthogonality are isometries

Proceedings of the Royal Society of Edinburgh Section A Mathematics (Impact Factor: 0.78). 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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    • "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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    • "It has been proved by Koldobsky [16] (for real spaces) and Blanco and Turnšek [5] (for real and complex ones) that a linear mapping f : X → X preserving the Birkhoff orthogonality, i.e., satisfying x⊥ B y =⇒ fx⊥ B fy, x, y ∈ X has to be a similarity (scalar multiple of an isometry). The same assertion can also be derived for linear mappings preserving semi-orthogonality, i.e., satisfying x⊥ s y =⇒ fx⊥ s fy, x, y ∈ X with respect to some semi-inner-product in X (cf. "
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