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


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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    • "[13] "
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    ABSTRACT: We investigate orthonormality-preserving, C⁎-conformal and conformal module mappings on full Hilbert C⁎-modules to obtain their general structure. Orthogonality-preserving bounded module maps T act as a multiplication by an element λ of the center of the multiplier algebra of the C⁎-algebra of coefficients combined with an isometric module operator as long as some polar decomposition conditions for the specific element λ are fulfilled inside that multiplier algebra. Generally, T always fulfills the equality 〈T(x),T(y)〉=2|λ|〈x,y〉 for any elements x, y of the Hilbert C⁎-module. At the contrary, C⁎-conformal and conformal bounded module maps are shown to be only the positive real multiples of isometric module operators.
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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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    ABSTRACT: We answer many open questions regarding approximately orthogonality preserving mappings (in Birkhoff–James sense) in normed spaces. In particular, we show that every approximately orthogonality preserving linear mapping (in Chmieliński sense) is necessarily a scalar multiple of an ε-isometry. Thus, whenever ε-isometries are close to isometries we obtain stability. An example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries, that is, stability does not hold.
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