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A variant of Wigner’s functional equation. (English) Zbl 1321.39025
The article deals with a new proof of the known Wigner’s theorem about mappings \(f:\;{\mathcal H} \to {\mathcal K}\) between inner product spaces \({\mathcal H}\) and \({\mathcal K}\) over the fields \({\mathbb R}\), \({\mathbb C}\), and \({\mathbb H}\). Two mappings \(f\) and \(g\) between \({\mathcal H}\) and \({\mathcal K}\) are phase equivalent if \(f(x) = \sigma(x)g(x)\) with \(|\sigma(x)| = 1\). Wigner’s theorem states that a mapping \(f\) satisfying the equation \[ |\langle f(x),f(y) \rangle| = |\langle x,y \rangle| \] is phase equivalent to a linear isometry in the case \({\mathbb F} = {\mathbb R}\) and \(\dim {\mathcal H} \geq 2\), is phase equivalent to either a linear isometry or to an anti-linear isometry in the case \({\mathbb F} = {\mathbb C}\) and \(\dim {\mathcal H} \geq 2\), and is phase equivalent to a linear isometry in the case \({\mathbb F} = {\mathbb H}\) and \(\dim {\mathcal H} \geq 3\). The proof presented in the article is simpler (the author’s opinion) than earlier known proofs.

39B05 General theory of functional equations and inequalities
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)
47J05 Equations involving nonlinear operators (general)
Full Text: DOI
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