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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.

##### MSC:
 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)
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