Phase-isometries between normed spaces. (English) Zbl 1458.39019
The main goal of this paper is to give a positive answer to a question of G. Maksa and Z. Páles [Publ. Math. 81, No. 1–2, 243–249 (2012; Zbl 1299.39017)] on a Wigner’s type result for real normed spaces. More precisely, it is proven that if \(X\) and \(Y\) are real normed spaces and \(f : X \to Y\) is a surjective mapping, then \(f\) satisfies \(\{f(x) + f(y), f(x) - f(y)\} = \{x + y, x - y \}\), for every \(x\) and \(y\in X\) if and only if \(f\) is phase equivalent to a surjective linear isometry, that is, \(f = \sigma U\), where \(U : X \to Y\) is a surjective linear isometry and \(\sigma : X \to \{-1, 1\}\).
Reviewer: Elói M. Galego (São Paulo)
MSC:
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 46C50 | Generalizations of inner products (semi-inner products, partial inner products, etc.) |
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 46B04 | Isometric theory of Banach spaces |
| 47J05 | Equations involving nonlinear operators (general) |
| 53A20 | Projective differential geometry |
Citations:
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