Abstract
Let X and Y be normed spaces over \({\mathbb {F}} \in \{{\mathbb {R}}, {\mathbb {C}}\}\) and \(f :X \rightarrow Y\) a surjective map**. Suppose that \(|\phi _{f(y)}(f(x))|=|\phi _y(x)|\) holds for all \(x,y\in X\) and all support functionals \(\phi _{f(y)}\) at f(y) and \(\phi _y\) at y, or equivalently, suppose that for all semi-inner products on X and Y, compatible with given norms, \(\vert [f(x), f(y)] \vert = \vert [x, y] \vert \) holds for all \(x,y \in X\). Then \(f=\sigma U\), where \(\sigma :X \rightarrow {\mathbb {F}}\) is a phase function, and \(U :X \rightarrow Y\) is a linear or a conjugate linear isometry.
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Dijana Ilišević has been fully supported by the Croatian Science Foundation [project number IP-2016-06-1046]. Aleksej Turnšek was supported in part by the Ministry of Science and Education of Slovenia, Grants P1-0222, J1-8133.
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Ilišević, D., Turnšek, A. A Variant of Wigner’s Theorem in Normed Spaces. Mediterr. J. Math. 18, 148 (2021). https://doi.org/10.1007/s00009-021-01791-9
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DOI: https://doi.org/10.1007/s00009-021-01791-9