Abstract
In this note we generalize the well-known Wigner’s unitary-antiunitary theorem. For smooth normed spaces X and Y and a surjective mapping \(f :X\rightarrow Y\) such that \(|[f(x),f(y)]|=|[x,y]|\), \(x,y\in X\), where \([\cdot ,\cdot ]\) is the unique semi-inner product, we show that f is phase equivalent to either a linear or an anti-linear surjective isometry. When X and Y are smooth real normed spaces and Y is strictly convex, we show that Wigner’s theorem is equivalent to \(\{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\), \(x,y\in X\).
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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 J1-8133 and P1-0222.
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Ilišević, D., Turnšek, A. On Wigner’s theorem in smooth normed spaces. Aequat. Math. 94, 1257–1267 (2020). https://doi.org/10.1007/s00010-020-00727-0
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DOI: https://doi.org/10.1007/s00010-020-00727-0