On normed spaces with the Wigner property (Q781692)

A version of the classical theorem of Wigner, for Hilbert spaces, identifies transformations \(T\) that satisfy \(|\langle T(x),T(y)\rangle|\) = \(|\langle x,y \rangle|\) for all \(x,y\). In this paper the authors consider real Banach spaces for which every surjective phase isometry is phase equivalent to a linear surjective phase isometry (called Wigner property). In [``Phase-isometries on real normed spaces'', Preprint (2019), \url{arxiv:1905.01637}], \textit{X.-J. Tan} and \textit{D.-N. Huang} have proved that smooth spaces have this property. The main result of this interesting paper shows that strictly convex spaces have the Wigner property. This is first established for spaces of dimension~2 in Section~3 and the general result in Section~4.