On Wigner's theorem in complex smooth normed spaces

In this note, we present a generalization of Wigner's theorem. Let X and Y be complex normed spaces with Y being smooth. We show that a surjective mapping f : X -> Y satisfies {II f (x) + beta f (y)II : beta is an element of T-n} = {II x + beta yII : beta is an element of T-n}, x, y is an element of X, where n >= 3 is a positive integer and T-n is the set of the n th roots of unity, if and only if there exists a phase function sigma : X -> T-n such that sigma<middle dot> f is a linear or an anti -linear isometry. (c) 2024 Elsevier Inc. All rights reserved.