On Wigner's theorem in complex smooth normed spaces

Abstract

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{‖f(x)+βf(y)‖:β∈Tn}={‖x+βy‖:β∈Tn},x,y∈X, where n≥3 is a positive integer and Tn is the set of the nth roots of unity, if and only if there exists a phase function σ:X→Tn such that σ⋅f is a linear or an anti-linear isometry.