A non-injective version of Wigner's theorem

Abstract

Let $H$ be a complex Hilbert space and let ${\mathcal F}_{s}(H)$ be the real vector space formed by all self-adjoint finite rank operators on $H$. We prove the following non-injective version of Wigner's theorem: every linear operator on ${\mathcal F}_{s}(H)$ sending rank one projections to rank one projections (without any additional assumption) is induced by a linear or conjugate-linear isometry or it is constant on the set of rank one projections