A non-injective version of Wigner's theorem

TL;DR

This paper extends Wigner's theorem to a non-injective setting, showing that linear operators on self-adjoint finite rank operators preserving rank one projections are either isometries or constant.

Contribution

It proves a non-injective version of Wigner's theorem for linear operators on self-adjoint finite rank operators, characterizing those that preserve rank one projections.

Findings

Operators are either induced by isometries or are constant.

No additional assumptions are needed for the preservation of rank one projections.

The result generalizes classical Wigner's theorem to a broader class of linear operators.

Abstract

Let $H$ be a complex Hilbert space and let ${\mathcal F}_{s}(H)$ be the real vector space of 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 either induced by a linear or conjugate-linear isometry or constant on the set of rank one projections.