Wigner's type theorem in terms of linear operators which send projections of a fixed rank to projections of other fixed rank

TL;DR

This paper characterizes linear operators on finite-rank self-adjoint operators in complex Hilbert spaces that map rank-k projections to rank-m projections, extending Wigner's theorem to broader settings with specific intersection conditions.

Contribution

It generalizes Wigner's theorem by describing all such linear operators under conditions involving fixed ranks and intersection dimensions, especially in infinite-dimensional spaces.

Findings

Operators are induced by isometries or conjugate-isometries in most cases.

The paper identifies conditions under which operators correspond to classical Wigner's theorem.

It extends the theorem to operators with intersection dimension constraints in infinite-dimensional spaces.

Abstract

Let $H$ be a complex Hilbert space whose dimension is not less than $3$ and let ${\mathcal F}_{s}(H)$ be the real vector space formed by all self-adjoint operators of finite rank on $H$. For every non-zero natural $k<\dim H$ we denote by ${\mathcal P}_{k}(H)$ the set of all rank $k$ projections. Let $H'$ be other complex Hilbert space of dimension not less than $3$ and let $L:{\mathcal F}_{s}(H)\to {\mathcal F}_{s}(H')$ be a linear operator such that $L({\mathcal P}_{k}(H))\subset {\mathcal P}_{m}(H')$ for some natural $k,m$ and the restriction of $L$ to ${\mathcal P}_{k}(H)$ is injective. If $H=H'$ and $k=m$, then $L$ is induced by a linear or conjugate-linear isometry of $H$ to itself, except the case $\dim H=2k$ when there is another one possibility (we get a classical Wigner's theorem if $k=m=1$). If $\dim H\ge 2k$, then $k\le m$. The main result describes all linear operators $L$ satisfying the above conditions under the assumptions that $H$ is infinite-dimensional and for any $P,Q\in {\mathcal P}_{k}(H)$ the dimension of the intersection of the images of $L(P)$ and $L(Q)$ is not less than $m-k$.