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

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 $m-k$.