  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 $m-k$.
