Additive maps preserving rank-bounded sets of matrices

Abstract

Let $2\le k\le n$ be integers and ${\mathrm{Mat}}_n\left(F\right)$ be the linear space of $n\times n$ matrices over a field $F$ of characteristic different from 2. Denote by ${\Gamma }^{\ge k}$ the set of matrices in ${\mathrm{Mat}}_n\left(F\right)$ of rank greater than or equal to k. The main goal of the present paper is to obtain a characterization of additive maps $f:{\mathrm{Mat}}_n\left(F\right)\to {\mathrm{Mat}}_n\left(F\right)$ satisfying $f\left({\Gamma }^{\ge k}\right)={\Gamma }^{\ge k}$ with either $n<2k-2$ or $F$ has characteristic $\mathrm{char}\left(F\right)=0$ or $\mathrm{char}\left(F\right)\ge k$.