A variant of Šemrl's preserver theorem for singular matrices

For positive integers $1\le k\le n$ let $M_n$ be the algebra of all $n\times n$ complex matrices and $M_n^{\le k}$ its subset consisting of all matrices of rank at most k. We first show that whenever $k>\frac{n}{2}$, any continuous spectrum-shrinking map $\varphi :M_n^{\le k}\to M_n$ (i.e. $\mathrm{sp}\left(\varphi \right(X\left)\right)\subseteq \mathrm{sp}\left(X\right)$ for all $X\in M_n^{\le k}$) either preserves characteristic polynomials or takes only nilpotent values. Moreover, for any k there exists a real analytic embedding of $M_n^{\le k}$ into the space of $n\times n$ nilpotent matrices for all sufficiently large n. This phenomenon cannot occur when ϕ is injective and either $k>n-\sqrt{n}$ or the image of ϕ is contained in $M_n^{\le k}$. We then establish a main result of the paper – a variant of Šemrl's preserver theorem for $M_n^{\le k}$: if $n\ge 3$, any injective continuous map $\varphi :M_n^{\le k}\to M_n^{\le k}$ that preserves commutativity and shrinks spectrum is of the form $\varphi (\cdot )=T(\cdot )T^{-1}$ or $\varphi (\cdot )=T{(\cdot )}^t{T}^{-1}$, for some invertible matrix $T\in M_n$. Moreover, when $k=n-1$, which corresponds to the set of singular $n\times n$ matrices, this result extends to maps ϕ which take values in $M_n$. Finally, we discuss the indispensability of assumptions in our main result.