标题： 奇异矩阵的Šemrl保持定理的一个变种 显示英文标题

标题： A variant of Šemrl's preserver theorem for singular matrices

摘要： 对于正整数$1 \leq k \leq n$，令$M_n$为所有$n \times n$阶复矩阵的代数，而$M_n^{\le k}$为其由所有秩至多为$k$的矩阵组成的子集。 我们首先证明，当$k>\frac{n}{2}$时，任何连续的谱收缩映射$\phi : M_n^{\le k} \to M_n$（即） $\mathrm{sp}(\phi(X)) \subseteq \mathrm{sp}(X)$对所有$X \in M_n^{\le k}$) 要么保持特征多项式，要么仅取幂零值。 此外，对于任何$k$，存在一个实解析嵌入$M_n^{\le k}$到所有足够大的$n$的$n\times n$幂零矩阵空间中。 这种现象在$\phi$为单射且$k > n - \sqrt{n}$或$\phi$的像包含在$M_n^{\le k}$中时不会发生。 我们然后建立本文的主要结果——Šemrl保子定理的一个变体，针对$M_n^{\le k}$：如果$n \geq 3$，任何保持交换性和缩小谱的单射连续映射$\phi :M_n^{\le k} \to M_n^{\le k}$都是形式为$\phi(\cdot)=T(\cdot)T^{-1}$或$\phi(\cdot)=T(\cdot)^tT^{-1}$，其中某个可逆矩阵$T\in M_n$。 此外，当$k=n-1$对应于奇异$n\times n$矩阵的集合时，此结果扩展到取值于$M_n$的映射$\phi$。 最后，我们讨论主要结果中假设的不可替代性。 显示英文摘要

摘要： For positive integers $1 \leq k \leq n$ 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 $\phi : M_n^{\le k} \to M_n$ (i.e. $\mathrm{sp}(\phi(X)) \subseteq \mathrm{sp}(X)$ 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 $\phi$ is injective and either $k > n - \sqrt{n}$ or the image of $\phi$ is contained in $M_n^{\le k}$. We then establish a main result of the paper -- a variant of \v{S}emrl's preserver theorem for $M_n^{\le k}$: if $n \geq 3$, any injective continuous map $\phi :M_n^{\le k} \to M_n^{\le k}$ that preserves commutativity and shrinks spectrum is of the form $\phi(\cdot)=T(\cdot)T^{-1}$ or $\phi(\cdot)=T(\cdot)^tT^{-1}$, 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 $\phi$ which take values in $M_n$. Finally, we discuss the indispensability of assumptions in our main result.