任意域上矩阵空间之间的非满射零积保持映射
Nonsurjective zero product preservers between matrix spaces over an arbitrary field
AI总结：
本文研究了任意域上不同维数矩阵代数之间的非满射线性零积保持映射，给出了其具体结构刻画，并推广至双重零积保持映射及具有平凡乘法的方阵子空间刻画。
摘要：
矩阵之间的映射 $Φ$ 被称为零积保持的，如果 $$ Φ(A)Φ(B) = 0 \quad \text{whenever}\quad AB = 0. $$ 在本文中，我们给出了任意域 $\mathbb{F}$ 上不同维数矩阵代数之间的加性/线性零积保持映射 $Φ: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_r(\mathbb{F})$ 的具体描述。特别地，我们证明如果 $Φ$ 是线性的且保持零积，则 $$ Φ(A)= S\begin{pmatrix} R_1 \otimes A & 0 \cr 0 & Φ_0(A)\end{pmatrix} S^{-1}, $$ 其中 $R_1 \in {\bf M}_k(\mathbb{F})$，$S \in {\bf M}_r(\mathbb{F})$ 为可逆矩阵，$Φ_0: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_{r-nk}(\mathbb{F})$ 为映入幂零矩阵的零积保持线性映射。若 $Φ(I_n)$ 可逆，则 $Φ_0$ 为空。一般而言，$Φ_0$ 的结构可能相当任意，特别是当 $Φ_0({\bf M}_n(\mathbb{F}))$ 具有平凡乘法时，即对所有 $X, Y \in {\bf M}_n(\mathbb{F})$，$Φ_0(X)Φ_0(Y) = 0$。我们证明若 $Φ_0(I_n) = 0$ 或 $r-nk \le n+1$，则 $Φ_0({\bf M}_n(\mathbb{F}))$ 确实具有平凡乘法。更一般地，我们刻画了满足对任意 $X, Y \in {\bf V}$ 均有 $XY = 0$ 的方阵子空间 ${\bf V}$。我们还获得了双重零积保持映射的类似结果。
英文摘要：
A map $Φ$ between matrices is said to be zero product preserving if $$ Φ(A)Φ(B) = 0 \quad \text{whenever}\quad AB = 0. $$ In this paper, we give concrete descriptions of an additive/linear zero product preserver $Φ: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_r(\mathbb{F})$ between matrix algebras of different dimensions over an arbitrary field $\mathbb{F}$. In particular, we show that if $Φ$ is linear and preserves zero products then $$ Φ(A)= S\begin{pmatrix} R_1 \otimes A & 0 \cr 0 & Φ_0(A)\end{pmatrix} S^{-1}, $$ for some invertible matrices $R_1$ in ${\bf M}_k(\mathbb{F})$, $S$ in ${\bf M}_r(\mathbb{F})$ and a zero product preserving linear map $Φ_0: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_{r-nk}(\mathbb{F})$ into nilpotent matrices. If $Φ(I_n)$ is invertible, then $Φ_0$ is vacuous. In general, the structure of $Φ_0$ could be quite arbitrary, especially when $Φ_0({\bf M}_n(\mathbb{F}))$ has trivial multiplication, i.e., $Φ_0(X)Φ_0(Y) = 0$ for all $X, Y$ in ${\bf M}_n(\mathbb{F})$. We show that if $Φ_0(I_n) = 0$ or $r-nk \le n+1$, then $Φ_0({\bf M}_n(\mathbb{F}))$ indeed has trivial multiplication. More generally, we characterize subspaces ${\bf V}$ of square matrices satisfying $XY = 0$ for any $X, Y \in {\bf V}$. Similar results for double zero product preserving maps are obtained.
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