论文标题
在任意字段上的矩阵空间之间的非偶性零产品保存器
Nonsurjective zero product preservers between matrix spaces over an arbitrary field
论文作者
论文摘要
如果矩阵之间的地图$φ$如果$$φ（a）φ（b）= 0 \ quad \ text {ever} \ quad ab = 0。 {\ bf m} _r（\ mathbb {f}）$在任意字段$ \ mathbb {f} $上的矩阵代数之间的矩阵代数之间。特别是，我们表明，如果$φ$是线性的，并且保留零产品，则$$φ（a）= s \ s \ begin {pmatrix} r_1 \ otimes a＆0 \ cr 0＆φ_0（a）\ end＆end＆end End {pmatrix} s^{ - 1} m} _k（\ mathbb {f}）$，$ s $ in $ {\ bf m} _r（\ mathbb {f}）$和零产品保存线性映射$φ_0：{\ bf m} _n（\ bf m} _n（ m} _ {r-nk}（\ mathbb {f}）$中的nilpotent矩阵。如果$φ（i_n）$是可逆的，则$φ_0$是空置的。通常，$φ_0$的结构可能是非常任意的，尤其是当$φ_0（{\ bf m} _n（\ mathbb {f}）））$具有微不足道的乘法，即$φ_0（x）φ_0（x）φ_0（y）= 0 $ for $ x，y $ in $ x，y $ in $ x，y $ x，y in $ x，$ {\ bf m {我们表明，如果$φ_0（i_n）= 0 $或$ r-nk \ le n+1 $，则$φ_0（{\ bf m} _n（\ mathbb {f}））$的确具有琐碎的乘数。更一般而言，我们表征了Square矩阵的子空间$ {\ bf v} $满足$ xy = 0 $的任何$ x，y \ in {\ bf v} $。获得双零产品保存图的相似结果。
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.
