Nonsurjective zero product preservers between matrix spaces over an arbitrary field

TL;DR

This paper characterizes additive and linear zero product preserving maps between matrix spaces over any field, revealing their structure and conditions under which their images have trivial multiplication.

Contribution

It provides a concrete description of zero product preservers between matrix algebras of different dimensions, including their structure and conditions for trivial multiplication.

Findings

Linear zero product preservers have a block structure involving tensor products and nilpotent maps.

If the image of the identity is invertible, the nilpotent part is absent.

Subspaces with trivial multiplication are characterized.

Abstract

A map $\Phi$ between matrices is said to be zero product preserving if $$ \Phi(A)\Phi(B) = 0 \quad \text{whenever}\quad AB = 0. $$ In this paper, we give concrete descriptions of an additive/linear zero product preserver $\Phi: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_r(\mathbb{F})$ between matrix algebras of different dimensions over an arbitrary field $\mathbb{F}$. In particular, we show that if $\Phi$ is linear and preserves zero products then $$ \Phi(A)= S\begin{pmatrix} R_1 \otimes A & 0 \cr 0 & \Phi_0(A)\end{pmatrix} S^{-1}, $$ 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 $\Phi_0: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_{r-nk}(\mathbb{F})$ into nilpotent matrices. If $\Phi(I_n)$ is invertible, then $\Phi_0$ is vacuous. In general, the structure of $\Phi_0$ could be quite arbitrary, especially when $\Phi_0({\bf M}_n(\mathbb{F}))$ has trivial multiplication, i.e., $\Phi_0(X)\Phi_0(Y) = 0$ for all $X, Y$ in ${\bf M}_n(\mathbb{F})$. We show that if $\Phi_0(I_n) = 0$ or $r-nk \le n+1$, then $\Phi_0({\bf M}_n(\mathbb{F}))$ 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.