# Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

## TL;DR

This paper characterizes linear maps between rectangular matrix spaces that preserve disjointness, triple products, or norms, revealing their specific structural forms and extending to nonsurjective maps.

## Contribution

It provides a complete characterization of linear disjointness-preserving maps and applies these results to maps preserving triple products and matrix norms.

## Key findings

- Disjointness-preserving maps have a specific block-diagonal form.
- Characterization of nonsurjective maps preserving JB*-triple products.
- Maps preserving Schatten p-norms or Ky Fan k-norms are characterized.

## Abstract

Let $M_{m,n}$ be the space of $m\times n$ real or complex rectangular matrices. Two matrices $A, B \in M_{m,n}$ are disjoint if $A^*B = 0_n$ and $AB^* = 0_m$. In this paper, a characterization is given for linear maps $\Phi: M_{m,n} \rightarrow M_{r,s}$ sending disjoint matrix pairs to disjoint matrix pairs, i.e., $A, B \in M_{m,n}$ are disjoint ensures that $\Phi(A), \Phi(B) \in M_{r,s}$ are disjoint. More precisely, it is shown that $\Phi$ preserves disjointness if and only if $\Phi$ is of the form $$\Phi(A) = U\begin{pmatrix} A \otimes Q_1 & 0 & 0 \cr 0 & A^t \otimes Q_2 & 0 \cr 0 & 0 & 0 \cr\end{pmatrix}V$$ for some unitary matrices $U \in M_{r,r}$ and $V\in M_{s,s}$, and positive diagonal matrices $Q_1, Q_2$, where $Q_1$ or $Q_2$ may be vacuous. The result is used to characterize nonsurjective linear maps that preserve the $JB^*$-triple product, or just the zero triple product, on rectangular matrices, defined by $\{A,B,C\} = \frac{1}{2}(AB^*C+CB^*A)$. The result is also applied to characterize linear maps between rectangular matrix spaces of different sizes preserving the Schatten $p$-norms or the Ky Fan $k$-norms.