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Let $M_{m,n}$ be the space of $m\times n$ real or complex rectangularmatrices. 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 matrixpairs, i.e., $A, B \in M_{m,n}$ are disjoint ensures that $\Phi(A), \Phi(B) \inM_{r,s}$ are disjoint. More precisely, it is shown that $\Phi$ preservesdisjointness 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 $V\inM_{s,s}$, and positive diagonal matrices $Q_1, Q_2$, where $Q_1$ or $Q_2$ maybe vacuous. The result is used to characterize nonsurjective linear maps thatpreserve the $JB^*$-triple product, or just the zero triple product, onrectangular matrices, defined by $\{A,B,C\} = \frac{1}{2}(AB^*C+CB^*A)$. Theresult is also applied to characterize linear maps between rectangular matrixspaces of different sizes preserving the Schatten $p$-norms or the Ky Fan$k$-norms.

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