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

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 $A^{\ast }B=0_n$ and $A{B}^{\ast }=0_m$. In this paper, a characterization is given for linear maps $\Phi :M_{m,n}\to M_{r,s}$ 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 $$\Phi (A)=U\left(\begin{array}{ccc}A\otimes Q_1& 0& 0\\ 0& A^t\otimes Q_2& 0\\ 0& 0& 0\end{array}\right)V$$ 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 $J{B}^{\ast }$-triple product, or just the zero triple product, on rectangular matrices, defined by $\{A,B,C\}=\frac{1}{2}(A{B}^{\ast }C+C{B}^{\ast }A)$. 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.

This paper will appear in JOT