Journal of Operator Theory

Volume 83, Issue 1, Winter 2020  pp. 27-53.

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

Authors:  Chi-Kwong Li (1), Mimg-Cheng Tsai (2), Ya-Shu Wang (3), Ngai-Ching Wong (4)
Author institution:(1) Department of Mathematics, The College of William \& Mary, Williamsburg, VA 13185, U.S.A.
(2) General Education Center, Taipei University of Technology 10608, Taiwan
(3) Department of Applied Mathematics, National Chung Hsing University, Taichung 40227, Taiwan
(4) Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

Summary: Let $\mathbf{M}_{m,n}$ be the space of $m\times n$ real or complex rectangular matrices. Two matrices $A, B \in \mathbf{M}_{m,n}$ are disjoint if $A^*B = 0_n$ and $AB^* = 0_m$. We show that a linear map $\Phi: \mathbf{M}_{m,n} \rightarrow \mathbf{M}_{r,s}$ preserving disjointness exactly when $$\Phi(A) = U \left( \begin{array}{ccc} A \otimes Q_1 & 0 & 0 \\ 0 & A^\mathrm t \otimes Q_2 \\ 0 & 0 & 0 \\ \end{array} \right)V, \quad\forall A\in \mathbf{M}_{m,n},$$ for some unitary matrices $U \in \mathbf{M}_{r,r}$ and $V\in \mathbf{M}_{s,s}$, 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 between rectangular matrix spaces preserving (zero) JB$^*$-triple products, the Schatten $p$-norms or the Ky--Fan $k$-norms.

DOI: http://dx.doi.org/10.7900/jot.2018may14.2238
Keywords: orthogonality preservers, matrix spaces, norm preservers, Ky--Fan $k$-norms, Schatten $p$-norms, $\mathrm{JB}$*-triples