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
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