Malaysian Journal of Mathematical Sciences, June 2025, Vol. 19, No. 2
Compound-Commuting Mappings on Skew-Hermitian Matrices
Zheng, W. S., Ng, W. S., and Chan, T. C.
Corresponding Email: ngws@utar.edu.my
Received date: 4 July 2024
Accepted date: 15 January 2025
Abstract:
Let $\mathbb{F}$ be a field with proper involution $-$ and let $r,s$ be even integers with $r,s>2$.
Let $\mathcal{SH}_r(\mathbb{F})$ and $C_{r-1}( M)$ denote the set of all $r\times r$ skew-Hermitian matrices over the field $\mathbb{F}$ and the $(r-1)$-th compound of a matrix $ M$, respectively.
In this study, we investigate the characterization of a mapping $\zeta\colon\mathcal{SH}_r(\mathbb{F})\rightarrow\mathcal{SH}_s(\mathbb{F})$ that satisfies,
\begin{align*}
\zeta(C_{r-1}( M+\gamma N))=C_{s-1}(\zeta( M)+\gamma\zeta( N)),
\end{align*}
for any $ M, N\in\mathcal{SH}_r(\mathbb{F})$ and $\gamma\in\mathbb{F}^-$, where $\mathbb{F}^-=\{x\in\mathbb{F}\mid \overline{x}=x\}$.
Keywords: preserver problems; compound-commuting mappings; skew-Hermitian matrices; rank
