## Abstract and Applied Analysis

### Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent Tensor Products

Li Yang, Wei Zhang, and Jinli Xu

#### Abstract

Suppose $m,n\ge \mathrm{2}$ are positive integers. Let ${\mathrm{\scr T}}_{n}$ be the space of all $n{\times}n$ complex upper triangular matrices, and let $\varphi$ be an injective linear map on ${\mathrm{\scr T}}_{m}\otimes {\mathrm{\scr T}}_{n}$. Then $\varphi (A\otimes B)$ is an idempotent matrix in ${\mathrm{\scr T}}_{m}\otimes {\mathrm{\scr T}}_{n}$ whenever $A\otimes B$ is an idempotent matrix in ${\mathrm{\scr T}}_{m}\otimes {\mathrm{\scr T}}_{n}$ if and only if there exists an invertible matrix $P\in {\mathrm{\scr T}}_{m}\otimes {\mathrm{\scr T}}_{n}$ such that $\varphi (A\otimes B)=P({\xi }_{\mathrm{1}}(A)\otimes {\xi }_{\mathrm{2}}(B)){P}^{-\mathrm{1}}, \forall A\in {\mathrm{\scr T}}_{m},\mathrm{ B}\in {\mathrm{\scr T}}_{n}$, or when $m=n$, $\varphi (A\otimes B)=P({\xi }_{\mathrm{1}}(B)\otimes {\xi }_{\mathrm{2}}(A)){P}^{-\mathrm{1}}, \forall A\in {\mathrm{\scr T}}_{m},\mathrm{ B}\in {\mathrm{\scr T}}_{m}$, where ${\xi }_{\mathrm{1}}([{a}_{ij}])=[{a}_{ij}]$ or ${\xi }_{\mathrm{1}}([{a}_{ij}])=[{a}_{m-i+\mathrm{1},m-j+\mathrm{1}}]$ and ${\xi }_{\mathrm{2}}([{b}_{ij}])=[{b}_{ij}]$ or ${\xi }_{\mathrm{2}}([{b}_{ij}])=[{b}_{n-i+\mathrm{1},n-j+\mathrm{1}}].$

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 148321, 8 pages.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.aaa/1395857995

Digital Object Identifier
doi:10.1155/2014/148321

Mathematical Reviews number (MathSciNet)
MR3166569

Zentralblatt MATH identifier
1142.11027

#### Citation

Yang, Li; Zhang, Wei; Xu, Jinli. Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent Tensor Products. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 148321, 8 pages. doi:10.1155/2014/148321. https://projecteuclid.org/euclid.aaa/1395857995