Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent Tensor Products

Abstract

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