Goodearl-Menal Pairs of Linear Transformations

Abstract:

      In this short note we discuss the GM property of some special linear transformation pairs over infinite-dimensional vector spaces. In particular, it is proved that if $R={\rm End}(V_D)$ is the endomorphism ring of an infinite-dimensional right vector space $V$ over a division ring $D$ with $|C(D)|>3$ and $g\in R$, then $(a_0+a_1g,\,g)$ is a GM pair for any $a_0,\,a_1\in C(D)$. Furthermore, two existing results are obtained as immediate consequences.