Power commuting additive maps on rank-k linear transformations

ABSTRACT

Let $\mathbb{D}$ be a division ring, let M be a right vector space over $\mathbb{D}$ and let $\text{End}\left(M_{\mathbb{D}}\right)$ be the ring of all $\mathbb{D}$-linear transformations from M into M. Suppose that R is a dense subring of $\text{End}\left(M_{\mathbb{D}}\right)$ consisting of finite rank transformations and $f:R\to \text{End}\left(M_{\mathbb{D}}\right)$ is an additive map. We show that if $f\left(x\right)x^{m\left(x\right)}=x^{m\left(x\right)}f\left(x\right)$ for every rank-k transformation $x\in R$, where k is a fixed integer with $1<k<\text{dim}{M}_{\mathbb{D}}$ and $m\left(x\right)\ge 1$ is an integer depending on x, then there exist $\lambda \in Z\left(\mathbb{D}\right)$ and an additive map $\mu :R\to Z\left(\mathbb{D}\right)I$ such that $f\left(x\right)=\lambda x+\mu \left(x\right)$ for all $x\in R$, where I denotes the identity transformation on M. This gives a natural generalization of the recent results obtained in [Franca, Commuting maps on rank-k matrices. Linear Alg Appl. 2013;438:2813–2815; Liu and Yang, Power commuting additive maps on invertible or singular matrices. Linear Alg Appl. 2017;530:127–149] and can be regarded as an infinite-dimensional version of the Franca theorem obtained in [Franca, Commuting maps on rank-k matrices. Linear Alg Appl. 2013;438:2813–2815].

KEYWORDS:

• Commuting map
• functional identity
• matrix
• linear transformation

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A03
• 15A04
• 16R60

Acknowledgments

The authors would like to thank the referee for the very thorough reading of the paper and valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.