Commuting maps on rank-$k$ matrices

TL;DR

This paper characterizes additive maps on matrix rings that commute with all rank-$k$ matrices, showing they are essentially scalar multiples plus central maps, with exceptions at rank-1 matrices.

Contribution

It provides a complete description of commuting additive maps on matrices for rank-$k$ matrices, extending known results and highlighting differences at rank-1.

Findings

Additive maps commuting with rank-$k$ matrices are of form λx + μ(x).

Counterexamples exist for rank-1 matrices not fitting this form.

Discussion includes m-additive maps and their properties.

Abstract

Let $n\geq2$ be a natural number. Let $M_n(\mathbb{K})$ be the ring of all $n \times n$ matrices over a field $\mathbb{K}$. Fix natural number $k$ satisfying $1<k\leq n$. Under a mild technical assumption over $\mathbb{K}$ we will show that additive maps $G:M_n(\mathbb{K})\to M_n(\mathbb{K})$ such that $[G(x),x]=0$ for every rank-$k$ matrix $x\in M_n(\mathbb{K})$ are of form $\lambda x + \mu(x)$, where $\lambda\in Z$, $\mu:M_n(\mathbb{K})\to Z$, and $Z$ stands for the center of $M_n(\mathbb{K})$. Furthermore, we shall see an example that there are additive maps such that $[G(x),x]=0$ for all rank-1 matrices that are not of the form $\lambda x + \mu(x)$. We will also discuss the $m$-additive case.