Power commuting additive maps on rank-k linear transformations

Let D be a division ring, let M be a right vector space over D and let End(M _D ) be the ring of all D-linear transformations from M into M. Suppose that R is a dense subring of End(MD) consisting of finite rank transformations and f : R → End(M _D ) is an additive map. We show that if f(x)x ^m(x) = x ^m(x) f(x) for every rank-k transformation x ∈ R, where k is a fixed integer with 1 < k < dimM _D and m(x) ≥ 1 is an integer depending on x, then there exist λ ∈ Z(D) and an additive map μ : R → Z(D)I such that f (x) = λx + μ(x) for all x ∈ 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].