ABSTRACT
Let be a division ring, let M be a right vector space over
and let
be the ring of all
-linear transformations from M into M. Suppose that R is a dense subring of
consisting of finite rank transformations and
is an additive map. We show that if
for every rank-k transformation
, where k is a fixed integer with
and
is an integer depending on x, then there exist
and an additive map
such that
for all
, 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].
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.