Maps on upper-triangular matrix algebras preserving k-potences

Let Mn be the space of all n×n complex matrices and Tn the subset of Mn consisting of all upper-triangular matrices. Denote by Γn the subset of Mn consisting of all n×nk-potent matrices, TΓn the subset of Γn consisting of all upper-triangular matrices. In this paper we describe the map ϕ:Tn→Mn satisfying A-λB∈TΓn if and only if ϕ(A)-λϕ(B)∈Γn for every A,B∈Tn and λ∈C.