Maps on upper-triangular matrix algebras preserving k-potences

Let M_n be the space of all n × n complex matrices and T_n the subset of M_n consisting of all upper-triangular matrices. Denote by Γ_n the subset of M_n consisting of all n × n k-potent matrices, T Γ_n the subset of Γ_n consisting of all upper-triangular matrices. In this paper we describe the map φ{symbol} : T_n → M_n satisfying A - λ B ∈ T Γ_n if and only if φ{symbol} (A) - λ φ{symbol} (B) ∈ Γ_n for every A, B ∈ T_n and λ ∈ C.