k-Potence preserving maps without the linearity and surjectivity assumptions Original Research Article

Let Mn be the space of all n × n complex matrices, and let Γn be the subset of Mn consisting of all n × n k-potent matrices. We denote by Ψn the set of all maps on Mn satisfying A − λB set membership, variant Γn if and only if phi(A) − λphi(B) set membership, variant Γn for every A,B set membership, variant Mn and λ set membership, variant C. It was shown that phi set membership, variant Ψn if and only if there exist an invertible matrix P set membership, variant Mn and c set membership, variant C with ck−1 = 1 such that either phi(A) = cPAP−1 for every A set membership, variant Mn, or phi(A) = cPATP−1 for every A set membership, variant Mn.