Additive rank-1 preservers between hermitian matrix spaces and applications

Denote the set of n × n complex hermitian matrices by Hn. We characterize the additive maps from Hn into Hm that preserve rank 1 matrices when the rank of the image of In is equal to n. As applications, the forms of the additive preserver of rank, the additive preserver of rank-additivity and the additive preserver of adjoint matrix between the spaces of hermitian matrices are also given.