| Suppose R is the real number field, C is the complex number field, and m, n are integers with min {m,n}≥2. Denote by S_n(R) (respectively, H_n(C)) the Rlinear space of all n x n real symmetric (respectively, complex Hermitian) matrices. Recently, owing to their wide applications in many different fields,the preserver problems between different sets of matrices have been active, and one of important techniques in the study of preserver problems is to reduce many new preserver problems to the known ones, such as idempotence preserver,rank one preserver,and so on,then the problems are solved . This implies that the rank-1 preserver problem on different sets of matrices I study plays an important role in studying preserver problems. I study the problem by searching some particular matrices in this paper. At first, in Chapter 2,we describe the structure of all additive rank-1 preservers from S_m(R) to H_n(C), and thereby, the general form of all additive rank preservers from S_m(R) to H_n(C) is determined as an application. Based on Chapter 2, the general form of all additive rank-1 preservers from H_m(C) to H_n(C) is characterized in Chapter 3; as an application, the general form of all additive rank preservers from H_m(C) to H_n(C) is also given. |
