Linear On The Complex Hermitian Matrix To Maintain

Suppose C is complex field, R is real number field, n and m be two arbitrary positive integers. Let M_n(C) and H_n(C) be the vector spaces of all n×n full matrices and all n×n complex hermitian matrices over R, respectively. T₁ expresses the idempotent preserving linear maps from H_n(C) to M_m(C), and the symbol (?)₂(H_n(C), M_m(C)denotes the set of T₁. T₂ expresses the tripotent preserving linear maps from H_n(C) to M_m(C), and the symbol (?)₃(H_n(C),M_m(C)denotes the set ofT₂.Lincar(Additive) preserve problem has been active, and complex hermitian matricesplay a very important part in matrix theory because of its particularity. Recently,the linear(additive) preserve problem about complex hermitian matrices has obtained many good result on rank one , rank additive preserver and so on.But the invariant which touch upon idempotence is only between the same matrice spaces. So I study it basing on this problem in the paper. First, in the chapter 2, the forms of the idempotent preserving linear operators from H_n(C) to M_m(C) are characterized , and thereby the forms of the linear operators from H_n(C) to H_m(C) preserving idempotenc are characterized by retricting the range of image of operators to Hm(F). In chapter 3, the forms of the tripotent preserving linear maps from H_n(C) to M_m(C) and from H_n(C) to H_m(C) are characterized by using the similar methods of chapter 2. As a deduction, the forms of the group inverse preserving linear maps from H_n(C) to M_m(C) are also given.