Mp Inverse Of Upper Triangular Matrix Space Remain

The study of invariable and Preserver Problems about all kinds of invariable is an observational problem in the area of mathematics. At present, some researchers are more interested in Preserver Problems. As the generalized inverses of a matrix have wide applications in many areas, it has become one of the important studying fields in the world since the middle of the last century. M－P inverses is a special generalized inverses of a matrix. The main purpose of this paper is to investigate the maps preserving M－P inverses of matrices on upper trangular matrix space over fields.Suppose F is a field. We denote by M_n(F) and T_n(F) the space of n×n full matrices and the space of n×n upper trangular matrices over F, respectively. Some related references have shown that the preserver problem about M－P inverses of upper trangular matrices is still an open problem. In terms of the particularity and complication of M－P inverses of matrices, similar to other generalized inverses, reducing the linear(additive) operators preserving M－P inverses of matrices to the idempotent preserver is difficult. so I study the problem by searching particular matrices directly in this paper.In the chapter 2, first, the linear maps from T_n(F) to M_n(F) preserving M－P inverses of matrices are characterized, and thereby the linear maps from T_n(F) to T_n(F) preserving M－P inverses of matrices are characterized by retricting the range of image of maps to T_n(F), later, the additive maps from T_n(F) to T_n(F) preserving M－P inverses of matrices are described by using the linear results. As an application, the forms of the linear maps from T_n(F) to M_n(F)(T_n(F)) preserving {1,3}({1,4}) inverses of matrices and the additive maps from T_n(F) to T_n(F) preserving {1,3}({1,4}) inverses of matrices are also given.