Inverse-Preserving Additive Maps On Hermitian Matrix Spaces

For a long time, studying all kinds of invariants and the mappings which preserve the invariants has been valuable and interesting problem to be considered. The characterization of linear operators on matrix spaces that leave certain fuction, subsets, relations, etc., invariant is called"linear preserver problem". In past decade,"linear preserver problem"has been one of the most active subjects in matrix theory. The reason is that it's valuable for theoretical research on one way; and on the other way this theory has widely practical applications in several researching fields such as differential equations, systematic control theory and mathematical statistics, etc.. The problem that inverse-preserving between matices spaces has many results, for example, on full matrix space, symmetric matrix space, and upper triangular marix space. Suppose is complex field, n is a positive integer≥2. Let ( )M n and ( )H n be the n×n full matrix space and Hermitian matrix space over , repectively. This paper characterizes all additive maps from Hermitian matrix space H_n(C) to full matrix space M_n(C) preserving inverses of matrices. It is shown that every additive map f:H_n(C)→M_n(C) preserving inverses of matrices is of the form f(X)=eP^-1X^σP for all (?)X∈H_n(C) orf(X) =eP^-1XT^σP for all X∈H_n(C), where e∈{-1,1},σis an injective homomorphism on . Thereby, all additive maps from H_n(C) to itself preserving inverses of matrices are characterized, i.e. every additive map f:H_n(C)→H_n(C) preserving inverses of matrices is of the form f(X)=ePXP^?or f(X) =ePXTP(?), (?)X∈H_n(C), where e∈{-1,1}, unitary matrix P∈GL_n(C). Finally, it is known that inverse-...