Linear Maps Preserving Rank-additivity And Rank-sum-minimal On Tensor Products Of Matrix Spaces

In the field of pure mathematics research,it is important to discuss invariants and the mapping and transformation of invariants.The problems of preserving invariants mapping in a given set of matrices are called the preserving problems.In recent decades,the problems of matrix preserving have become a core research field.Many scholars have studied many important and meaningful results for these problems,so the results of matrix preserving problems in general matrix spaces have become more and more perfect.In recent years,we have shifted our research direction to tensor products matrix spaces.Since the tensor products of matrix spaces are different from the general matrix spaces,the difficulty of research also increases with it.In 2012,Professor Chi-Kwong Li publicly raised the problem of linear maps preserving rank on the tensor products of matrix spaces.This problem was then solved by Baodong Zheng and Jinli Xu in 2015.This conclusion enriches the preserving problem on the tensor products of matrix spaces.Let V be a matrix space over field F.If the matrices pair A,B ?V satisfying R(A+B)=R(A)+R(B)or R(A+B)=|R(A)-R(B)|,it is said that A and B satisfies rank-additivity or rank-sum-minimal.For the linear mapping ? on V and the matrices pair A,B in V,if R(A+B)=R(A)+R(B)infers R?(A+B)=R?(A)+R?(B),it is said that?preserves rank-additivity.If R(A+B)=|R(A)-R(B)| infers R?(A+B)=|R?(A)-R?(B)|,it is said that ? preserves the rank-sum-minimal.If ?(Aad)-(?(A))ad,then ? is called preserving adjoint matrix.In this paper,we characterize the linear maps of the tensor products of square matrix spaces and the tensor products of Hermite matrix spaces,which preserves the rank-additivity and rank-sum-minimal.As an application,the linear maps problem of preserving adjoint matrix in tensor products of Hermite matrix spaces are also described.