Several Special Linear Preserver Problems

Studying some invariants and mappings associated with invariants has always been a concern for scholars in the field of mathematics.The problem of maintaining a linear mapping of some invariants on operator algebras or matrix algebras is called the linear preserver problem.The preserver problem can simplify many problems in the field of mathematics.The study of the preserver problem provides a feasible solution to the construction of the algebraic group problem.The preserver problem can provide new ideas and directions for the basic problems in the matrix space and the solution of their anti-problems.At the same time,the preserver problems have broad space for development in computer image processing,cryptography,quantum mechanics,cybernetics,and other disciplines.The research content of this thesis is about kinds of linear preserver problems.They are the problem of preserving the numerical radius,the problem of the guaranteed Hadamard product and the permanent formula in the matrix space.The thesis is mainly divided into three parts.In the first part,the basic definitions and properties of the rank one projection operator are introduced.Through these properties,the numerical range preserver problem in Hilbert space is studied.The second part is about the linear preserver problem of numerical radius.According to a definition method about linear diameter maintenance and C*-isomorphism,the two forms about the linear mapping of preserving numerical radius are obtained.The third part deals with the preserver problem in matrix space.The definition and properties of the Hadamard product and the permanent formula in the matrix space are mainly introduced.And we give an assertion about the preserver problem of Hadamard product.And the types of matrix spaces that satisfy the permanent formula preserver problem are analyzed.