Maps Preserving Numerical Radius Or Cross Norms On 2×2 Matrix Spaces

The linear preserver problem on matrix algebras can be dated back to 19th century and the similar questions on operator algebras on infinite dimensional spaces have also been studied extensively since twety years ago. Recently, the study of linear preserver problems have been generalized to the study of non-linear preserver problems. In this thesis, we discussing the question of characterizing non-linear maps that preserve the numerical radius or a cross norm of product.(1) The unital surjective maps on n×n symmetric matix space with n≥2 which preserve the numerical radius of Jordan semi-product of matrices or preserving a cross norm of Jordan semi-products of matrices are characterized.(2) The unital surjective maps on 2×2 self-adjoint matix space which preserve the numer-ical radius of product or Jordan semi-product of self-adjoint matrices, or preserves a cross norm of products or Jordan semi-products of self-adjoint matrices are characterized. These supplement the corresponding results for n×n self-adjoint matrix space with n≥3.(3) The unital surjective maps on 2×2 matix algebra which preserve the numerical radius of Jordan semi-product of matrices or preserving a cross norm of Jordan semi-products of matrices are characterized. These supplement the corresponding results for n×n matrix algebras with n≥3.