Preservers Of Lie Products On Self-Adjoint Operators And 2×2 Matrices

Let H be a complex separable Hilbert space of dimension at least 3, BS(H) be the space of all bounded self-adjoint operators on H. A classification of bijective maps for all diagnolizable operators on BS(H) preserving functional value of Lie products is given. The theorem is then used to charater surjective maps on self-adjoint matrices preserving functional value of Lie products and surjective maps on BS(H) preserving p-norm of Lie products. Let M2 be the algebra of all 2×2 complex matrices. An 1-1 correspondence between the set of 3×3 complex orthogonal matrices and the set of similarity transformations of M2 is established, and then maps preserves the determinant (resp. the spectrum, the peripheral spectrum) of Lie products on M2 is also given.