To Keep The Jordan Product Norm And Reversible Mapping. B_s (h)

We study the maps leaving some properties of elements on the space of self-adjoint operators, which are bijective maps preserving norms of Jordan products and additive maps preserving invertibility on the space of self-adjoint operators. Our results show that such maps must be isomorphisms or anti-isomorphisms on the space of self-adjoint operators. This paper contains three chapters:In chapter 1, we introduce some definitions and some theorems which are to be used in this paper.In chapter 2, a characterization is given for bijective maps preserving norms of Jordan products on BS(H) with dim H≥2. LetΦbe a bijective map on BS(H) with‖ΦA)Φ(B)+Φ(B)Φ(A)‖=‖AB+BA‖for all A,B∈BS(H). Clearly,Φpreserves Jordan zero-product on the space of self-adjoint operators in both directions. So we characterize the maximality of the ker(δA) in the Lemma 2.2.1 firstly. Take any nonzero operator A∈BS(H), the ker(δA) is maximal if and only if rank(A)=1 or there is a nonzero numberλ∈R such that under the decomposition of the space H with H=H1(?)H2(?)H3,Hi≠{0}(i=1,2). On the basis, we characterize the norms of Jordan products for further study. At last, we obtain that such maps must be isomorphisms or anti-isomorphisms.In chapter 3, the additive maps preserving invertibility on the space of self-adjoint operators are dicussed. We know that the key steps of preserver problems on operator algebras are to characterize the rank-1 operators. Therefore, we character-ize the rank-1 self-adjoint operators in the Lemma 3.2.1. The operator A is the rank-1 self-adjoint operator if and only if for any T∈BS(H),σ(T+A)∩σ(T+2A)(?)σ(T). By characterizing the structure of the rank-1 self-adjoint operators and using the mind of preserving invertibility on the algebra of all bounded linear operators on H, we obtain that such maps must be isomorphisms or anti-isomorphisms on the space of self-adjoint operators.