Maps Preserving Projections Of Jordan Products Of Operators On The Space Of Self-adjoint Operators

The preserving problem on operator algebra is one of the most concerned prob-lem in operator theory.Research on mappings that preserve nonzero projection of operator products has attracted wide attention of scholars.Let H be a complex Hilbert space with dim K? 3,B(H)be the algebra of all bounded linear operators on H and Bs(H)be the real linear space of all self-adjoint operators on H.For all A,B?B(H),the Jordan product of A and B is defined as A?B=1/2(AB+BA).In this text,we study the nonlinear maps on the space of self-adjoint operators preserve nonzero projections of Jordan products of two operators in both directions.Firstly,we give the equivalent characterization of the equality of two nonzero operators on Then we let be a surjective map on Bs(H),if ? preserves nonzero projections of Jordan products of two operators in both directions,then we can get that is a bijective map which preserves rank-n projections,the order and the orthogonality of projections in both directions.Secondly,we restrict the continuity of and get the concrete form of continuous surjective map ? which preserves nonzero projections of Jordan products of two operators in both directions by space decomposition and Uhlhorn's theorem.There exist a unitary or an anti-unitary operator U on H and a constant ? with ?2=1 such that ?(A)=?U*AU for all A?Bs(H).The conclusion makes the study of maps preserving nonzero projection of operator products on the space of self-adjoint operators more perfect,thus,we can have a better understanding of the structure of the self-adjoint operator space.