自伴算子空间上保持Jordan积范数的映射

Maps preserving norms of Jordan products on the space of self-adjoint operators

设H是复数域C上的Hilbert空间且dimH≥2,Bs(H)是H上所有自伴算子全体。设Φ是Bs(H)上的双射,如果Φ满足对任意A,B∈Bs(H),都有‖Φ(A)Φ(B)+Φ(B)Φ(A)‖=‖AB+BA‖,则存在一个酉算子或反酉算子U和泛函h:B(H)→{1,-1}使得对任意X∈B(H),有Φ(X)=h(X)UXU*。

Let H be a complex Hilbert space with dimH≥2 and let Bs（H） be the space of all self-adjoint operators on H.Let Φ be a bijective map on Bs（H）.It is proved that if ‖Φ（A）Φ（B）＋Φ（B）Φ（A）‖=‖AB＋BA‖ for all A,B∈Bs（H）,then there is a unitary operator or a conjugate unitary operator U and a functional h from Bs（H） to {1,-1} such that Φ（X）=h（X）UXU for all X∈Bs（H）.