自伴算子空间上满足[Φ(A^2),Φ(A)]=0的可加满射

Additive Maps Satisfying [Φ(A^2),Φ(A)]=0 on the Space of Self-adjoint Operators

令H为维数大于2的复Hilbert空间,B_s(H)为H上所有有界自伴算子构成的实线性空间.该文给出B_s(H)上满足[Φ(A^2),Φ(A)]=0对所有A∈B_s(H)成立的可加双射Φ的刻画,在Φ(F_s(H))■RI或RI■Φ(RI)的条件下证明了上述Φ具有形式Φ(A)=cUAU*+f(A)I,A∈B_s(H),其中c∈R,c≠0,U:H→H是酉算子或共轭酉算子,而f是B_s(H)上的可加泛函.

Let H be a complex Hilbert space with dimension greater than 2 and B_s（H） thespace of all self-adjoint operators in B（H）.A characterization is given for additive bijective mapΦon B_s（H） satisfying[Φ（A^2）,Φ（A）]=0 for all A∈B_s（H）.It is showed that,ifΦ（F_s（H））■RIor RI■Φ（RI）,thenΦhas the formΦ（A）=cU AU~＊＋f（A）I,A∈B_s（H）,where c∈R,c≠0,U：H→H is is an unitary or conjugate unitary operator,and f is an additive real functional ofB_s（H）.