数值域的函数演算

A functional calculus for numerical ranges

目的主要刻画复Hilbert空间H上的有界线性算子A,B都是单位算子的常数倍。方法利用解析函数的性质及算子分块的性质。结果与结论证明了对复Hilbert空间H上的有界线性算子A,B∈B(H)和B的数值域W(B)上的非线性解析函数g,若对任意的单位向量x∈H,有(Ax,x)=g((Bx,x)),则A和B都是单位算子的常数倍。

Aim It is described that two bounded linear operators A,B on a complex Hilbert space H are multiples of identity.Methods The properties of analytic function and block of operators.Results and Conclusion It is proved that for any bounded linear operators A and B on a complex Hilbert space H and a non-linear analytic function g on the numerical range W（B） of B,if（Ax,x）=g（（Bx,x）） for all unit vectors x∈H,then both A and B are multiples of identity.