Hilbert空间中数值域半径

Numerical radius of operator in Hilbert space

设B(H)是Hilbert空间H上全体有界线性算子构成的代数,I是H上的恒等算子,N(⋅)是B(H)上的任意范数,T∈B(H)。利用近似D-正交性给出数值域半径的推广,wN−D−ε(T)=sup{|ζ|:ζ∈C,I⊥ND−ε(T−ζI)}。证明了wN−D−ε(⋅)是B(H)上的一个半范数,给出wN−D−ε(⋅)成为B(H)上范数的一个充分必要条件。当wN−D−ε(⋅)是B(H)上的一个范数时,讨论了赋范线性空间(B(H),wN−D−ε(⋅))的几何结构以及相关性质。

Let B(H)be the algebra of all bounded linear operators on a Hilbert space H,I be the identity operator on H,T∈B(H).Let N(·)be an arbitrary norm on B(H).An extension of the numerical radius based on the approximate D-orthogonality by wN−D−ε(T)=sup{|ζ|:ζ∈C,I⊥ND−ε(T−ζI)}is given.It is proved that wN-D-ε(·)is a semi-norm on B(H).It is also given a necessary and sufficient condition that wN-D-ε(·)is a norm on B(H).When wN-D-ε(·)is a norm,the geometry and related properties of the normed linear space(B(H),wN-D-ε(·))are investigated.