有界线性算子的单值扩张性质的摄动

设H是复可分无限维Hilbert空间,B(H)为H上的有界线性算子的全体。Hilbert空间H中一个算子T称作有单值扩张性质(简写为SVEP,记作T∈(SVEP)),若对任意一个开集U∈C,满足方程(T-λI)f(λ)=0(∀λ∈U)的唯一的解析函数为零函数,其中C代表复数集。T∈B(H)称为满足单值扩张性质的紧摄动,若对任意的紧算子K∈K(H),T+K满足单值扩张性质。 讨论了有界线性算子满足单值扩张性质的紧摄动的判定条件,同时给出了2×2上三角算子矩阵满足单值扩张性质的紧摄动的充要条件。

The perturbation of the single valued extension property for bounded linear operators

Let H be an infinite dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. An operator T∈B(H) is said to have the single-valued extension property(SVEP for brevity, write T∈ (SVEP)), if for every open set U∈C, the only analytic solution f:U→X of the equation (T-λI)f(λ)=0 for all λ∈U is zero function on U, where C denotes the complex number set. T∈B(H) is said to have the perturbations of the single valued extension property if T+K have the single-valued extension property for every compact operator K∈K(H). The perturbations of the single valued extension property for bounded linear operators are discussed, and the sufficient necessary condition for is given 2×2 upper triangular operator matrices for which the single valued extension property is stable under compact perturbations.