有界线性算子的单值扩张性质的摄动

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

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。

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 allλ∈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.