算子及其函数的(UW_(π))性质与(ω)性质

The property (UW_(π)) and property (ω) for operators and their functions

设H为无限维复可分的Hilbert空间,B(H)为H上的有界线性算子的全体.T∈B(H)称为满足(UW_(π))性质,若σ_(a)(T)\σ_(ea)(T)=P_(D)(T),其中P_(D)(T)=σ(T)\σ_(D)(T)={λ∈σ(T):T-λI是Drazin可逆算子},σ(T)={λ∈C:T-λI不是可逆算子};若σ_(a)(T)\σ_(ea)(T)=π_(00)(T),称T满足(ω)性质,其中σ_(a)(T)和σ_(ea)(T)分别表示算子T的逼近点谱和本质逼近点谱,π_(00)(T)={λ∈isoσ(T):0<dim N(T-λI)<∞}.该文首先给出了有界线性算子同时满足(UW_(π))性质以及(ω)性质的充要条件;之后研究了算子函数同时满足(UW_(π))性质以及(ω)性质的判定方法.

Let H be an infinite dimensional complex separable Hilbert space and B(H)be the algebra of all bounded linear operators on H.T∈B(H)is said to satisfy property(UW_(π))ifσ_(a)(T)\σ_(ea)(T)=P_(D)(T),where P_(D)(T)=σ(T)\σ_(D)(T)={λ∈σ(T):T-λI is Drazin invertible operator}.Ifσ_(a)(T)\σ_(ea)(T)=π_(00)(T),T is said to satisfy property(ω),where σ_(a)(T)and σ_(ea)(T)denote the approximate point spectrum and essential approximate point spectrum of T respectively,and π_(00)(T)={λ∈isoσ(T):0<dim N(T-λI)<∞}.Firstly,the necessary and sufficient conditions for which the property(UWπ)and the property(ω)hold for bounded linear operators are given.Then,the judgements for operator functions satisfying the property(UW_(π))and the property(ω)are studied.