有界线性算子及其函数的（R）性质

设H为无限维复可分的Hilbert空间，B(H)为H中有界线性算子的全体。若{\sigma }_{\mathrm{a}}(T)\backslash {\sigma }_{\mathrm{a}\mathrm{b}}(T)?{\pi }_{\mathrm{00}}(T)，则称T\in B(H)满足(R_{\mathrm{1}})性质，其中{\sigma }_{\mathrm{a}}(T)和{\sigma }_{\mathrm{a}\mathrm{b}}(T)分别表示算子T的逼近点谱和Browder本质逼近点谱，；若{\sigma }_{\mathrm{a}}(T)\backslash {\sigma }_{\mathrm{a}\mathrm{b}}(T)={\pi }_{\mathrm{00}}(T)，则称T满足(R)性质。给出了有界线性算子满足(R_{\mathrm{1}})性质或(R)性质的充要条件，研究了算子函数满足(R_{\mathrm{1}})性质或(R)性质的判定方法，并讨论了完全*-paranormal算子及其函数的(R_{\mathrm{1}})性质或(R)性质。

Property (R) for bounded linear operator and its functions

Let H be an infinite dimensional complex separable Hilbert space and B(H) be the algebra of all bounded linear operators on H. T\in B(H) is said to satisfy property (R_{\mathrm{1}}) if {\sigma }_{\mathrm{a}}(T)\backslash {\sigma }_{\mathrm{a}\mathrm{b}}(T)?{\pi }_{\mathrm{00}}(T), where {\sigma }_{\mathrm{a}}(T) and {\sigma }_{\mathrm{a}\mathrm{b}}(T) denote the approximate point spectrum and the Browder essential approximate point spectrum of T respectively, and . If {\sigma }_{\mathrm{a}}(T)\backslash {\sigma }_{\mathrm{a}\mathrm{b}}(T)={\pi }_{\mathrm{00}}(T), T is said to satisfy property (R). In this paper, we give the necessary and sufficient conditions for which the property (R_{\mathrm{1}}) or property (R) holds for bounded linear operators. In addition, we characterize the judgements for operator functions satisfying property (R_{\mathrm{1}}) or property (R) and explored the property (R_{\mathrm{1}}) or property (R) for totally *-paranormal operators.