A Note on Property $(\omega)$

DOI：10.3770/j.issn:1000-341X.2011.01.011

 作者 单位 王济荣 山西运城学院应用数学系, 山西 运城 044000 曹小红 陕西师范大学数学与信息科学学院, 陕西 西安 710062

在本文中, 根据新定义的谱集,我们研究了由Rako$\mathrm{\breve{c}}$evi$\mathrm{\grave{c}}$介绍的Weyl定理的一种变化性质:性质$(\omega)$. 给出了Banach空间上有界线性算子同时满足性质$(\omega)$和逼近Weyl定理的充要条件. 利用所得的结论, 我们研究了$\lambda-$弱$-H(p)$算子的性质$(\omega)$和逼近Weyl定理.

In this note we study the property $(\omega)$, a variant of Weyl's theorem introduced by Rako\v{c}evi\`{c}, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property $(\omega)$ and approximate Weyl's theorem hold. As a consequence of the main result, we study the property $(\omega)$ and approximate Weyl's theorem for a class of operators which we call the $\lambda$-weak-$H(p)$ operators.