A Note on Property $(\omega)$
Received:August 10, 2009  Revised:April 27, 2010
Key Word: approximate Weyl's theorem   property $(\omega)$   Browder operator.
Fund ProjectL:Supported by the Fundamental Research Funds for the Central Universities (Grant No.GK200901015), the Support Plan of the New Century Talented Person of Ministry of Education (2006), P.R. China and by Major Subject Foundation of Shanxi.
 Author Name Affiliation Ji Rong WANG Department of Mathematics, Yuncheng University, Shanxi 044000, P. R. China Xiao Hong CAO College of Mathematics and Information Science, Shaanxi Normal University, Shaanxi 710062, P. R. China
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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.