Property $(\omega)$ and Its Perturbations
Received:December 30, 2010  Revised:December 19, 2011
Key Word: property $(\omega)$   perturbation   consistency in Fredholm and index.
Fund ProjectL:Supported by the Fundamental Research Funds for the Central Universities (Grant No.GK200901015).
 Author Name Affiliation Qiaoling XIN College of Mathematics and Information Science, Shaanxi Normal University, Shaanxi 710062, P. R. China Xiaohong CAO College of Mathematics and Information Science, Shaanxi Normal University, Shaanxi 710062, P. R. China
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In the note, we establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for the stability of property $(\omega)$ by means of the variant of the essential approximate point spectrum and the induced spectrum of consistency in Fredholm and index. In addition, the stability of property $(\omega)$ for $H(P)$ operators is considered.