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Mohammed Berkani, Department of Mathematics, Science Faculty of Oujda, University Mohammed I, Team EQUITOMI, SFO, Laboratory MATSI, EST, e-mail: berkanimo@aim.com; Hassan Zariouh, Department of Mathematics, Science Faculty of Meknes, University Moulay Ismail, e-mail: h.zariouh@yahoo.fr
Abstract: An operator $T$ acting on a Banach space $X$ possesses property $(\gw)$ if $\sigma_a(T)\setminus\sigma_{\SBF_+^-}(T)= E(T), $ where $\sigma_a(T)$ is the approximate point spectrum of $T$, $\sigma_{\SBF_+^-}(T)$ is the essential semi-B-Fredholm spectrum of $T$ and $E(T)$ is the set of all isolated eigenvalues of $T.$ In this paper we introduce and study two new properties $(\b)$ and $(\gb)$ in connection with Weyl type theorems, which are analogous respectively to Browder's theorem and generalized Browder's theorem. Among other, we prove that if $T$ is a bounded linear operator acting on a Banach space $X$, then property $(\gw)$ holds for $T$ if and only if property $(\gb)$ holds for $T$ and $E(T)=\Pi(T),$ where $\Pi(T)$ is the set of all poles of the resolvent of $T.$
Keywords: B-Fredholm operator, Browder's theorem, generalized Browder's theorem, property $(\b)$, property $(\gb)$
Classification (MSC 2000): 47A53, 47A10, 47A11
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