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Keywords:
B-Fredholm operator; Browder's theorem; generalized Browder's theorem; property $({\rm b})$; property $({\rm gb})$
B-Fredholm operator; Browder's theorem; generalized Browder's theorem; property $({\rm b})$; property $({\rm gb})$
Summary:
An operator $T$ acting on a Banach space $X$ possesses property $({\rm gw})$ if $\sigma _a(T)\setminus \sigma _{{\rm SBF}_+^-}(T)= E(T), $ where $\sigma _a(T)$ is the approximate point spectrum of $T$, $\sigma _{{\rm SBF} _+^-}(T)$ 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 $({\rm b})$ and $({\rm gb})$ in connection with Weyl type theorems, which are analogous respectively to Browder's theorem and generalized Browder's theorem. \endgraf Among other, we prove that if $T$ is a bounded linear operator acting on a Banach space $X$, then property $({\rm gw})$ holds for $T$ if and only if property $({\rm gb})$ holds for $T$ and $E(T)=\Pi (T),$ where $\Pi (T)$ is the set of all poles of the resolvent of $T.$
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