Bulletin of the
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Bull. Korean Math. Soc. 2012; 49(5): 1027-1040

Printed September 30, 2012

https://doi.org/10.4134/BKMS.2012.49.5.1027

Copyright © The Korean Mathematical Society.

Weyl type-theorems for direct sums

Mohammed Berkani and Hassan Zariouh

Operator theory team, SFO, University Moulay Ismail

Abstract

The aim of this paper is to study the Weyl type-theorems for the orthogonal direct sum $S\oplus T$, where $S$ and $T$ are bounded linear operators acting on a Banach space $X$. Among other results, we prove that if both $T$ and $S$ possesses property $(gb)$ and if $\Pi(T)\subset\sigma_a(S)$, $\Pi(S)\subset\sigma_a(T)$, then $S\oplus T$ possesses property $(gb)$ if and only if $\sigma_{SBF_+^-}(S\oplus T)=\sigma_{SBF_+^-}(S)\cup \sigma_{SBF_+^-}(T)$. Moreover, we prove that if $T$ and $S$ both satisfies generalized Browder's theorem, then $S\oplus T$ satisfies generalized Browder's theorem if and only if $\sigma_{BW}(S\oplus T)=\sigma_{BW}(S)\cup \sigma_{BW}(T)$.

Keywords: property $(gb)$, property $(b)$, property $(gw)$, direct sums, essential semi-B-Fredholm spectrum

MSC numbers: Primary 47A53, 47A10, 47A11

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