Weyl type-theorems for direct sums
Bull. Korean Math. Soc. 2012 Vol. 49, No. 5, 1027-1040
https://doi.org/10.4134/BKMS.2012.49.5.1027
Printed September 30, 2012
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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