Geodesic
A-Browder-type theorems for direct sums of operators
Mathematica Bohemica, Tome 141 (2016) no. 1, pp. 99-108.

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We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $(\rm SBaw)$, $(\rm SBab)$, $(\rm SBw)$ and $(\rm SBb)$ are not preserved under direct sums of operators. \endgraf However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $(\rm SBab)$, then $S\oplus T$ has the property $(\rm SBab)$ if and only if $\sigma _{\rm SBF_+^-}(S\oplus T)=\sigma _{\rm SBF_+^-}(S)\cup \sigma _{\rm SBF_+^-}(T)$, where $\sigma _{\rm SBF_{+}^{-}}(T)$ is the upper semi-B-Weyl spectrum of $T$. \endgraf We obtain analogous preservation results for the properties $(\rm SBaw)$, $(\rm SBb)$ and $(\rm SBw)$ with extra assumptions.
DOI : 10.21136/MB.2016.8
Classification : 47A10, 47A11, 47A53, 47A55
Keywords: property $(\rm SBaw)$; property $(\rm SBab)$; upper semi-B-Weyl spectrum; direct sum 
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Berkani, Mohammed; Sarih, Mustapha; Zariouh, Hassan. A-Browder-type theorems for direct sums of operators. Mathematica Bohemica, Tome 141 (2016) no. 1, pp. 99-108. doi : 10.21136/MB.2016.8. http://geodesic.mathdoc.fr/articles/10.21136/MB.2016.8/

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