A-Browder-type theorems for direct sums of operators

summary: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