A-Browder-type theorems for direct sums of operators

摘要

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 $(m SBaw)$, $(m SBab)$, $(m SBw)$ and $(m SBb)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $(m SBab)$, then $Søplus T$ has the property $(m SBab)$ if and only if $sigma_{m SBF_+^-}(Søplus T)=sigma_{m SBF_+^-}(S)cupsigma_{m SBF_+^-}(T)$, where $sigma_{m SBF_+^-}(T)$ is the upper semi-B-Weyl spectrum of $T$. We obtain analogous preservation results for the properties $({m SBaw)$, $(m SBb)$ and $(m SBw)$ with extra assumptions.}