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A-Browder-type theorems for direct sums of operators. (English) Zbl 1389.47044
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 $$(\text{SBaw})$$, $$(\text{SBab})$$, $$(\text{SBw})$$ and $$(\text{SBb})$$ 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 $$(\text{SBab})$$, then $$S\oplus T$$ has the property $$(\text{SBab})$$ if and only if $$\sigma_{\text{SBF}_+^-}(S\oplus T)=\sigma_{\text{SBF}_+^-}(S)\cup\sigma_{\text{SBF}_+^-}(T)$$, where $$\sigma_{\text{SBF}_{+}^{-}}(T)$$ is the upper semi-B-Weyl spectrum of $$T$$.
We obtain analogous preservation results for the properties $$(\text{SBaw})$$, $$(\text{SBb})$$ and $$(\text{SBw})$$ with extra assumptions.
##### MSC:
 47A53 (Semi-) Fredholm operators; index theories 47A55 Perturbation theory of linear operators 47A10 Spectrum, resolvent 47A11 Local spectral properties of linear operators
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