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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.
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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