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Spectra of upper triangular operator matrices. (English) Zbl 1067.47005
Summary: Let \(X, Y\) be given Banach spaces. For \(A\in{\mathcal L}(X),\,B\in{\mathcal L}(Y)\) and \(C\in{\mathcal L}(Y,X)\), let \(M_C\) be the operator defined on \(X\oplus Y\) by \( M_C = \left[\begin{smallmatrix} A & C\\ 0 & B \end{smallmatrix}\right]\). We give sufficient conditions on \(C\) to get \(\Sigma(M_C) = \Sigma(M_0),\) where \(\Sigma\) runs over a large class of spectra. We also discuss the case of some spectra for which the latter equality fails.

MSC:
47A11 Local spectral properties of linear operators
47A10 Spectrum, resolvent
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