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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
##### Keywords:
spectra; local spectral theory; operator matrices
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