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Local spectral properties of a square matrix of operators. (Propriétés spectrales locales d’une matrice carrée des opérateurs.) (French) Zbl 0970.47003
Summary: If \(X\) and \(Y\) are complex Banach spaces, then for \(A\in{\mathcal L}(X)\), \(B\in{\mathcal L}(Y)\) and \(C\in{\mathcal L}(Y, X)\) we denote by \(M_C\) the operator defined on \(X\oplus Y\) by \[ M_C= \begin{pmatrix} A & C\\ 0 & B\end{pmatrix}. \] When \(B\) has SVEP, we show that \(\sigma(M_C)= \sigma(A)\cup \sigma(B)\) for all \(C\in{\mathcal L}(Y, X)\). And in the Hilbert space setting, this result gives a partial positive answer to the question 3 posed in [Hong-Ke Du and Jin Pan, Proc. Am. Math. Soc. 121, No. 3, 761-766 (1994; Zbl 0814.47016)].

47A11 Local spectral properties of linear operators
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47A10 Spectrum, resolvent
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)