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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)].

##### MSC:
 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.)