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Local spectral theory for \(2\times 2\) operator matrices. (English) Zbl 1060.47003

Authors’ abstract: We discuss the spectral properties of the operator \(M_C\in{\mathfrak I}(X\oplus Y)\), defined by \(M_C= {AC\choose 0B}\), where \(A\in{\mathfrak I}(X)\), \(B\in{\mathfrak I}(Y)\), \(C\in{\mathfrak I}(Y,X)\), and \(X\), \(Y\) are complex Banach spaces. We prove that \((S_{A^*}\cap S_B)\cup\sigma(M_C)= \sigma(A)\cup\sigma(B)\) for all \(C\in{\mathfrak I}(Y,X)\). This allows us to give a partial positive answer to Question 3 of H.–K. Du and P. Jin [Proc. Am. Math. Soc. 121, No. 3, 761–766 (1994; Zbl 0814.47016)] and generalizations of some results of M. Houimdi and H. Zguitti [Acta Math. Vietnam. 25, No. 2, 137–144 (2000; Zbl 0970.47003)]. Some applications to the similarity problem are also given.

MSC:

47A10 Spectrum, resolvent
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