## 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

### Keywords:

Banach space; linear operator; spectrum; operator matrix

### Citations:

Zbl 0814.47016; Zbl 0970.47003
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