Journal of the
Korean Mathematical Society JKMS

Let ${\mathcal S}$ be the collection of the operator matrices $\left(\begin{smallmatrix} A & C \cr Z & B\end{smallmatrix}\right)$ where the range of $C$ is closed. In this paper, we study the properties of operator matrices in the class ${\mathcal S}$. We first explore various local spectral relations, that is, the property $(\beta)$, decomposable, and the property $(C)$ between the operator matrices in the class $\mathcal{S}$ and their component operators. Moreover, we investigate Weyl and Browder type spectra of operator matrices in the class $\mathcal S$, and as some applications, we provide the conditions for such operator matrices to satisfy $a$-Weyl's theorem and $a$-Browder's theorem, respectively.

Keywords: $2\times 2$ operator matrices, the property $(\beta)$, decomposable, the property $(C)$, Browder essential approximate point spectrum, Weyl's theorem, a-Weyl's theorem, a-Browder's theorem

MSC numbers: Primary 47A53, 47A55, 47A10, 47B40

Supported by: The first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(2017R1C1B1006538).
The second author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(2019R1F1A1058633).
The third author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2019R1A2C1002653).