Properties of operator matrices
J. Korean Math. Soc. 2020 Vol. 57, No. 4, 893-913
https://doi.org/10.4134/JKMS.j190439
Published online July 1, 2020
Il Ju An, Eungil Ko, Ji Eun Lee
Kyung Hee University; Ewha Womans University; Sejong University
Abstract : 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).
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