Operator matrices and their Weyl type theorems

Abstract

We denote the collection of the $2\times 2$ operator matrices with $(1, 2)$-entries having closed range by $\mathcal{S}$.

In this paper, we study the relations between the operator matrices in the class $\mathcal{S}$ and their component operators in terms of the Drazin spectrum and left Drazin spectrum, respectively. As some application of them, we investigate how the generalized Weyl's theorem and the generalized $a$-Weyl's theorem hold for operator matrices in $\mathcal{S}$, respectively. In addition, we provide a simple example about an operator matrix in $\mathcal S$ satisfying such Weyl type theorems.