Operator Matrices and Their Weyl Type Theorems

Abstract

We denote the collection of the 2 x 2 operator matrices with (1, 2)-entries having closed range by S. In this paper, we study the relations between the operator matrices in the class 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 S, respectively. In addition, we provide a simple example about an operator matrix in S satisfying such Weyl type theorems.