On the Drazin inverse for upper triangular operator matrices. (English) Zbl 1312.47003

Zguitti, Hassane

Bull. Math. Anal. Appl. 2, No. 2, 27-33 (2010).

Summary: In this paper we investigate the stability of Drazin spectrum \(\sigma_D(.)\) for upper triangular operator matrices \(M_C= \left[\begin{matrix} A& C\\ 0 & B\end{matrix}\right]\) using tools from local spectral theory. We show that \(\sigma_D(M_C) \cup [\mathcal{S}(A^\ast)\cap\mathcal{S}(B)]=\sigma_D(A)\cup\sigma_D(B)\) where \(\mathcal{S}(.)\) is the set where an operator fails to have the SVEP. As application we explore how the generalized Weyl’s theorem survives for \(M_C\).

MSC:

47A05	General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A11	Local spectral properties of linear operators

Keywords:

operator matrices; Drazin spectrum; single-valued extension property