Defect set theory and generalized Drazin invertibility of operator matrices

The defect set, a fundamental concept in local spectral theory, serves as a well-established criterion in the study of generalized Drazin invertibility. This paper emphasizes its significance by examining the left and right generalized Drazin invertibility of upper triangular operator matrices within Banach spaces. Additionally, it aims to build on the recent advancements made by Bahloul and Walha [Generalized Drazin invertibility of operator matrices, Numer. Funct. Anal. Opt. 43(16) (2022) 1836-1847]. Our approach focuses on establishing sufficient conditions, using the defect set within local spectral theory, to explore the relationship between the generalized Drazin-type spectra of 3 x 3 upper triangular block operator matrices and those associated with their diagonal entries. Specifically, this contribution addresses the questions raised by Zguitti [A note on Drazin invertibility for upper triangular block operators, Mediterr. J. Math. 10 (2013) 1497-1507] and tackles the challenging problem outlined by Campbell [The Drazin inverse and systems of second order linear differential equations, Linear Multilinear Algebra 14 (1983) 195-198] regarding the representation of left and right generalized Drazin spectra for 3 x 3 block operator matrices.