On The Spectral Properties for Drazin Invertible Operator

Abstract

This dissertation presents a comprehensive study of the local spectral properties of Drazin invertible operators on complex Banach spaces, building upon the foundational work of Aiena and Triolo. We systematically investigate the transmission of key spectral properties including the single valued extension property (SVEP), Dunford's property (C), Bishop's property (β), and property (Q) between an operator R and its Drazin inverse S. The analytical framework relies heavily on the core nilpotent decomposition, which enables an independent treatment of the invertible and nilpotent parts. We establish a reciprocal relationship for the non-zero points of the local spectrum and prove that the aforementioned properties are indeed transmission of the polaroid and a-polaroid properties, and discuss the algebraic and spectral implications for related operators such as RS and SR. This work aims to provide a self-contained, detailed exposition of these results, supplemented by the necessary background material and illustrative examples.