Common local spectral properties for the operator products

Abstract

In their paper [Generalized Clines Formula for g-Drazin Inverse in a Ring, Filomat 35(8) (2021) 2573–2583], Chen and Abdolyousefi examined the common spectral properties of operators [Formula: see text] and [Formula: see text], subject to the following conditions: [Formula: see text] [Formula: see text] In this note, we explore the common local spectral properties of these operators. We demonstrate that [Formula: see text] and [Formula: see text] shared the single-valued extension property, the Bishop property [Formula: see text], the property [Formula: see text], the decomposition property [Formula: see text], and decomposability. Furthermore, we investigate the closedness of the analytic core and the quasinilpotent part. We also delve into the Dunford’s property [Formula: see text] and the property [Formula: see text]