Common local spectral properties for the operators AC and BD

Abstract

In [6], Chen and Abdolyousefi studied the common spectral properties of operators AC and BD satisfying (AC)²A = ACDBA = DBACA = (DB)²A; (AC)²D = ACDBD = DBACD = (DB)²D. In this note, we continue studying the common local spectral properties of these operators. We show that AC and BD shared the single-valued extension property, the Bishop property (β), the property (β_ϵ), the decomposition property (δ), and decomposability. Furthermore, we investigate the closedness of the analytic core and the quasinilpotent part. We also examine the Dunford’s property (C) and the property (Q).