Back to the common spectral properties of operators satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

Abstract

In [A. Gupta and A. Kumar, Common spectral properties of linear operators [Formula: see text] and [Formula: see text] satisfying [Formula: see text] and [Formula: see text], Asian-European J. Math. 12(5) (2019) 1950084], Gupta and Kumar studied the common spectral properties for two Banach bounded linear operators [Formula: see text] and [Formula: see text] satisfying [Formula: see text] and [Formula: see text] for some positive integer [Formula: see text]. In this note, we study the common local spectral properties for such operators by showing that [Formula: see text] and [Formula: see text] share the single valued extension property, the Bishop property [Formula: see text] the property [Formula: see text] the decomposition property [Formula: see text] and decomposability. The closedness of analytic core and the quasinilpotent part and then the Dunford’s property [Formula: see text] and the property [Formula: see text] are also investigated.