Common properties of bounded linear operators AC and BA: Local spectral theory

Let X, Y be Banach spaces, A : X → Y and B , C : Y → X be bounded linear operators satisfying operator equation A B A = A C A . In this paper, we show that AC and BA share the local spectral properties such as Bishopʹs property (β), subscalarity and Dunfordʹs property (C). Also, the quasi-nilpotent part and the analytic core for AC and BA are studied.