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 ABA=ACA. In this paper, we show that AC and BA share the local spectral properties such as Bishop's property (β), subscalarity and Dunford's property (C). Also, the quasi-nilpotent part and the analytic core for AC and BA are studied.