Common properties of the operator products in local spectral theory

Let X, Y be Banach spaces, A,D: X . Y and B,C: Y . X be the bounded linear operators satisfying operator equation set $\left\{ ACD=DBD,DBA=ACA. \right. $. In this paper, we show that AC and BD share some basic operator properties such as the injectivity and the invertibility. Moreover, we show that AC and BD share many common local spectral properties including SVEP, Bishop property (.) and Dunford property (C).