Further common local spectral properties for bounded linear operators

In this note, we study common local spectral properties for bounded linear operators $A\in \mathcal{L}(X,Y)$ and $B,C\in \mathcal{L}(Y,X)$ such that $$A(BA{)}^2=ABACA=ACABA=(AC{)}^{2}A\mathrm{.}$$ We prove that $AC$ and $BA$ share the single valued extension property, the Bishop property $(\beta )$, the property $({\beta }_{ϵ})$, the decomposition property $(\delta )$ and decomposability. Closedness of analytic core and quasinilpotent part are also investigated. Some applications to Fredholm operators are given.