New results on common properties of the products AC and BA

Let X, Y   be Banach spaces, A:X⟶YA:X⟶Y and B,C:Y⟶XB,C:Y⟶X be bounded linear operators satisfying operator equation ABA=ACAABA=ACA. In this paper, we show that the products AC and BA   share the spectral properties such as Drazin invertibility, polaroidness and B-Fredholmness. As an application, we show that generalized Weyl's theorem holds for the Aluthge transform T˜ of an algebraically (n,k)(n,k)-quasiparanormal operator T  . Also, Cline's formula for Drazin inverse in a ring with identity is established in the case when aba=acaaba=aca, and in this case we establish Cline's formula for generalized Drazin inverse in the setting of Banach algebra.