New results on common properties of the products AC and BA

Abstract

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 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)-quasiparanormal operator T. Also, Cline's formula for Drazin inverse in a ring with identity is established in the case when aba=aca, and in this case we establish Cline's formula for generalized Drazin inverse in the setting of Banach algebra.