Drazin Invertibility of Operators AB and BA

Abstract:

      In this note an alternative proof of the equivalence of Drazin invertibility of operators $AB$ and $BA$ is given. As an application, we will prove that $\sigma_D(AB)=\sigma_D(BA)$ and $\sigma_D(A)=\sigma_D(\widetilde{A})$, where $\sigma_D(M)$ and $\widetilde{M}$ denote the Drazin spectrum and the Aluthge transform of an operator $M\in{\cal B}({\cal H})$, respectively.