On New Properties of the Drazin-Star and the Star-Drazin Inverses

The aim of this work is to study new properties of the Drazin-Star and the Star-Drazin inverses of a bounded finite potent operator on a Hilbert space. Given a bounded finite potent operator φ ∈ Endₖ ( H) φ ∈ End k ( H ) , we prove that the pseudo-characteristic polynomials of φ D,* φ D , ∗ and φ *,D φ ∗ , D coincide. Accordingly, we obtain that σ (φ D,*) = σ (φ *,D) σ ( φ D , ∗ ) = σ ( φ ∗ , D ) , TrH (φ D,*) = TrH (φ *,D) Tr H ( φ D , ∗ ) = Tr H ( φ ∗ , D ) and detH (Id + φ D,*) = detH (Id + φ *,D) det H ( Id + φ D , ∗ ) = det H ( Id + φ ∗ , D ) . In particular, these results hold for a finite square complex matrix A . Moreover, we offer the explicit characterization of the AST-decompositions of