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 \(\varphi \in \operatorname {End}_k (\mathcal H)\) , we prove that the pseudo-characteristic polynomials of \(\varphi ^{D,*}\) and \(\varphi ^{*,D}\) coincide. Accordingly, we obtain that \(\sigma (\varphi ^{D,*}) = \sigma (\varphi ^{*,D})\) , \(\operatorname {Tr}_\mathcal {H} (\varphi ^{D,*}) = \operatorname {Tr}_\mathcal {H} (\varphi ^{*,D})\) and \(\operatorname {det}_\mathcal {H} (\text {Id} + \varphi ^{D,*}) = \operatorname {det}_\mathcal {H} (\text {Id} + \varphi ^{*,D})\) . In particular, these results hold for a finite square complex matrix A. Moreover, we offer the explicit characterization of the AST-decompositions of \(\mathcal H\) induced by the Group-Star and the Star-Group inverses of a bounded linear operator \(\psi \) on \(\mathcal H\) with \(i(\psi )\le 1\) .
