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

Abstract

[EN]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 $\varphi \in \ed_k (\mathcal H)$, we prove that the pseudo-characteristic polynomials of $\varphi^{D,*}$ and $\varphi^{*,D}$ coincide. Accordingly, we obtain that $\sigma (\varphi^{D,*}) = \sigma (\varphi^{*,D})$, $\tr_{\mathcal H} (\varphi^{D,*}) = \tr_{\mathcal H} (\varphi^{*,D})$ and $\Det_{\mathcal H} (\text{Id} + \varphi^{D,*}) = \Det_{\mathcal H} (\text{Id} + \varphi^{*,D})$. 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 $i(\psi)\leq 1$.