On New Properties of the Drazin-Star and the Star-Drazin Inverses
[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 $\varphi^{D,*}$...
| Authors: | , |
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| Format: | article |
| Status: | Published version |
| Publication Date: | 2025 |
| Country: | España |
| Institution: | Universidad de Salamanca (USAL) |
| Repository: | GREDOS. Repositorio Institucional de la Universidad de Salamanca |
| OAI Identifier: | oai:gredos.usal.es:10366/168475 |
| Online Access: | http://hdl.handle.net/10366/168475 |
| Access Level: | Open access |
| Keyword: | Spectrum Trace Determinant Star-Drazin inverse Drazin-Star inverse Bounded operator Hilbert space Finite potent endomorphism Square matrix 12 Matemáticas |
| Summary: | [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 $i(\psi)\leq 1$. |
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