Drazin and group invertibility in algebras spanned by two idempotents
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916245094268928 |
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| author | Biswas, Rounak Roy, Falguni |
| author_facet | Biswas, Rounak Roy, Falguni |
| contents | For two given idempotents $p\text{ and }q$ from an associative algebra $\mathcal{A},$ in this paper, we offer a comprehensive classification of algebras spanned by the idempotents $p\text{ and }q$. This classification is based on the condition that $p\text{ and }q$ are not tightly coupled and satisfies $(pq)^{m-1}=(pq)^{m}$ but $(pq)^{m-2}p\neq (pq)^{m-1}p$ for some $m(\geq2)\in\mathbb{N}.$ Subsequently, we categorized all the group invertible elements and established an upper bound for Drazin index of any elements in these algebras spanned by $p,q$. Moreover, we formulate a new representation for the Drazin inverse of $(αp+q)$ under two different assumptions, $(pq)^{m-1}=(pq)^m$ and $λ(pq)^{m-1}=(pq)^m,$ here $α$ is a non-zero and $λ$ is a non-unit real or complex number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11460 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Drazin and group invertibility in algebras spanned by two idempotents Biswas, Rounak Roy, Falguni Functional Analysis Operator Algebras Rings and Algebras 15A30, 15A09, 16S15, 17C27 For two given idempotents $p\text{ and }q$ from an associative algebra $\mathcal{A},$ in this paper, we offer a comprehensive classification of algebras spanned by the idempotents $p\text{ and }q$. This classification is based on the condition that $p\text{ and }q$ are not tightly coupled and satisfies $(pq)^{m-1}=(pq)^{m}$ but $(pq)^{m-2}p\neq (pq)^{m-1}p$ for some $m(\geq2)\in\mathbb{N}.$ Subsequently, we categorized all the group invertible elements and established an upper bound for Drazin index of any elements in these algebras spanned by $p,q$. Moreover, we formulate a new representation for the Drazin inverse of $(αp+q)$ under two different assumptions, $(pq)^{m-1}=(pq)^m$ and $λ(pq)^{m-1}=(pq)^m,$ here $α$ is a non-zero and $λ$ is a non-unit real or complex number. |
| title | Drazin and group invertibility in algebras spanned by two idempotents |
| topic | Functional Analysis Operator Algebras Rings and Algebras 15A30, 15A09, 16S15, 17C27 |
| url | https://arxiv.org/abs/2402.11460 |