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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 52 Issue: 4

Generalized invertibility in two semigroups of Banach algebras

Wende Lı(Southeast University)0000-0003-2558-9117
Jianlong Chen(Southeast University)0000-0002-6798-488X
Yuanyuan Ke(Jianghan University)0000-0003-1123-5289
Pages
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Publication Date
2023-08-15
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Then this result leads to a remarkable behavior of pseudo Drazin invertibility (resp., generalized Drazin invertibility) between the operators in the semigroup $AA^{-}\mathscr{B}(Y)AA^{-}+I_Y-AA^{-}$ and the semigroup $A^{=}A\mathscr{B}(X)A^{=}A+I_X-A^{=}A$, where $A^{-}, A^{=}\in \mathscr{B}(Y,X)$ are inner inverses of $A\in \mathscr{B}(X,Y)$.

📌 3 Key Takeaways

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Objective & Hypothesis:Motivated by the results involving Drazin inverses of Patrício and Puystjens, we investigate the relevant results for pseudo Drazin invertibility and generalized Drazin invertibility in two semigroups of Banach algebras.
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Key Findings & Data:Given a Banach algebra $\mathcal{A}$ and $e^2=e\in \mathcal{A}$, we firstly establish the relation between pseudo Drazin invertibility (resp., generalized Drazin invertibility) of elements in $e\mathcal{A}e$ and $e\mathcal{A}e+1-e$.
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Conclusion & Impact:Then this result leads to a remarkable behavior of pseudo Drazin invertibility (resp., generalized Drazin invertibility) between the operators in the semigroup $AA^{-}\mathscr{B}(Y)AA^{-}+I_Y-AA^{-}$ and the semigroup $A^{=}A\mathscr{B}(X)A^{=}A+I_X-A^{=}A$, where $A^{-}, A^{=}\in \mathscr{B}(Y,X)$ are inner inverses of $A\in \mathscr{B}(X,Y)$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:generalized Drazin invertibility · pseudo Drazin invertibility · Banach algebra

Abstract

Motivated by the results involving Drazin inverses of Patrício and Puystjens, we investigate the relevant results for pseudo Drazin invertibility and generalized Drazin invertibility in two semigroups of Banach algebras. Given a Banach algebra $\mathcal{A}$ and $e^2=e\in \mathcal{A}$, we firstly establish the relation between pseudo Drazin invertibility (resp., generalized Drazin invertibility) of elements in $e\mathcal{A}e$ and $e\mathcal{A}e+1-e$. Then this result leads to a remarkable behavior of pseudo Drazin invertibility (resp., generalized Drazin invertibility) between the operators in the semigroup $AA^{-}\mathscr{B}(Y)AA^{-}+I_Y-AA^{-}$ and the semigroup $A^{=}A\mathscr{B}(X)A^{=}A+I_X-A^{=}A$, where $A^{-}, A^{=}\in \mathscr{B}(Y,X)$ are inner inverses of $A\in \mathscr{B}(X,Y)$.

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Cite This Article

Lı, W., Chen, J., Ke, Y. (2023). Generalized invertibility in two semigroups of Banach algebras. Hacettepe Journal of Mathematics and Statistics, Volume: 52 Issue: 4. https://doi.org/10.15672/hujms.1099257

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