A note on common properties of the products ac and ba

Yanxun Ren; Lining Jiang; Qiaoling Xin

doi:10.1007/s11587-025-00959-9

A note on common properties of the products ac and ba

Yanxun Ren^*
, Lining Jiang
, Qiaoling Xin

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

Let R be an associative ring with unit 1, and let a,b,c∈R satisfy (ac)2a=abaca=acaba=a(ba)2. We prove that if α=1-ba is generalized Drazin invertible, then 1-ac is generalized Drazin invertible. This extends the results given by Chen and Abdolyousefi (Comm. Algebra, 49 (2021) 3263-3272) from Banach algebras to rings. Moreover, Jacobson’s lemma for generalized Fredholm elements relative to an ideal and Fredholm elements relative to a trace ideal is investigated in rings and in semisimple Banach algebras, respectively. Applying the above results, norm closure of hypercyclic operators is considered.

Original language	English
Pages (from-to)	2949-2964
Number of pages	16
Journal	Ricerche di Matematica
Volume	74
Issue number	5
DOIs	https://doi.org/10.1007/s11587-025-00959-9
Publication status	Published - Nov 2025
Externally published	Yes

Keywords

Drazin inverses
Fredholm elements
Generalized Fredholm elements
Hypercyclic operator
Jacobson’s lemma

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10.1007/s11587-025-00959-9

Cite this

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title = "A note on common properties of the products ac and ba",

abstract = "Let R be an associative ring with unit 1, and let a,b,c∈R satisfy (ac)2a=abaca=acaba=a(ba)2. We prove that if α=1-ba is generalized Drazin invertible, then 1-ac is generalized Drazin invertible. This extends the results given by Chen and Abdolyousefi (Comm. Algebra, 49 (2021) 3263-3272) from Banach algebras to rings. Moreover, Jacobson{\textquoteright}s lemma for generalized Fredholm elements relative to an ideal and Fredholm elements relative to a trace ideal is investigated in rings and in semisimple Banach algebras, respectively. Applying the above results, norm closure of hypercyclic operators is considered.",

keywords = "Drazin inverses, Fredholm elements, Generalized Fredholm elements, Hypercyclic operator, Jacobson{\textquoteright}s lemma",

author = "Yanxun Ren and Lining Jiang and Qiaoling Xin",

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doi = "10.1007/s11587-025-00959-9",

language = "English",

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T1 - A note on common properties of the products ac and ba

AU - Ren, Yanxun

AU - Jiang, Lining

AU - Xin, Qiaoling

N1 - Publisher Copyright: © Università degli Studi di Napoli "Federico II" 2025.

PY - 2025/11

Y1 - 2025/11

N2 - Let R be an associative ring with unit 1, and let a,b,c∈R satisfy (ac)2a=abaca=acaba=a(ba)2. We prove that if α=1-ba is generalized Drazin invertible, then 1-ac is generalized Drazin invertible. This extends the results given by Chen and Abdolyousefi (Comm. Algebra, 49 (2021) 3263-3272) from Banach algebras to rings. Moreover, Jacobson’s lemma for generalized Fredholm elements relative to an ideal and Fredholm elements relative to a trace ideal is investigated in rings and in semisimple Banach algebras, respectively. Applying the above results, norm closure of hypercyclic operators is considered.

AB - Let R be an associative ring with unit 1, and let a,b,c∈R satisfy (ac)2a=abaca=acaba=a(ba)2. We prove that if α=1-ba is generalized Drazin invertible, then 1-ac is generalized Drazin invertible. This extends the results given by Chen and Abdolyousefi (Comm. Algebra, 49 (2021) 3263-3272) from Banach algebras to rings. Moreover, Jacobson’s lemma for generalized Fredholm elements relative to an ideal and Fredholm elements relative to a trace ideal is investigated in rings and in semisimple Banach algebras, respectively. Applying the above results, norm closure of hypercyclic operators is considered.

KW - Drazin inverses

KW - Fredholm elements

KW - Generalized Fredholm elements

KW - Hypercyclic operator

KW - Jacobson’s lemma

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DO - 10.1007/s11587-025-00959-9

M3 - Article

AN - SCOPUS:105003469165

SN - 0035-5038

VL - 74

SP - 2949

EP - 2964

JO - Ricerche di Matematica

JF - Ricerche di Matematica

IS - 5

ER -

A note on common properties of the products ac and ba

Abstract

Keywords

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