TY - JOUR
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
UR - https://www.scopus.com/pages/publications/105003469165
U2 - 10.1007/s11587-025-00959-9
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 -