{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:33:19Z","timestamp":1760239999104,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,17]],"date-time":"2019-01-17T00:00:00Z","timestamp":1547683200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Based on the conditions     a  b 2  = 0     and      b \u03c0   ( a b )  \u2208   A  d     , we derive that      ( a b )  n    ,      ( b a )  n    , and     a b + b a     are all generalized Drazin invertible in a Banach algebra    A   , where     n \u2208 N     and a and b are elements of    A   . By using these results, some results on the symmetry representations for the generalized Drazin inverse of     a b + b a     are given. We also consider that additive properties for the generalized Drazin inverse of the sum     a + b    .<\/jats:p>","DOI":"10.3390\/sym11010105","type":"journal-article","created":{"date-parts":[[2019,1,17]],"date-time":"2019-01-17T11:30:27Z","timestamp":1547724627000},"page":"105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra"],"prefix":"10.3390","volume":"11","author":[{"given":"Yonghui","family":"Qin","sequence":"first","affiliation":[{"name":"College of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China"}]},{"given":"Xiaoji","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3222-3036","authenticated-orcid":false,"given":"Julio","family":"Ben\u00edtez","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46022 Valencia, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.laa.2004.11.001","article-title":"Additive perturbation results for the Drazin inverse","volume":"397","year":"2005","journal-title":"Linear Algebra Appl."},{"key":"ref_2","unstructured":"Ben-Israel, A., and Greville, T.N.E. 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