{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T04:26:33Z","timestamp":1779337593333,"version":"3.51.4"},"reference-count":12,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,25]],"date-time":"2023-08-25T00:00:00Z","timestamp":1692921600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Natural Science Foundation of China","award":["12061031"],"award-info":[{"award-number":["12061031"]}]},{"name":"the National Natural Science Foundation of China","award":["2021JM-189"],"award-info":[{"award-number":["2021JM-189"]}]},{"name":"the National Natural Science Foundation of China","award":["120MS030"],"award-info":[{"award-number":["120MS030"]}]},{"name":"the National Natural Science Foundation of China","award":["120QN250"],"award-info":[{"award-number":["120QN250"]}]},{"name":"the Natural Science Basic Research Plan in Shaanxi Province of China","award":["12061031"],"award-info":[{"award-number":["12061031"]}]},{"name":"the Natural Science Basic Research Plan in Shaanxi Province of China","award":["2021JM-189"],"award-info":[{"award-number":["2021JM-189"]}]},{"name":"the Natural Science Basic Research Plan in Shaanxi Province of China","award":["120MS030"],"award-info":[{"award-number":["120MS030"]}]},{"name":"the Natural Science Basic Research Plan in Shaanxi Province of China","award":["120QN250"],"award-info":[{"award-number":["120QN250"]}]},{"name":"the Natural Science Basic Research Plan in the Hainan Province of China","award":["12061031"],"award-info":[{"award-number":["12061031"]}]},{"name":"the Natural Science Basic Research Plan in the Hainan Province of China","award":["2021JM-189"],"award-info":[{"award-number":["2021JM-189"]}]},{"name":"the Natural Science Basic Research Plan in the Hainan Province of China","award":["120MS030"],"award-info":[{"award-number":["120MS030"]}]},{"name":"the Natural Science Basic Research Plan in the Hainan Province of China","award":["120QN250"],"award-info":[{"award-number":["120QN250"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, using the technique of operator matrix, we consider the positive solution of the system of operator equations AX=B,XC=D in the framework of the Hilbert space; here, the ranges R(A) of A and R(C) of C are not necessarily closed. Firstly, we provide a new necessary and sufficient condition for the existence of positive solutions of AX=B and also provide a representation of positive solutions, which generalize previous conclusions. Furthermore, using the above result, a condition of equivalence for the existence of common positive solutions of AX=B,XC=D is given, as well as the general forms of positive solutions.<\/jats:p>","DOI":"10.3390\/axioms12090818","type":"journal-article","created":{"date-parts":[[2023,8,28]],"date-time":"2023-08-28T06:57:35Z","timestamp":1693205855000},"page":"818","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Positive Solutions of Operator Equations AX = B, XC = D"],"prefix":"10.3390","volume":"12","author":[{"given":"Haiyan","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanni","family":"Dou","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shaanxi Normal University, Xi\u2019an 710062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weiyan","family":"Yu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1137\/0131050","article-title":"Hermitian and nonnegative definite solutions of linear matrix equations","volume":"31","author":"Khatri","year":"1976","journal-title":"SIAM J. Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1016\/j.jmaa.2006.11.016","article-title":"Positive solutions to equations AX = C and XB = D for Hilbert space operators","volume":"333","author":"Koliha","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1779","DOI":"10.1016\/j.laa.2008.05.012","article-title":"Equations ax = c and xb = d in rings and rings with involution with applications to Hilbert space operators","volume":"429","author":"Koliha","year":"2008","journal-title":"Linear Algebra Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1090\/S0002-9939-1966-0203464-1","article-title":"On majorization, factorization, and range inclusion of operators on Hilbert spaces","volume":"17","author":"Douglas","year":"1966","journal-title":"Proc. Am. Math. 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Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.laa.2008.01.030","article-title":"Common hermitian and positive solutions to the adjointable operator equations AX = C,XB = D","volume":"429","author":"Xu","year":"2008","journal-title":"Linear Algebra Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.laa.2005.02.025","article-title":"Explicit expressions of the generalized inverses and condensed Cramer rules","volume":"404","author":"Ji","year":"2005","journal-title":"Linear Algebra Appl."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Halmos, P.R. (1982). A Hilbert Space Problem Book, Spinger.","DOI":"10.1007\/978-1-4684-9330-6"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/9\/818\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:39:24Z","timestamp":1760128764000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/9\/818"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,25]]},"references-count":12,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["axioms12090818"],"URL":"https:\/\/doi.org\/10.3390\/axioms12090818","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,25]]}}}