{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:38:27Z","timestamp":1760146707579,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,3]],"date-time":"2024-12-03T00:00:00Z","timestamp":1733184000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12301149","62272079","232102210187","23B210005"],"award-info":[{"award-number":["12301149","62272079","232102210187","23B210005"]}]},{"name":"Henan Provincial Department of Science and Technology Research Project","award":["12301149","62272079","232102210187","23B210005"],"award-info":[{"award-number":["12301149","62272079","232102210187","23B210005"]}]},{"name":"Key Scientific Research Projects of Colleges and Universities in Henan Province","award":["12301149","62272079","232102210187","23B210005"],"award-info":[{"award-number":["12301149","62272079","232102210187","23B210005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Motivated by recent advancements in the quantum detection problem employing both discrete and continuous frames, this paper delves into a quantum detection problem utilizing g-frames within the context of quaternionic Hilbert spaces. We offer several equivalent representations of injective g-frames in separable quaternionic Hilbert spaces. By normalizing the trace, we establish a classification for the g-frame injectivity problem. Additionally, we propose a method to derive an injective g-frame by leveraging an injective frame within quaternionic Hilbert spaces. Furthermore, we demonstrate that the injectivity of a g-frame remains intact under a linear isomorphism, while injective g-frames exhibit instability in infinite-dimensional scenarios.<\/jats:p>","DOI":"10.3390\/axioms13120851","type":"journal-article","created":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T10:07:10Z","timestamp":1733306830000},"page":"851","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on Injective g-Frames in Quaternionic Hilbert Spaces"],"prefix":"10.3390","volume":"13","author":[{"given":"Jianxia","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Intelligent Engineering, Henan Institute of Technology, Xinxiang 453003, China"},{"name":"School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fugen","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9382-5412","authenticated-orcid":false,"given":"Guoqing","family":"Hong","sequence":"additional","affiliation":[{"name":"School of Science, Henan Institute of Technology, Xinxiang 453003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Khokulan, M., Thirulogasanthar, K., and Srisatkunarajah, S. 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