{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:10:39Z","timestamp":1760148639237,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,5,26]],"date-time":"2023-05-26T00:00:00Z","timestamp":1685059200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Distinguished Scientist Fellowship Program, King Saud University, Riyadh, Saudi Arabia","award":["RSP2023R187"],"award-info":[{"award-number":["RSP2023R187"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper presents new results related to Bombieri\u2019s generalization of Bessel\u2019s inequality in a semi-inner product space induced by a positive semidefinite operator A. Specifically, we establish new inequalities that generalize the classical Bessel inequality and extend previous results in this area. Furthermore, our findings have applications to the study of operators on positive semidefinite inner product spaces, also known as semi-Hilbert spaces, and contribute to a deeper understanding of their properties and applications. Our work has implications for various fields, including functional analysis and operator theory.<\/jats:p>","DOI":"10.3390\/axioms12060522","type":"journal-article","created":{"date-parts":[[2023,5,27]],"date-time":"2023-05-27T16:18:43Z","timestamp":1685204323000},"page":"522","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Bombieri-Type Inequalities and Their Applications in Semi-Hilbert Spaces"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7442-8841","authenticated-orcid":false,"given":"Najla","family":"Altwaijry","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2902-6805","authenticated-orcid":false,"given":"Silvestru Sever","family":"Dragomir","sequence":"additional","affiliation":[{"name":"Mathematics, College of Sport, Health and Engineering, Victoria University, P.O. Box 14428, Melbourne City, VIC 8001, Australia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9326-4173","authenticated-orcid":false,"given":"Kais","family":"Feki","sequence":"additional","affiliation":[{"name":"Faculty of Economic Sciences and Management of Mahdia, University of Monastir, Mahdia 5111, Tunisia"},{"name":"Laboratory Physics-Mathematics and Applications (LR\/13\/ES-22), Faculty of Sciences of Sfax, University of Sfax, Sfax 3018, Tunisia"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/BF01875849","article-title":"On Bessel\u2019s and Gaur\u2019s inequality in prehilbertian spaces","volume":"29","author":"Dragomir","year":"1994","journal-title":"Period. Math. Hung."},{"key":"ref_2","first-page":"77","article-title":"Some remarks on Bessel\u2019s inequality in inner product spaces, Studia Univ","volume":"37","author":"Dragomir","year":"1992","journal-title":"Babes-Bolyai Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Mitrinovi\u2019c, D.S., Pe\u010dari\u2019c, J.E., and Fink, A.M. (1993). Classical and New Inequalities in Analysis, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-017-1043-5_18"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"401","DOI":"10.4064\/aa-18-1-401-404","article-title":"A note on the large sieve","volume":"18","author":"Bombieri","year":"1971","journal-title":"Acta Arith."},{"key":"ref_5","first-page":"456","article-title":"On Selberg\u2019s inequality","volume":"349","author":"Haroske","year":"2009","journal-title":"J. Math. Anal. 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Mathematics, 11.","DOI":"10.3390\/math11102293"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1460","DOI":"10.1016\/j.laa.2007.09.031","article-title":"Partial isometries in semi-Hilbertian spaces","volume":"428","author":"Arias","year":"2008","journal-title":"Linear Algebra Appl."},{"key":"ref_12","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 in Hilbert space","volume":"17","author":"Douglas","year":"1966","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s00020-008-1613-6","article-title":"Metric properties of projections in semi-Hilbertian spaces","volume":"62","author":"Arias","year":"2008","journal-title":"Integral Equ. Oper. Theory"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Altwaijry, N., Feki, K., and Minculete, N. (2023). A new seminorm for d-tuples of A-bounded operators and its applications. Mathematics, 11.","DOI":"10.3390\/math11030685"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Bhunia, P., Dragomir, S.S., Moslehian, M.S., and Paul, K. (2022). Lectures on Numerical Radius Inequalities, Springer.","DOI":"10.1007\/978-3-031-13670-2"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s43037-022-00185-7","article-title":"Some A-spectral radius inequalities for A-bounded Hilbert space operators","volume":"16","author":"Feki","year":"2022","journal-title":"Banach J. Math. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1016\/j.laa.2018.06.021","article-title":"Joint numerical ranges of operators in semi-Hilbertian spaces","volume":"555","author":"Baklouti","year":"2018","journal-title":"Linear Algebra Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1007\/s43034-020-00064-y","article-title":"Spectral radius of semi-Hilbertian space operators and its applications","volume":"11","author":"Feki","year":"2020","journal-title":"Ann. Funct. Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/522\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:42:34Z","timestamp":1760125354000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/522"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,26]]},"references-count":18,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["axioms12060522"],"URL":"https:\/\/doi.org\/10.3390\/axioms12060522","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,5,26]]}}}