{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T00:02:30Z","timestamp":1779062550565,"version":"3.51.4"},"reference-count":25,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,29]],"date-time":"2022-11-29T00:00:00Z","timestamp":1669680000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002383","name":"King Saud University","doi-asserted-by":"publisher","award":["RSP-2021\/187"],"award-info":[{"award-number":["RSP-2021\/187"]}],"id":[{"id":"10.13039\/501100002383","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we have developed new estimates of some estimates involving the Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space H\u03a9. The uniqueness or novelty of this article consists of new estimates of Berezin numbers for different types of operators. These estimates improve the upper bounds of the Berezin numbers obtained by other similar papers. We give several upper bounds for berr(S*T), where T,S\u2208B(H\u03a9) and r\u22651. We also present an estimation of ber2r\u2211i=1dTi where Ti\u2208B(H\u03a9), i=1,d\u00af and r\u22651. Some of the obtained inequalities represent improvements to earlier ones. In this work, the ideas and methodologies presented may serve as a starting point for future investigation in this field.<\/jats:p>","DOI":"10.3390\/axioms11120683","type":"journal-article","created":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T05:45:22Z","timestamp":1669787122000},"page":"683","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Some New Estimates for the Berezin Number of Hilbert Space Operators"],"prefix":"10.3390","volume":"11","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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0716-293X","authenticated-orcid":false,"given":"Nicu\u015for","family":"Minculete","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Transilvania University of Brasov, 500091 Brasov, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bhunia, P., Dragomir, S.S., Moslehian, M.S., and Paul, K. 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USSR-Izv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1070\/IM1974v008n05ABEH002140","article-title":"Quantizations","volume":"8","author":"Berezin","year":"1974","journal-title":"Math. USSR-Izv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"997","DOI":"10.21136\/CMJ.2018.0048-17","article-title":"Some Berezin number inequalities for operator matrices","volume":"68","author":"Bakherad","year":"2018","journal-title":"Czech. Math. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2327","DOI":"10.1090\/S0002-9939-04-07354-X","article-title":"Functional analysis proofs of Abels theorems","volume":"132","author":"Karaev","year":"2004","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_8","first-page":"185","article-title":"Some results on Berezin symbols","volume":"50","author":"Karaev","year":"2005","journal-title":"Complex Var. Theory Appl."},{"key":"ref_9","first-page":"362","article-title":"Boundary values of Berezin symbols","volume":"73","author":"Nordgren","year":"1994","journal-title":"Oper. Theory Adv. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1007\/s11785-012-0232-z","article-title":"Reproducing kernels and Berezin symbols techniques in various questions of operator theory","volume":"7","author":"Karaev","year":"2013","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1216\/rmj.2022.52.1039","article-title":"Berezin number inequalities of operators on reproducing kernel Hilbert spaces","volume":"52","author":"Sen","year":"2022","journal-title":"Rocky Mt. J. Math."},{"key":"ref_12","unstructured":"Bhunia, P., Sen, A., and Paul, K. (2022). Development of the Berezin number inequalities. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.jfa.2006.04.030","article-title":"Berezin symbol and invertibility of operators on the functional Hilbert spaces","volume":"238","author":"Karaev","year":"2006","journal-title":"J. Funct. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Moslehian, M.S. (2023). On some Berezin number and norm inequalities for operators in Hilbert and semi-Hilbert spaces. Matrix and Operator Equations and Applications, Springer. Accepted for publication.","DOI":"10.1007\/978-3-031-25386-7"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Bhunia, P., Paul, K., and Sen, A. (2021). Inequalities involving Berezin norm and Berezin number. arXiv.","DOI":"10.1007\/s11785-022-01305-9"},{"key":"ref_16","first-page":"1117","article-title":"Cauchy\u2013Schwarz type inequalities and applications to numerical radius inequalities","volume":"23","author":"Kittaneh","year":"2020","journal-title":"Math. Ineq. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1007\/BF01343117","article-title":"Notes on some inequalities for linear operators","volume":"125","author":"Kato","year":"1952","journal-title":"Math. Ann."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Halmos, P.R. (1982). A Hilbert Space Problem Book, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4684-9330-6"},{"key":"ref_19","unstructured":"Furuta, T., Mi\u0107i\u0107, J., Pe\u010dari\u0107, J., and Seo, Y. (2005). Mound\u2013Pe\u010dari\u0107 Method in Operator Inequalities, Element."},{"key":"ref_20","first-page":"405","article-title":"Generalizzazione della diseguaglianza di Cauchy-Schwarz (Italian)","volume":"31","author":"Buzano","year":"1974","journal-title":"Rend. Sem. Mat. Univ. Politech. Torino"},{"key":"ref_21","first-page":"282","article-title":"Some inequalities for complex numbers","volume":"1","year":"1971","journal-title":"Math. Balk."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"946","DOI":"10.1080\/03081087.2020.1810200","article-title":"New estimates for the numerical radius of Hilbert space operators","volume":"69","author":"Omidvar","year":"2021","journal-title":"Linear Multilinear Algebra"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.jmaa.2009.08.059","article-title":"Improved Young and Heinz inequalities for matrices","volume":"36","author":"Kittaneh","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Altwaijry, N., Feki, K., and Minculete, N. (2022). Further Inequalities for the Weighted Numerical Radius of Operators. 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