{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:28:12Z","timestamp":1760059692604,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T00:00:00Z","timestamp":1751241600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["PNURSP2025R514"],"award-info":[{"award-number":["PNURSP2025R514"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper provides a new proof of the operator norm identity \u2225Q\u2225\u00a0=\u00a0\u2225I\u2212Q\u2225, where Q is a bounded idempotent operator on a complex Hilbert space, and I is the identity operator. We also derive explicit lower and upper bounds for the distance from an arbitrary idempotent operator to the set of orthogonal projections. Our approach simplifies existing proofs.<\/jats:p>","DOI":"10.3390\/axioms14070509","type":"journal-article","created":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T13:06:17Z","timestamp":1751288777000},"page":"509","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["New Results on Idempotent Operators in Hilbert Spaces"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7928-5624","authenticated-orcid":false,"given":"Salma","family":"Aljawi","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5699-4740","authenticated-orcid":false,"given":"Cristian","family":"Conde","sequence":"additional","affiliation":[{"name":"Instituto de Ciencias, Universidad Nacional de General Sarmiento, Los Polvorines B1613, Argentina"},{"name":"Consejo Nacional de Investigaciones Cient\u00edficas y T\u00e9cnicas, Buenos Aires B1425, Argentina"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9326-4173","authenticated-orcid":false,"given":"Kais","family":"Feki","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts, Najran University, Najran 66462, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9929-0954","authenticated-orcid":false,"given":"Shigeru","family":"Furuichi","sequence":"additional","affiliation":[{"name":"Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo 156-8550, Japan"},{"name":"Department of Mathematics, Saveetha School of Engineering, SIMATS, Thandalam, Chennai 602105, Tamilnadu, India"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gal\u00e1ntai, A. 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