{"title":"On conjugations concerning idempotents","authors":"Xiao-Ming Xu, Yi Yuan, Yuan Li, Yong Chen","doi":"10.1007/s43034-024-00354-9","DOIUrl":null,"url":null,"abstract":"<div><p>We introduce the <i>C</i>-decomposition property for reducible bounded linear operators on a Hilbert space, and prove that an arbitrary idempotent operator has the <i>C</i>-decomposition property with respect to a particular space decomposition, which is related to Halmos’ two projections theory. Using this, we obtain a general explicit description for all the conjugations <i>C</i> such that a given idempotent operator is a <i>C</i>-projection. We also present a characterization of the ranges of <i>C</i>-projections for any conjugation <i>C</i>.</p></div>","PeriodicalId":48858,"journal":{"name":"Annals of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s43034-024-00354-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce the C-decomposition property for reducible bounded linear operators on a Hilbert space, and prove that an arbitrary idempotent operator has the C-decomposition property with respect to a particular space decomposition, which is related to Halmos’ two projections theory. Using this, we obtain a general explicit description for all the conjugations C such that a given idempotent operator is a C-projection. We also present a characterization of the ranges of C-projections for any conjugation C.
期刊介绍：
Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
Ann. Funct. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and all modern related topics (e.g., operator theory). Ann. Funct. Anal. normally publishes original research papers numbering 18 or fewer pages in the journal’s style. Longer papers may be submitted to the Banach Journal of Mathematical Analysis or Advances in Operator Theory.
Ann. Funct. Anal. presents the best paper award yearly. The award in the year n is given to the best paper published in the years n-1 and n-2. The referee committee consists of selected editors of the journal.
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