{"title":"On Operator Valued Haar Unitaries and Bipolar Decompositions of R-diagonal Elements","authors":"","doi":"10.1007/s00020-024-02760-z","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>In the context of operator valued W<span> <span>\\(^*\\)</span> </span>-free probability theory, we study Haar unitaries, R-diagonal elements and circular elements. Several classes of Haar unitaries are differentiated from each other. The term bipolar decomposition is used for the expression of an element as <em>vx</em> where <em>x</em> is self-adjoint and <em>v</em> is a partial isometry, and we study such decompositions of operator valued R-diagonal and circular elements that are free, meaning that <em>v</em> and <em>x</em> are <span> <span>\\(*\\)</span> </span>-free from each other. In particular, we prove, when <span> <span>\\(B={\\textbf{C}}^2\\)</span> </span>, that if a <em>B</em>-valued circular element has a free bipolar decomposition with <em>v</em> unitary, then it has one where <em>v</em> normalizes <em>B</em>.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"2015 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Integral Equations and Operator Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00020-024-02760-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the context of operator valued W\(^*\)-free probability theory, we study Haar unitaries, R-diagonal elements and circular elements. Several classes of Haar unitaries are differentiated from each other. The term bipolar decomposition is used for the expression of an element as vx where x is self-adjoint and v is a partial isometry, and we study such decompositions of operator valued R-diagonal and circular elements that are free, meaning that v and x are \(*\)-free from each other. In particular, we prove, when \(B={\textbf{C}}^2\), that if a B-valued circular element has a free bipolar decomposition with v unitary, then it has one where v normalizes B.
摘要 在无算子值 W （^*\）概率论的背景下，我们研究了哈尔单元、R 对角元素和圆元素。我们将几类 Haar 单元相互区分开来。双极分解这一术语用于表达一个元素为 vx，其中 x 是自共轭的，v 是部分等距的，我们研究了算子值 R 对角元素和圆元素的这种分解，它们是自由的，这意味着 v 和 x 彼此是 \(*\) -free 的。特别是，我们证明，当 \(B={\textbf{C}}^2\) 时，如果一个 B 值圆周元素有一个自由的双极分解，且 v 是单元的，那么它就有一个 v 使 B 正常化的双极分解。
期刊介绍：
Integral Equations and Operator Theory (IEOT) is devoted to the publication of current research in integral equations, operator theory and related topics with emphasis on the linear aspects of the theory. The journal reports on the full scope of current developments from abstract theory to numerical methods and applications to analysis, physics, mechanics, engineering and others. The journal consists of two sections: a main section consisting of refereed papers and a second consisting of short announcements of important results, open problems, information, etc.
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