{"title":"Geometric Characterization of the Numerical Range of Parallel Sum of Two Orthogonal Projections","authors":"Weiyan Yu, Ran Wang, Chen Zhang","doi":"10.1155/2024/1448498","DOIUrl":null,"url":null,"abstract":"Let <svg height=\"9.25986pt\" style=\"vertical-align:-0.2455397pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.01432 13.1092 9.25986\" width=\"13.1092pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg> be a complex separable Hilbert space and <svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 34.5353 11.5564\" width=\"34.5353pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,12.35,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,16.848,0)\"><use xlink:href=\"#g198-9\"></use></g><g transform=\"matrix(.013,0,0,-0.013,29.809,0)\"></path></g></svg> be the algebra of all bounded linear operators from <svg height=\"9.25986pt\" style=\"vertical-align:-0.2455397pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.01432 13.1092 9.25986\" width=\"13.1092pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-9\"></use></g></svg> to <span><svg height=\"9.25986pt\" style=\"vertical-align:-0.2455397pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.01432 13.1092 9.25986\" width=\"13.1092pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-9\"></use></g></svg>.</span> Our goal in this article is to describe the closure of numerical range of parallel sum operator <span><svg height=\"10.9105pt\" style=\"vertical-align:-2.15716pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.75334 14.622 10.9105\" width=\"14.622pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,11.658,0)\"></path></g></svg><span></span><svg height=\"10.9105pt\" style=\"vertical-align:-2.15716pt\" version=\"1.1\" viewbox=\"18.204183800000003 -8.75334 17.203 10.9105\" width=\"17.203pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,18.254,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,25.573,0)\"></path></g></svg></span> for two orthogonal projections <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.15071 8.68572\" width=\"8.15071pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-81\"></use></g></svg> and <svg height=\"10.7866pt\" style=\"vertical-align:-2.150701pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.52083 10.7866\" width=\"9.52083pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg> in <svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 34.5353 11.5564\" width=\"34.5353pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-3\"></use></g><g transform=\"matrix(.013,0,0,-0.013,12.35,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.848,0)\"><use xlink:href=\"#g198-9\"></use></g><g transform=\"matrix(.013,0,0,-0.013,29.809,0)\"><use xlink:href=\"#g113-42\"></use></g></svg> as a closed convex hull of some explicit ellipses parameterized by points in the spectrum.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/1448498","RegionNum":4,"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
Let be a complex separable Hilbert space and be the algebra of all bounded linear operators from to . Our goal in this article is to describe the closure of numerical range of parallel sum operator for two orthogonal projections and in as a closed convex hull of some explicit ellipses parameterized by points in the spectrum.
期刊介绍：
Journal of Mathematics is a broad scope journal that publishes original research articles as well as review articles on all aspects of both pure and applied mathematics. As well as original research, Journal of Mathematics also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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