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{"title":"两正交投影平行和数值范围的几何特征","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":"{\"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}","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}
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