Abdelkader Segres, Ahmed Bachir, Sid Ahmed Ould Ahmed Mahmoud
{"title":".算子的相似性算子与超不变子空间问题","authors":"Abdelkader Segres, Ahmed Bachir, Sid Ahmed Ould Ahmed Mahmoud","doi":"10.1155/2024/9943902","DOIUrl":null,"url":null,"abstract":"In the present paper, we first show that the existence of the solutions of the operator equation <span><svg height=\"10.3089pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"-0.0498162 -10.1025 40.024 10.3089\" width=\"40.024pt\" 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(.0091,0,0,-0.0091,6.136,-5.741)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,12.215,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,20.678,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,32.393,0)\"></path></g></svg><span></span><svg height=\"10.3089pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"43.6061838 -10.1025 10.133 10.3089\" width=\"10.133pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,43.656,0)\"></path></g></svg></span> is related to the similarity of operators of class <span><svg height=\"11.927pt\" style=\"vertical-align:-3.291101pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 15.8622 11.927\" width=\"15.8622pt\" 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=\"#g113-68\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,8.619,3.132)\"><use xlink:href=\"#g50-50\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,13.051,3.132)\"></path></g></svg>,</span> and then we give a sufficient condition for the existence of nontrivial hyperinvariant subspaces. These subspaces are the closure of <svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 42.6255 12.7178\" width=\"42.6255pt\" 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,-0.013,4.823,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,10.413,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,17.59,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,25.157,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,29.655,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,37.94,0)\"></path></g></svg> for some singular inner functions <span><svg height=\"9.39034pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.69399 9.39034\" width=\"7.69399pt\" 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=\"#g113-253\"></use></g></svg>.</span> As an application, we prove that every <span><svg height=\"11.927pt\" style=\"vertical-align:-3.291101pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 18.1457 11.927\" width=\"18.1457pt\" 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=\"#g113-68\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,8.619,3.132)\"><use xlink:href=\"#g50-50\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,13.051,3.132)\"></path></g></svg>-</span>quasinormal operator and <span><svg height=\"11.927pt\" style=\"vertical-align:-3.291101pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 18.1457 11.927\" width=\"18.1457pt\" 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=\"#g113-68\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,8.619,3.132)\"><use xlink:href=\"#g50-50\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,13.051,3.132)\"><use xlink:href=\"#g50-49\"></use></g></svg>-</span>centered operator, under suitable conditions, have nontrivial hyperinvariant subspaces.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Similarity of : Operators and the Hyperinvariant Subspace Problem\",\"authors\":\"Abdelkader Segres, Ahmed Bachir, Sid Ahmed Ould Ahmed Mahmoud\",\"doi\":\"10.1155/2024/9943902\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the present paper, we first show that the existence of the solutions of the operator equation <span><svg height=\\\"10.3089pt\\\" style=\\\"vertical-align:-0.2063999pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -10.1025 40.024 10.3089\\\" width=\\\"40.024pt\\\" 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(.0091,0,0,-0.0091,6.136,-5.741)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,12.215,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,20.678,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,32.393,0)\\\"></path></g></svg><span></span><svg height=\\\"10.3089pt\\\" style=\\\"vertical-align:-0.2063999pt\\\" version=\\\"1.1\\\" viewbox=\\\"43.6061838 -10.1025 10.133 10.3089\\\" width=\\\"10.133pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,43.656,0)\\\"></path></g></svg></span> is related to the similarity of operators of class <span><svg height=\\\"11.927pt\\\" style=\\\"vertical-align:-3.291101pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 15.8622 11.927\\\" width=\\\"15.8622pt\\\" 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=\\\"#g113-68\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,8.619,3.132)\\\"><use xlink:href=\\\"#g50-50\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,13.051,3.132)\\\"></path></g></svg>,</span> and then we give a sufficient condition for the existence of nontrivial hyperinvariant subspaces. These subspaces are the closure of <svg height=\\\"12.7178pt\\\" style=\\\"vertical-align:-3.42947pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 42.6255 12.7178\\\" width=\\\"42.6255pt\\\" 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,-0.013,4.823,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,10.413,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,17.59,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,25.157,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,29.655,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,37.94,0)\\\"></path></g></svg> for some singular inner functions <span><svg height=\\\"9.39034pt\\\" style=\\\"vertical-align:-3.42943pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -5.96091 7.69399 9.39034\\\" width=\\\"7.69399pt\\\" 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=\\\"#g113-253\\\"></use></g></svg>.</span> As an application, we prove that every <span><svg height=\\\"11.927pt\\\" style=\\\"vertical-align:-3.291101pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 18.1457 11.927\\\" width=\\\"18.1457pt\\\" 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=\\\"#g113-68\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,8.619,3.132)\\\"><use xlink:href=\\\"#g50-50\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,13.051,3.132)\\\"></path></g></svg>-</span>quasinormal operator and <span><svg height=\\\"11.927pt\\\" style=\\\"vertical-align:-3.291101pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 18.1457 11.927\\\" width=\\\"18.1457pt\\\" 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=\\\"#g113-68\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,8.619,3.132)\\\"><use xlink:href=\\\"#g50-50\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,13.051,3.132)\\\"><use xlink:href=\\\"#g50-49\\\"></use></g></svg>-</span>centered operator, under suitable conditions, have nontrivial hyperinvariant subspaces.\",\"PeriodicalId\":54214,\"journal\":{\"name\":\"Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-04-17\",\"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/10.1155/2024/9943902\",\"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/10.1155/2024/9943902","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Similarity of : Operators and the Hyperinvariant Subspace Problem
In the present paper, we first show that the existence of the solutions of the operator equation is related to the similarity of operators of class , and then we give a sufficient condition for the existence of nontrivial hyperinvariant subspaces. These subspaces are the closure of for some singular inner functions . As an application, we prove that every -quasinormal operator and -centered operator, under suitable conditions, have nontrivial hyperinvariant subspaces.
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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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