{"title":"关于半希尔伯特算子 A 数程的说明","authors":"Anirban Sen, Riddhick Birbonshi, Kallol Paul","doi":"10.1016/j.laa.2024.09.008","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we explore the relation between the <em>A</em>-numerical range and the <em>A</em>-spectrum of <em>A</em>-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of <em>A</em>-normal operator and prove that closure of the <em>A</em>-numerical range of an <em>A</em>-normal operator is the convex hull of the <em>A</em>-spectrum. We further prove Anderson's theorem for the sum of <em>A</em>-normal and <em>A</em>-compact operators which improves and generalizes the existing result on Anderson's theorem for <em>A</em>-compact operators. Finally we introduce strongly <em>A</em>-numerically closed class of operators and along with other results prove that the class of <em>A</em>-normal operators is strongly <em>A</em>-numerically closed.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A note on the A-numerical range of semi-Hilbertian operators\",\"authors\":\"Anirban Sen, Riddhick Birbonshi, Kallol Paul\",\"doi\":\"10.1016/j.laa.2024.09.008\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper we explore the relation between the <em>A</em>-numerical range and the <em>A</em>-spectrum of <em>A</em>-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of <em>A</em>-normal operator and prove that closure of the <em>A</em>-numerical range of an <em>A</em>-normal operator is the convex hull of the <em>A</em>-spectrum. We further prove Anderson's theorem for the sum of <em>A</em>-normal and <em>A</em>-compact operators which improves and generalizes the existing result on Anderson's theorem for <em>A</em>-compact operators. Finally we introduce strongly <em>A</em>-numerically closed class of operators and along with other results prove that the class of <em>A</em>-normal operators is strongly <em>A</em>-numerically closed.</p></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Linear Algebra and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0024379524003707\",\"RegionNum\":3,\"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":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003707","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
摘要
在本文中，我们探讨了半希尔伯特结构背景下 A 有界算子的 A 数值范围与 A 频谱之间的关系。我们引入了 A-正则算子的新定义，并证明了 A-正则算子的 A-数值范围的闭包是 A-谱的凸壳。我们进一步证明了 A 正算子与 A 紧算子之和的安德森定理，该定理改进并推广了关于 A 紧算子的安德森定理的现有结果。最后，我们引入了强 A 数闭算子类，并与其他结果一起证明了 A 正算子类是强 A 数闭的。
A note on the A-numerical range of semi-Hilbertian operators
In this paper we explore the relation between the A-numerical range and the A-spectrum of A-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of A-normal operator and prove that closure of the A-numerical range of an A-normal operator is the convex hull of the A-spectrum. We further prove Anderson's theorem for the sum of A-normal and A-compact operators which improves and generalizes the existing result on Anderson's theorem for A-compact operators. Finally we introduce strongly A-numerically closed class of operators and along with other results prove that the class of A-normal operators is strongly A-numerically closed.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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