{"title":"几乎可交换自伴随算子和迭代换向子估计","authors":"Jakob Geisler","doi":"10.1016/j.laa.2025.09.004","DOIUrl":null,"url":null,"abstract":"<div><div>Given two almost commuting self-adjoint operators, a new method for finding exactly commuting operators is presented. For this, a differential equation for self-adjoint Hilbert-Schmidt operators is introduced. Quantitative results are proven that the exactly commuting operators are close to the old ones in the Hilbert-Schmidt norm. The proof relies on a novel estimate in which the norm of the commutator is bounded from above by the norm of the iterated commutators times a constant. This inequality is proven in finite dimensions and lower bounds for the optimal constants are given.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 388-411"},"PeriodicalIF":1.1000,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Almost commuting self-adjoint operators and iterated commutator estimates\",\"authors\":\"Jakob Geisler\",\"doi\":\"10.1016/j.laa.2025.09.004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Given two almost commuting self-adjoint operators, a new method for finding exactly commuting operators is presented. For this, a differential equation for self-adjoint Hilbert-Schmidt operators is introduced. Quantitative results are proven that the exactly commuting operators are close to the old ones in the Hilbert-Schmidt norm. The proof relies on a novel estimate in which the norm of the commutator is bounded from above by the norm of the iterated commutators times a constant. This inequality is proven in finite dimensions and lower bounds for the optimal constants are given.</div></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":\"727 \",\"pages\":\"Pages 388-411\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2025-09-05\",\"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/S0024379525003702\",\"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/S0024379525003702","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Almost commuting self-adjoint operators and iterated commutator estimates
Given two almost commuting self-adjoint operators, a new method for finding exactly commuting operators is presented. For this, a differential equation for self-adjoint Hilbert-Schmidt operators is introduced. Quantitative results are proven that the exactly commuting operators are close to the old ones in the Hilbert-Schmidt norm. The proof relies on a novel estimate in which the norm of the commutator is bounded from above by the norm of the iterated commutators times a constant. This inequality is proven in finite dimensions and lower bounds for the optimal constants are given.
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.