关于araki型迹不等式

IF 1.1 3区 数学 Q1 MATHEMATICS
Po-Chieh Liu , Hao-Chung Cheng
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On the other hand, if <span><math><mi>s</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> and the map <span><math><mi>x</mi><mo>↦</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>s</mi></mrow></msup><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is positive and decreasing, then <span><math><mi>Tr</mi><mo>[</mo><mi>g</mi><mo>(</mo><mi>A</mi><mo>)</mo><msup><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mi>B</mi><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mo>]</mo><mo>≤</mo><mi>Tr</mi><mspace></mspace><mrow><mo>[</mo><mi>g</mi><mo>(</mo><mi>A</mi><mo>)</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>]</mo></mrow></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 320-330"},"PeriodicalIF":1.1000,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Araki-type trace inequalities\",\"authors\":\"Po-Chieh Liu ,&nbsp;Hao-Chung Cheng\",\"doi\":\"10.1016/j.laa.2025.08.023\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper, we prove a trace inequality <span><math><mi>Tr</mi><mspace></mspace><mrow><mo>[</mo><mi>f</mi><mo>(</mo><mi>A</mi><mo>)</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>]</mo></mrow><mo>≤</mo><mi>Tr</mi><mo>[</mo><mi>f</mi><mo>(</mo><mi>A</mi><mo>)</mo><msup><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mi>B</mi><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mo>]</mo></math></span> for any positive and monotonically increasing function <em>f</em>, <span><math><mi>s</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, and positive semi-definite matrices <em>A</em> and <em>B</em>. 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引用次数: 0

摘要

本文证明了一个迹不等式Tr[f(a) asb]≤Tr[f(a) asb]对于任意正单调递增的函数f, s∈[0,1]和正半定矩阵a, b。另一方面,如果s∈[0,1]且映射x∈xsg(x)是正递减的,则Tr[g(a)(A1/2BA1/2)s]≤Tr[g(a) asb]。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On Araki-type trace inequalities
In this paper, we prove a trace inequality Tr[f(A)AsBs]Tr[f(A)(A1/2BA1/2)s] for any positive and monotonically increasing function f, s[0,1], and positive semi-definite matrices A and B. On the other hand, if s[0,1] and the map xxsg(x) is positive and decreasing, then Tr[g(A)(A1/2BA1/2)s]Tr[g(A)AsBs].
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来源期刊
CiteScore
2.20
自引率
9.10%
发文量
333
审稿时长
13.8 months
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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