二阶轨迹公式

IF 0.8 3区 数学 Q2 MATHEMATICS
Arup Chattopadhyay, Soma Das, Chandan Pradhan
{"title":"二阶轨迹公式","authors":"Arup Chattopadhyay,&nbsp;Soma Das,&nbsp;Chandan Pradhan","doi":"10.1002/mana.202200295","DOIUrl":null,"url":null,"abstract":"<p>Koplienko [Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743] found a trace formula for perturbations of self-adjoint operators by operators of Hilbert–Schmidt class <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>B</mi>\n <mn>2</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>H</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathcal {B}_2(\\mathcal {H})$</annotation>\n </semantics></math>. Later, Neidhardt introduced a similar formula in the case of pairs of unitaries <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>U</mi>\n <mo>,</mo>\n <msub>\n <mi>U</mi>\n <mn>0</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(U,U_0)$</annotation>\n </semantics></math> via multiplicative path in [Math. Nachr. 138 (1988), 7–25]. In 2012, Potapov and Sukochev [Comm. Math. Phys. 309 (2012), no. 3, 693–702] obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon [Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2]. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>T</mi>\n <mo>,</mo>\n <msub>\n <mi>T</mi>\n <mn>0</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(T,T_0)$</annotation>\n </semantics></math>, where the initial operator <span></span><math>\n <semantics>\n <msub>\n <mi>T</mi>\n <mn>0</mn>\n </msub>\n <annotation>$T_0$</annotation>\n </semantics></math> is normal, via linear path by reducing the problem to a finite-dimensional one as in the proof of Krein's trace formula by Voiculescu [Oper. Theory Adv. Appl. 24 (1987) 329–332] and Sinha and Mohapatra [Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853] and [Integral Equations Operator Theory 24 (1996), no. 3, 285–297]. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko–Neidhardt trace formula for a class of pairs of contractions <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>T</mi>\n <mo>,</mo>\n <msub>\n <mi>T</mi>\n <mn>0</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(T,T_0)$</annotation>\n </semantics></math> via multiplicative path using the finite-dimensional approximation method.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 7","pages":"2581-2608"},"PeriodicalIF":0.8000,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Second-order trace formulas\",\"authors\":\"Arup Chattopadhyay,&nbsp;Soma Das,&nbsp;Chandan Pradhan\",\"doi\":\"10.1002/mana.202200295\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Koplienko [Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743] found a trace formula for perturbations of self-adjoint operators by operators of Hilbert–Schmidt class <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>B</mi>\\n <mn>2</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>H</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\mathcal {B}_2(\\\\mathcal {H})$</annotation>\\n </semantics></math>. Later, Neidhardt introduced a similar formula in the case of pairs of unitaries <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>U</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>U</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(U,U_0)$</annotation>\\n </semantics></math> via multiplicative path in [Math. Nachr. 138 (1988), 7–25]. In 2012, Potapov and Sukochev [Comm. Math. Phys. 309 (2012), no. 3, 693–702] obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon [Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2]. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>T</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>T</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(T,T_0)$</annotation>\\n </semantics></math>, where the initial operator <span></span><math>\\n <semantics>\\n <msub>\\n <mi>T</mi>\\n <mn>0</mn>\\n </msub>\\n <annotation>$T_0$</annotation>\\n </semantics></math> is normal, via linear path by reducing the problem to a finite-dimensional one as in the proof of Krein's trace formula by Voiculescu [Oper. Theory Adv. Appl. 24 (1987) 329–332] and Sinha and Mohapatra [Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853] and [Integral Equations Operator Theory 24 (1996), no. 3, 285–297]. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko–Neidhardt trace formula for a class of pairs of contractions <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>T</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>T</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(T,T_0)$</annotation>\\n </semantics></math> via multiplicative path using the finite-dimensional approximation method.</p>\",\"PeriodicalId\":49853,\"journal\":{\"name\":\"Mathematische Nachrichten\",\"volume\":\"297 7\",\"pages\":\"2581-2608\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-03-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Nachrichten\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.202200295\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202200295","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

科普连科 [Sib.Mat.25 (1984), 62-71; English transl.25 (1984), 62-71; English transl. in Siberian Math.25 (1984), 735-743] 发现了希尔伯特-施密特类算子对自相关算子扰动的迹公式。后来,奈德哈特在[数学通报 138 (1988),7-25]中通过乘法路径引入了一对单元的类似公式。2012 年,波塔波夫和苏科切夫[Comm. Math. Phys. 309 (2012),no. 3, 693-702]通过回答格兹特西、普什尼茨基和西蒙[Zh. Mat. Fiz. Anal. Geom. 4 (2008),no. 1, 63-107, 202; Open Question 11.2]提出的一个开放问题,得到了类似科普连科迹式的成对收缩迹式。在本文中,我们通过线性路径,将问题简化为有限维问题,提供了科普连科迹线公式在成对收缩情况下的新证明,其中初始算子是正常的,正如沃伊库勒斯库对克雷恩迹线公式的证明[Oper.24 (1987) 329-332] 以及 Sinha 和 Mohapatra [Proc. Indian Acad. Sci.因此,我们利用 Schäffer 矩阵单元扩张,得到了一类成对收缩的科普连科迹线公式。此外,我们还利用 Cayley 变换得到了一对自相关算子和最大耗散算子的科普连科迹线公式。最后,我们利用有限维近似法,通过乘法路径扩展了一类成对收缩的科普连科-奈德哈特迹公式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Second-order trace formulas

Koplienko [Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743] found a trace formula for perturbations of self-adjoint operators by operators of Hilbert–Schmidt class B 2 ( H ) $\mathcal {B}_2(\mathcal {H})$ . Later, Neidhardt introduced a similar formula in the case of pairs of unitaries ( U , U 0 ) $(U,U_0)$ via multiplicative path in [Math. Nachr. 138 (1988), 7–25]. In 2012, Potapov and Sukochev [Comm. Math. Phys. 309 (2012), no. 3, 693–702] obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon [Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2]. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions ( T , T 0 ) $(T,T_0)$ , where the initial operator T 0 $T_0$ is normal, via linear path by reducing the problem to a finite-dimensional one as in the proof of Krein's trace formula by Voiculescu [Oper. Theory Adv. Appl. 24 (1987) 329–332] and Sinha and Mohapatra [Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853] and [Integral Equations Operator Theory 24 (1996), no. 3, 285–297]. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko–Neidhardt trace formula for a class of pairs of contractions ( T , T 0 ) $(T,T_0)$ via multiplicative path using the finite-dimensional approximation method.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
1.50
自引率
0.00%
发文量
157
审稿时长
4-8 weeks
期刊介绍: Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信