{"title":"二阶轨迹公式","authors":"Arup Chattopadhyay, Soma Das, 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, Soma Das, 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}
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 . Later, Neidhardt introduced a similar formula in the case of pairs of unitaries 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 , where the initial operator 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 via multiplicative path using the finite-dimensional approximation method.
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