Functions of self-adjoint operators under relatively bounded and relatively trace class perturbations

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematische Nachrichten Pub Date : 2025-08-14 DOI:10.1002/mana.70000

A. B. Aleksandrov, V. V. Peller

{"title":"Functions of self-adjoint operators under relatively bounded and relatively trace class perturbations","authors":"A. B. Aleksandrov, V. V. Peller","doi":"10.1002/mana.70000","DOIUrl":null,"url":null,"abstract":"We study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbation. We introduce and study the class of relatively operator Lipschitz functions. An essential role is played by double operator integrals. We also consider the class of resolvent Lipschitz functions. Then we obtain a trace formula in the case of relatively trace class perturbations and show that the maximal class of function for which the trace formula holds in the case of relatively trace class perturbations coincides with the class of relatively operator Lipschitz functions. Our methods also give us a new approach to the inequality <math>\n <semantics>\n <mrow>\n <mrow>\n <mo>∫</mo>\n <mo>|</mo>\n <mrow>\n <mi>ξ</mi>\n </mrow>\n <mrow>\n <mo>(</mo>\n <mi>t</mi>\n <mo>)</mo>\n </mrow>\n <mo>|</mo>\n <mo>(</mo>\n <mn>1</mn>\n </mrow>\n <mo>+</mo>\n <msup>\n <mrow>\n <mo>|</mo>\n <mi>t</mi>\n <mo>|</mo>\n <mo>)</mo>\n </mrow>\n <mrow>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mspace></mspace>\n <mi>d</mi>\n <mi>t</mi>\n <mo><</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$\\int |\\bm{\\xi }(t)|(1+|t|)^{-1}\\,{\\rm d}t<\\infty$</annotation>\n </semantics></math> for the spectral shift function <math>\n <semantics>\n <mrow>\n <mi>ξ</mi>\n </mrow>\n <annotation>$\\bm{\\xi }$</annotation>\n </semantics></math> in the case of relatively trace class perturbations.","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 9","pages":"3027-3048"},"PeriodicalIF":0.8000,"publicationDate":"2025-08-14","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.70000","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

Abstract

We study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbation. We introduce and study the class of relatively operator Lipschitz functions. An essential role is played by double operator integrals. We also consider the class of resolvent Lipschitz functions. Then we obtain a trace formula in the case of relatively trace class perturbations and show that the maximal class of function for which the trace formula holds in the case of relatively trace class perturbations coincides with the class of relatively operator Lipschitz functions. Our methods also give us a new approach to the inequality $\int | ξ (t) | (1 + {| t |)}^{- 1} d t < \infty$ for the spectral shift function $ξ$ in the case of relatively trace class perturbations.

查看原文本刊更多论文

相对有界和相对迹类摄动下自伴随算子的函数

研究了自伴随算子函数在相对有界和相对迹类摄动下的行为。介绍并研究了一类相对算子Lipschitz函数。二重算子积分起着重要的作用。我们还考虑了一类可分解的Lipschitz函数。然后，我们得到了相对微扰情况下的迹公式，并证明了在相对微扰情况下，迹公式成立的最大函数类与相对算子Lipschitz函数类重合。我们的方法也给出了求解不等式∫| ξ (t) |（1 + |）的新方法T |)−1 d T ＆lt;∞$\int |\bm{\xi }(t)|(1+|t|)^{-1}\,{\rm d}t<\infty$对于谱移函数ξ $\bm{\xi }$ in相对微量类扰动的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Mathematische Nachrichten 数学-数学

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