{"title":"Anisotropic Gohberg lemmas for pseudodifferential operators on Abelian compact groups","authors":"Marius Măntoiu","doi":"10.1007/s00013-025-02160-8","DOIUrl":null,"url":null,"abstract":"<div><p>Classically, Gohberg-type lemmas provide lower bounds for the distance of suitable pseudodifferential operators acting in a Hilbert space to the ideal of compact operators, in terms of “the behavior of the symbol at infinity”. In this article, the pseudodifferential operators are associated to a compact Abelian group <span>\\(\\textsf{X}\\)</span> and an important role is played by its Pontryagin dual <span>\\({\\widehat{\\textsf{X}}}\\)</span> . Hörmander-type classes of symbols are not always available; they will be replaced by crossed product <span>\\(C^*\\)</span>-algebras involving a vanishing oscillation condition, which anyway is more general even in the particular cases allowing a full pseudodifferential calculus. In addition, the distance to a large class of operator ideals is controlled; the compact operators only form a particular case. This involves invariant closed subsets of certain compactifications of the dual group or, equivalently, invariant ideals of <span>\\(\\ell ^\\infty ({\\widehat{\\textsf{X}}})\\)</span> .</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 4","pages":"399 - 411"},"PeriodicalIF":0.5000,"publicationDate":"2025-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archiv der Mathematik","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00013-025-02160-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Classically, Gohberg-type lemmas provide lower bounds for the distance of suitable pseudodifferential operators acting in a Hilbert space to the ideal of compact operators, in terms of “the behavior of the symbol at infinity”. In this article, the pseudodifferential operators are associated to a compact Abelian group \(\textsf{X}\) and an important role is played by its Pontryagin dual \({\widehat{\textsf{X}}}\) . Hörmander-type classes of symbols are not always available; they will be replaced by crossed product \(C^*\)-algebras involving a vanishing oscillation condition, which anyway is more general even in the particular cases allowing a full pseudodifferential calculus. In addition, the distance to a large class of operator ideals is controlled; the compact operators only form a particular case. This involves invariant closed subsets of certain compactifications of the dual group or, equivalently, invariant ideals of \(\ell ^\infty ({\widehat{\textsf{X}}})\) .
期刊介绍:
Archiv der Mathematik (AdM) publishes short high quality research papers in every area of mathematics which are not overly technical in nature and addressed to a broad readership.