Anisotropic Gohberg lemmas for pseudodifferential operators on Abelian compact groups

IF 0.5 4区 数学 Q3 MATHEMATICS
Marius Măntoiu
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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 \(\ell ^\infty ({\widehat{\textsf{X}}})\) .

阿贝尔紧群上伪微分算子的各向异性Gohberg引理
经典地,gohberg型引理根据“符号在无穷远处的行为”提供了作用于希尔伯特空间中合适的伪微分算子到紧算子理想的距离的下界。在本文中,伪微分算子与紧阿贝尔群\(\textsf{X}\)相关联,其庞特里亚金对偶\({\widehat{\textsf{X}}}\)起着重要作用。Hörmander-type符号类并不总是可用的;它们将被交叉积\(C^*\) -代数所取代,包含一个消失的振荡条件,无论如何,它是更普遍的,即使在特殊情况下允许一个完整的伪微分学。此外,还控制了到一大类算子理想的距离;紧算符只构成一种特殊情况。这涉及对偶群的某些紧化的不变闭子集,或者等价地,\(\ell ^\infty ({\widehat{\textsf{X}}})\)的不变理想。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Archiv der Mathematik
Archiv der Mathematik 数学-数学
CiteScore
1.10
自引率
0.00%
发文量
117
审稿时长
4-8 weeks
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.
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