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NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS

Part of: Linear spaces and algebras of operators CR manifolds Spectral theory and eigenvalue problems Infinite-dimensional manifolds

Published online by Cambridge University Press:  23 January 2017

HEIKO GIMPERLEIN  and
MAGNUS GOFFENG
HEIKO GIMPERLEIN
Affiliation:
Maxwell Institute for Mathematical Sciences and Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK Institute for Mathematics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany; h.gimperlein@hw.ac.uk
MAGNUS GOFFENG
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden; goffeng@chalmers.se

Abstract

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We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$, where $P$ is an operator of order 0 with geometric origin and $f$ a multiplication operator by a function. When $f$ is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions $f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

MSC classification

Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 58B34: Noncommutative geometry (à la Connes)
Secondary: 32V20: Analysis on CR manifolds 47L20: Operator ideals
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e3
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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