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- Title
NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS.
- Authors
GIMPERLEIN, HEIKO; GOFFENG, MAGNUS
- Abstract
We consider the spectral behavior and noncommutative geometry of commutators [f U] 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 theWeyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.
- Subjects
NONCLASSICAL mathematical logic; CONTACT manifolds; NONCOMMUTATIVE differential geometry; COMMUTATORS (Operator theory); HANKEL operators; RIEMANNIAN manifolds
- Publication
Forum of Mathematics, Sigma, 2017, Vol 5, p1
- ISSN
2050-5094
- Publication type
Article
- DOI
10.1017/fms.2016.33