Nonclassical Spectral Asymptotics and Dixmier Traces: from Circles to Contact Manifolds
Journal article, 2017

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 Holder 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 Holder 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.

connes-dixmier

formulas

hankel-operators

lipschitz

geometry

mappings

heisenberg manifolds

integral-operators

cr

space

Author

Heiko Gimperlein

Magnus C H T Goffeng

Chalmers, Mathematical Sciences

University of Gothenburg

FORUM OF MATHEMATICS SIGMA

2050-5094 (ISSN)

Vol. 5 Article no e3 -

Subject Categories

Mathematics

Geometry

Mathematical Analysis

Roots

Basic sciences

DOI

10.1017/fms.2016.33

More information

Latest update

12/10/2018