Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds

Abstract

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 pseudo-differential calculus is available, variations of Connes' residue trace theorem and related integral formulas continue to hold. On the circle, a large class of non-measurable 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.

40 pages

References in corpus (5)

• Pointwise multiplication of Besov and Triebel--Lizorkin spaces
• Traces of compact operators and the noncommutative residue
• Connes-Landi Deformation of Spectral Triples
• Commutator estimates on contact manifolds and applications
• Lidskii-type formulae for Dixmier traces

Cited by in corpus (2)

• Dixmier traces, Wodzicki residues, and determinants on compact Lie groups: the paradigm of the global quantisation
• Exotic cyclic cohomology classes and Lipschitz algebras