Extended zeta-function residues on principal ideals
Yongqiang Tian, Alexandr Usachev
TL;DR
This paper investigates the relationship between extended zeta-function residues on principal ideals of compact operators and Dixmier traces, providing a formula and conditions for operator measurability based on eigenvalues and zeta-function asymptotics.
Contribution
It introduces a Lidskii-type formula for continuous singular traces and characterizes Dixmier measurability through eigenvalue and zeta-function asymptotics.
Findings
Established a Lidskii-type formula for singular traces
Derived necessary and sufficient conditions for Dixmier measurability
Linked zeta-function residues with eigenvalue asymptotics
Abstract
We study extended zeta-function residues on principal ideals of compact operators and their connections with Dixmier traces. We establish a Lidskii-type formula for continuous singular traces on these ideals. Using this formula, we obtain a necessary and sufficient conditions for an arbitrary operator being Dixmier measurable. These conditions are expressed in terms of eigenvalues of an operator and an asymptotic of its zeta-function.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAnalytic Number Theory Research · Graph theory and applications · Rings, Modules, and Algebras