Nonclassical Weyl laws and Connes' Integration for weak Lorentz ideals, II
Abstract
This is the second in a series of papers on Connes' integration in weak Lorentz ideals. Building on the Dixmier trace theory, Birman--Solomyak perturbation theory, and strong measurability developed in Part 1, we establish a spectral form of Pietsch's correspondence for traces on these ideals, extending to this setting results of Semenov--Sukochev--Usachev--Zanin for the weak trace-class. The correspondence describes the positive normalized traces in terms of Banach limits and yields a complete spectral characterization of strong measurability. We also introduce hypermeasurability (measurability with respect to every normalized trace). Unlike the ambient ideal, it depends on the specific regularly varying function chosen, and we characterize it spectrally by means of eigenvalue sums. We further show that hypermeasurability and spectral measurability are incomparable. Finally, we apply these results to examples arising from nonclassical Weyl laws in the sense of Simon, including the logarithm of the Laplacian on a closed manifold, the double Laplacian, multi-tensor products of Laplacians, and an example arising from Connes' approach to the Riemann Hypothesis.
Related
- Nonclassical Weyl laws and Connes' Integration for weak Lorentz ideals, I
- Lack of isomorphic embedding from Lorentz function spaces into Lorentz operator ideals
- The Baum-Connes conjecture: a concise course
- Connes-embeddability and twisted group von Neumann algebras
- Large eigenvalues of the Connes-Moscovici operator
BibTeX
@misc{ponge2026nonclassical,
title = {{Nonclassical Weyl laws and Connes' Integration for weak Lorentz ideals, II}},
author = {Raphael Ponge and Yongqiang Tian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09568},
eprint = {2609.09568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09568},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nonclassical Weyl laws and Connes' Integration for weak Lorentz ideals, II
AU - Raphael Ponge
AU - Yongqiang Tian
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09568
UR - https://arxiv.org/abs/2609.09568
AB - This is the second in a series of papers on Connes' integration in weak Lorentz ideals. Building on the Dixmier trace theory, Birman--Solomyak perturbation theory, and strong measurability developed in Part 1, we establish a spectral form of Pietsch's correspondence for traces on these ideals, extending to this setting results of Semenov--Sukochev--Usachev--Zanin for the weak trace-class. The correspondence describes the positive normalized traces in terms of Banach limits and yields a complete spectral characterization of strong measurability. We also introduce hypermeasurability (measurability with respect to every normalized trace). Unlike the ambient ideal, it depends on the specific regularly varying function chosen, and we characterize it spectrally by means of eigenvalue sums. We further show that hypermeasurability and spectral measurability are incomparable. Finally, we apply these results to examples arising from nonclassical Weyl laws in the sense of Simon, including the logarithm of the Laplacian on a closed manifold, the double Laplacian, multi-tensor products of Laplacians, and an example arising from Connes' approach to the Riemann Hypothesis.
ER -