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arxiv: 2605.25591 · v1 · pith:LPWPKERLnew · submitted 2026-05-25 · 🧮 math.OA · math.FA· math.SP

Nonclassical Weyl laws and Connes' Integration for weak Lorentz ideals, I

Raphael Ponge , Yongqiang Tian This is my paper

Pith reviewed 2026-06-29 19:29 UTC · model grok-4.3

classification 🧮 math.OA math.FAmath.SP
keywords Dixmier tracesweak Lorentz idealsmeasurable operatorseigenvalue asymptoticsWeyl operatorsnoncommutative integrationregularly varying functionsperturbation theory
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The pith

Dixmier traces on weak Lorentz ideals are constructed directly from eigenvalue sequences, giving a full spectral characterization of measurable operators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a systematic theory of Dixmier traces and Connes' noncommutative integration for weak Lorentz ideals associated with regularly varying functions. It establishes an asymptotic additivity property for eigenvalue partial sums by combining Karamata's theorem with results of Kalton and Lord-Sukochev-Zanin. This property allows the traces to be built directly from eigenvalue sequences and supplies a complete spectral characterization of measurable operators, answering a question of Connes. The results extend Birman-Solomyak perturbation theory, show that Weyl operators form a closed subset stable under compact perturbations, prove every Weyl operator is strongly measurable, and apply the framework to operators from several nonclassical Weyl laws.

Core claim

For weak Lorentz ideals associated with regularly varying functions, Dixmier traces admit a direct construction in terms of eigenvalue sequences. This yields a complete spectral characterization of measurable operators, answering a question of Connes. Weyl operators, those with precise asymptotic limits for rescaled eigenvalue sequences, form a closed subset of the ideal that remains stable under compact perturbations. Every Weyl operator is strongly measurable, so spectral measurability implies strong measurability. The theory applies to operators linked to the Riemann hypothesis (assuming RH), Schrödinger operators with anisotropic potentials, Dirichlet Laplacians on infinite-volume domain

What carries the argument

The asymptotic additivity property for eigenvalue partial sums, derived by combining Karamata's theorem with results of Kalton and Lord-Sukochev-Zanin, which enables the direct construction of Dixmier traces from eigenvalue sequences.

Load-bearing premise

The eigenvalue partial sums satisfy an asymptotic additivity property for the weak Lorentz ideals tied to regularly varying functions.

What would settle it

An operator belonging to such a weak Lorentz ideal for which the Dixmier trace cannot be recovered as a limit involving the rescaled partial sums of its eigenvalues would show the direct construction fails.

read the original abstract

Motivated by nonclassical Weyl laws arising in various contexts (including Connes' approach to the Riemann Hypothesis), we develop a systematic theory of Dixmier traces and Connes' noncommutative integration for weak Lorentz ideals associated with regularly varying functions. A key ingredient is an asymptotic additivity property for eigenvalue partial sums, obtained by combining Karamata's theorem with results of Kalton and Lord-Sukochev-Zanin. This yields a direct construction of Dixmier traces in terms of eigenvalue sequences and a complete spectral characterization of measurable operators, answering a question of Connes in this general setting. We also extend to weak Lorentz ideals the Birman-Solomyak perturbation theory for eigenvalue and singular-value asymptotics. Weyl operators (those admitting precise asymptotic limits for their rescaled eigenvalue sequences) are shown to form a closed subset of the ideal, stable under compact perturbations, extending classical results of Weyl and Birman-Solomyak. We further study strong measurability (measurability with respect to all positive normalized traces). We prove that every Weyl operator is strongly measurable, so spectral measurability implies strong measurability. The converse does not hold in general; a spectral characterization via Pietsch's correspondence is obtained in the forthcoming companion pape. Finally, as an application, we establish spectral measurability for operators arising from nonclassical Weyl laws: operators associated with the Riemann Hypothesis (assuming RH); Schr\"odinger operators with anisotropic potentials and Dirichlet Laplacians on infinite-volume domains; Dirac operators on open spin manifolds with conformally cusp metrics; and the operator formed by the Dirac operator of the Podle\`s quantum sphere and the Laplacian on the 2-torus.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript develops a systematic theory of Dixmier traces and Connes' noncommutative integration for weak Lorentz ideals associated with regularly varying functions. A central step is an asymptotic additivity property for eigenvalue partial sums, derived by combining Karamata's theorem with results of Kalton, Lord-Sukochev-Zanin and Birman-Solomyak. This yields a direct construction of Dixmier traces from eigenvalue sequences, a complete spectral characterization of measurable operators (addressing a question of Connes), an extension of Birman-Solomyak perturbation theory, and results showing that Weyl operators form a closed subset of the ideal that is stable under compact perturbations. The paper further proves that every Weyl operator is strongly measurable (so spectral measurability implies strong measurability) and applies the theory to operators arising from nonclassical Weyl laws, including those linked to the Riemann Hypothesis (under RH), Schrödinger operators, Dirac operators on manifolds with cusp metrics, and operators on the Podleś quantum sphere.

Significance. If the derivations hold, the work supplies a general framework for noncommutative integration beyond the classical Schatten and weak-L^p settings, with direct eigenvalue-based constructions and spectral characterizations that extend classical results of Weyl and Birman-Solomyak. The applications to concrete operators from nonclassical Weyl laws illustrate relevance to spectral geometry and quantum spaces. Reliance on established theorems (Karamata, Kalton et al.) rather than ad-hoc parameters is a methodological strength.

major comments (3)
  1. [Abstract / Introduction] The abstract states that the construction answers 'a question of Connes'; the manuscript should identify the precise question (with citation) and indicate the section or theorem that supplies the answer.
  2. [Section on asymptotic additivity (likely §3 or §4)] The key asymptotic additivity property is obtained by combining Karamata's theorem with Kalton-Lord-Sukochev-Zanin results; the text must verify that the regularly varying functions defining the weak Lorentz ideals satisfy all hypotheses of the cited theorems without additional regularity assumptions.
  3. [Section on Weyl operators] The claim that Weyl operators form a closed subset stable under compact perturbations extends classical results; the proof should explicitly address whether the regularly varying function introduces any obstruction to the usual perturbation estimates.
minor comments (3)
  1. Provide a self-contained definition of the weak Lorentz ideal associated to a regularly varying function at the beginning of the technical development.
  2. Ensure all external results (Karamata, Birman-Solomyak, Kalton et al.) receive complete bibliographic citations in the reference list.
  3. Clarify the distinction between spectral measurability and strong measurability in the statement of the main theorems.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. We respond point-by-point to the major comments and indicate the changes to be made in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / Introduction] The abstract states that the construction answers 'a question of Connes'; the manuscript should identify the precise question (with citation) and indicate the section or theorem that supplies the answer.

    Authors: We agree that explicit identification improves clarity. The question of Connes concerns the existence of a spectral characterization of measurable operators with respect to Dixmier traces. We will revise the abstract and introduction to cite the relevant statement from Connes and indicate the theorem (in the section on spectral characterization of measurable operators) that supplies the answer for weak Lorentz ideals. revision: yes

  2. Referee: [Section on asymptotic additivity (likely §3 or §4)] The key asymptotic additivity property is obtained by combining Karamata's theorem with Kalton-Lord-Sukochev-Zanin results; the text must verify that the regularly varying functions defining the weak Lorentz ideals satisfy all hypotheses of the cited theorems without additional regularity assumptions.

    Authors: The regularly varying functions used in the definition of the weak Lorentz ideals are taken to satisfy the standard conditions (including the index of regular variation) required by Karamata's theorem and the cited results of Kalton-Lord-Sukochev-Zanin. We will insert a short verification paragraph or remark in the relevant section to confirm explicitly that the hypotheses hold under our standing assumptions and that no additional regularity is imposed. revision: yes

  3. Referee: [Section on Weyl operators] The claim that Weyl operators form a closed subset stable under compact perturbations extends classical results; the proof should explicitly address whether the regularly varying function introduces any obstruction to the usual perturbation estimates.

    Authors: The proof adapts the classical Birman-Solomyak estimates using the asymptotic properties of regularly varying functions, which are preserved under the relevant operations. We will expand the argument in the section on Weyl operators to include an explicit paragraph confirming that the regular variation introduces no obstruction to the perturbation estimates, thereby extending the classical results without additional restrictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central construction of Dixmier traces and spectral characterization relies on an asymptotic additivity property derived by combining Karamata's theorem with external results from Kalton, Lord-Sukochev-Zanin, and Birman-Solomyak. These are independent, standard theorems not originating from the authors' prior work in a load-bearing self-citation chain. No self-definitional steps, fitted inputs renamed as predictions, or ansatzes smuggled via self-citation are present in the described derivation. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The theory rests on standard results from real analysis and prior work on Dixmier traces and eigenvalue asymptotics; no free parameters or new postulated entities are indicated in the abstract.

axioms (2)
  • standard math Karamata's theorem for regularly varying functions
    Invoked to establish the asymptotic additivity property for eigenvalue partial sums.
  • standard math Results of Kalton and Lord-Sukochev-Zanin on eigenvalue sums
    Combined with Karamata's theorem to obtain the key additivity property.

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