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All-dimensional spectral problem for BHEP operators

For every dimension d∈Nd\in\mathbb{N} and smoothing parameter β>0\beta>0, determine the complete spectral decomposition of the Henze–Wagner covariance-kernel integral operator A~β,d\widetilde A_{\beta,d} on its Gaussian-weighted Hilbert space: all eigenvalues with their multiplicities, all eigenfunctions, the null space, and all exceptional parameter cases. Equivalently, determine the full spectral data governing the limiting null distribution of the multivariate BHEP normality test.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Factorization into companion operators

    The Henze–Zirkler and Henze–Wagner operators can be represented as Xβ,d∗Xβ,d\mathcal{X}_{\beta,d}^{*}\mathcal{X}_{\beta,d} and Xβ,dXβ,d∗\mathcal{X}_{\beta,d}\mathcal{X}_{\beta,d}^{*}, respectively; hence their nonzero eigenvalues agree, including multiplicities.

    source: The Spectra of the Henze-Zirkler and Henze-Wagner Operators for BHEP Tests

References

Primary source

arXiv

Additional references

functional-analysisprobabilityspectral-theory
Posted · edited

Progress summary

Refreshed
Claimed solved

A new unrefereed preprint claims to give the complete answer in every dimension, but no independent verification has appeared.

The problem asks for the full spectral description underlying limiting null distributions of multivariate normality tests. A preprint by Bruno Ebner, Dominic Edelmann, Norbert Henze, Frédéric Ouimet, and Donald Richards claims a solution for every d∈Nd\in\mathbb{N} and β>0\beta>0.

September 2026 claimed resolution

The preprint claims complete eigenvalues, multiplicities, eigenfunctions, exceptional poles, and null spaces for both BHEP operators. It identifies matching positive spectra through Xβ,d∗Xβ,d\mathcal{X}_{\beta,d}^{*}\mathcal{X}_{\beta,d} and Xβ,dXβ,d∗\mathcal{X}_{\beta,d}\mathcal{X}_{\beta,d}^{*}, and gives the limiting distribution as Tβ(d)=D∑j=1∞λj(β,d)Nj2T_{\beta}(d)\stackrel{\mathcal D}{=}\sum_{j=1}^{\infty}\lambda_j(\beta,d)N_j^2. For the companion operator it claims ker⁡(A~β,d)=P≤2\ker(\widetilde A_{\beta,d})=\mathcal P_{\leq 2} with dimension (d+1)(d+2)2\frac{(d+1)(d+2)}{2}. The claim is unrefereed and independently unverified.

Current status (as of September 2026): A preprint claims to settle the all-dimensional spectral problem, including the positive spectrum and null space, but the result remains unverified.

Sources

Solutions 0

No solutions have been posted yet.