bibkey: garciavolcic2025hunter authors: S. R. Garcia and J. Volčič year: 2025 title: A noncommutative generalization of Hunter’s positivity theorem doi: 10.1090/proc/17480 url: https://arxiv.org/abs/2503.12376v2 claim: ‘Let $n,d\ge 2$. For all tuples of hermitian operators $X_1,\dots ,X_n$ on a Hilbert space, $\ker \left(H_{2d}(X_1,\dots ,X_n)-{\mu }_{n,d}(X_1^{2d}+\cdots +X_n^{2d})\right)=\ker X_1\cap \cdots \cap \ker X_n.$’ strata_touched:

• D5/S3/Analytic/Hunter/NCHunterPositivity
• D5/S3/Analytic/Hunter/NCHunterKernelRigidity license: citation-only triage: anchor

The noncommutative Hunter bound

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DOI: https://doi.org/10.1090/proc/17480

Source: https://arxiv.org/abs/2503.12376v2

Theorem 1.1(ii), pages 2–3, gives the sharp positivity constant. Example 4.4 and Conjecture 4.5 are on page 11 of arXiv v2. Conjecture 4.6 is on page 12. The published article is in Proceedings of the American Mathematical Society 154(2), pages 585–597.

Source definitions

“Let $\sigma :\mathbb{R}[x_1,\dots ,x_n]\to \mathbb{R}⟨x_1,\dots ,x_n⟩$ denote the linear map”

$$\sigma (m):=\frac{1}{|{\alpha }^{-1}(m)|}\sum _{w\in {\alpha }^{-1}(m)}w.$$

“The noncommutative complete homogeneous symmetric (NCHS) polynomial of degree $d$ in $n$ (noncommuting) variables is”

$$H_d(x_1,\dots ,x_n):=\sigma (h_d(x_1,\dots ,x_n)).$$

The coefficient of a word of length $k$ with multiplicities $m_i$ is ${\prod }_i{m}_i!/k!$, the reciprocal of its actual abelianization-fibre size. The Lean carrier reindexes letters by Fin n and uses the frozen occupation map to count them; evaluation is the ordered product, not a commutative one.

Conjecture 4.5

“Let $n,d\ge 2$. For all tuples of hermitian operators $X_1,\dots ,X_n$ on a Hilbert space,”

$$\ker \left(H_{2d}(X_1,\dots ,X_n)-{\mu }_{n,d}(X_1^{2d}+\cdots +X_n^{2d})\right)=\ker X_1\cap \cdots \cap \ker X_n.$$

The formal statement uses bounded complex-linear operators on a complete complex inner-product space. Unbounded self-adjoint operators and their domain questions are outside this carrier.

Scope

The source’s Theorem 1.1(ii) supplies the sharp positivity statement; its optimality assertion is attributed to the source and is not a new formal statement here. The proof uses a finite factorial Gram identity instead of simplex integration. This is a proof representation and does not claim that the scalar Vandermonde identity is new.

Conjecture 4.6 concerns entrywise nonnegative factorizations, a stronger property than positive semidefiniteness. It is untouched by the kernel proof.

Scalar factorial identity

The shifted scalar identity follows from Chu–Vandermonde, NIST Digital Library of Mathematical Functions, equation 15.4.24: https://dlmf.nist.gov/15.4.E24 . In $F(-a,-b;g+1;1)=(g+1+b{)}_a/(g+1{)}_a$, expand the terminating series and multiply by $(a+g)!(b+g)!/g!$. Taking products over letters gives the finite factorial Gram representation. Neither the scalar identity nor this product construction is claimed original.