On the existence of extremizers for the sum of eigenvalues of Toeplitz operators

Fabio Nicola; Federico Riccardi; Paolo Tilli

REVIEW 3 major objections 4 minor 2 cited by

On the existence of extremizers for the sum of eigenvalues of Toeplitz operators

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For any prescribed measure and rank k, an optimal set of Toeplitz eigenvalues exists.

desk verdict The Fock-space existence theorem for k>1 is new and looks correct, and the abstract extension is promising but needs a missing vanishing-at-infinity hypothesis before its broadest claims can be trusted. read the letter →

arxiv 2607.20965 v1 pith:7DZ7SH6I submitted 2026-07-23 math.FA

Fabio Nicola , Federico Riccardi , Paolo Tilli This is my paper · ORCID

classification math.FA MSC 47B3547A7549Q1049R0530H2047A30

keywords ToeplitzoperatorseigenvalueoptimizationoptimalsetsDonoho-StarkconjectureFockspacelocalizationwavelettransformsorthonormalsystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that for every $s>0$ and every $k\geq 1$, among all measurable subsets of the complex plane with Lebesgue measure $s$ there exists a set that maximizes the sum of the first $k$ eigenvalues of the associated Toeplitz operator on the Fock space. Equivalently, the supremum $S_k(s)$ is attained rather than merely approached. The case $k=1$ was known, with balls optimal; for $k>1$ the existence of maximizing sets was open. The same theorem is extended to localization operators built from abstract wavelet transforms, under a positivity condition, and this yields the existence of optimal sets for the classical band-limited concentration problem, with every maximizer a finite union of intervals.

What carries the argument

The load-bearing object is the concentration functional $J_s(F)$, defined by integrating the joint Husimi function $u_F$ over its superlevel set of measure $s$; the bathtub principle identifies the original eigenvalue supremum with the maximum of $J_s$ over orthonormal systems. The proof then runs on a geometric lemma in Hilbert space that repairs near-orthogonality of a weak limit by adjoining small correcting vectors, a disintegration lemma that rewrites $u_F+u_H$ as a convex combination of Husimi functions of mixed blocks, and a strict inequality that compares partial sums of lower rank with the full rank-$k$ sum. Together these force an asymptotically optimal block to coincide with the normalized weak limit, yielding a genuine maximizer. In the abstract wavelet setting the same chain works when localization operators are strictly positive on every set of positive measure.

What would settle it

Exhibit a square-integrable projective representation and an admissible wavelet for which some squared wavelet transform does not tend to zero at infinity; then the translation-and-compactness step has no basis and the abstract theorem has no candidate extremizer. In the Fock case, the corresponding test would be to find a positive measure and a rank for which the supremum is approached only by systems escaping to infinity, which the paper's conclusion rules out.

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Extended reading notes

Core claim

The central discovery is that every partial sum of Toeplitz eigenvalues in the Fock space achieves its extremal value: for prescribed measure $s$ and rank $k$, there is an orthonormal system of $k$ Fock-space functions whose joint Husimi function $u_F$ has a superlevel set of measure $s$ on which the integrated concentration $J_s(F)$ equals $S_k(s)$. The proof converts the eigenvalue problem into a variational problem for orthonormal systems, then shows that a maximizing sequence can be modified so that its weak limit is a true orthonormal system rather than a degenerate one; the same mechanism, with a strict positivity assumption in place of mere positivity, establishes existence of optimal sets for localization operators of abstract wavelet transforms. In the band-limited setting this gives the existence part of the concentration problem: optimal sets exist and, up to measure zero, are finite unions of intervals.

Load-bearing premise

The load-bearing premise is that joint Husimi wavelet-transform intensity functions vanish at infinity, so a maximizing sequence can be translated to peak at a fixed point and a nonzero weak limit can be extracted; the paper verifies this in the Fock space but invokes it in the abstract setting without proving it for all admissible square-integrable projective representations.

Editorial extensions

If this is right

The existence part of the band-limited concentration problem follows: for every prescribed measure $s>0$, some set maximizes the first eigenvalue of the time-frequency localization operator, and any such optimal set is a finite union of intervals up to measure zero.
In the Fock space, for every $k\geq 1$, optimal sets exist and are bounded up to sets of measure zero, since each is a superlevel set of a joint Husimi function that vanishes at infinity.
For localization operators of many abstract wavelet transforms, including short-time Fourier transforms with non-Gaussian windows and analytic wavelet transforms, the existence result is new even for $k=1$ whenever the strict positivity condition holds.
The theorem supplies the missing existence step that complements the recent determination of the optimal shape in the band-limited problem, so the two results together fully characterize the extremizers.
The min-max reformulation shows the same supremum is attained simultaneously for the set and for the orthonormal system, so questions about uniqueness and shape can be studied on a nonempty admissible class.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the strict positivity assumption (5.6) fails, the strict inequality in the splitting step can become an equality, and the induction selecting the surviving block may no longer force a nontrivial limit; in such settings the theorem might genuinely fail, making the assumption a substantive restriction rather than a technical convenience.
The Hilbert-space repair lemma is stated abstractly, so the method may transfer to orthonormal Strichartz inequalities and Riesz-potential bounds if their extremal problems admit a superlevel-set representation; testing that transfer is a natural next step.
A sharper version of the abstract theorem would replace the unlisted vanishing-at-infinity premise by a verifiable decay condition on the admissible wavelet; one concrete check is whether every square-integrable projective representation automatically has this decay, or whether a counterexample exists.

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Formalized claims in Lean

Claim #1: The central discovery is that every partial sum of Toeplitz eigenvalues in the Fock space achieves its extremal value: for prescribed measure $s$ and rank $k$, there is an orthonormal system of $k$ Fock-space functions whose joint Husimi function $u_F$ has a superlevel set of measure $s$ on which the integrated concentration $J_s(F)$ equals $S_k(s)$. The proof converts the eigenvalue problem into

/-- @claim 1 The central discovery is that every partial sum of Toeplitz eigenvalues in the Fock space achieves its extremal value: for prescribed measure $s$ and rank $k$, there is an orthonormal system of $k$ Fock-space functions whose joint Husimi function $u_F$ has a superlevel set of measure $s$ on which the integrated concentration $J_s(F)$ equals $S_k(s)$. The proof converts the eigenvalue problem into  -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main result is real: for the Fock space, the supremum of the sum of the first k Toeplitz eigenvalues over sets of fixed measure is attained, and this was open for k>1. The proof is a direct method with two genuinely useful new pieces: Lemma 3.1, a Hilbert-space lemma that repairs the orthonormality of a weak limit, and Lemma 2.6, which disintegrates the joint Husimi function into mixed blocks. I followed the Fock argument in Section 4 and it is coherent; the re-orthogonalization step, the splitting of the functional, and the induction using Lemma 2.8 all check out. Nothing is fitted and no external heavy result is imported. The k=1 case was known, but the k>1 existence is new, and the unified framework for STFT, wavelet, and Paley-Wiener settings is valuable even before the abstract extension is fully polished.

The soft spot is Section 5, and it is real. The proof of Theorem 5.5 uses the fact that Husimi functions vanish at infinity in three load-bearing places: translating maximizing sequences so that the maximum sits at the identity, producing the compact set K in the cross-term estimate, and ensuring optimal sets are bounded so disjoint translates exist in Lemma 2.8. For an arbitrary square-integrable projective representation of a noncompact locally compact group, W_psi f being in L^2 does not automatically give W_psi f in C_0(G). The Mackey obstruction group construction in Remark 5.1 shows isometry but not the C_0 property. The paper even says in Section 2 that these properties are 'straightforward to verify in the Fock setting' and then carries them to the abstract setting without a hypothesis or proof. That is a genuine gap in Theorem 5.5 as stated, and the stress-test note is right about it. It is likely fixable by adding an assumption or proving the embedding into the regular representation of the obstruction group, but it has to be done.

Two smaller issues: the claim in Theorem 1.3 that optimal sets are finite unions of intervals gets only a one-line justification from Remark 2.2 and analyticity; plausible, but it deserves a few lines. And the Donoho–Stark settlement is cited to [4], a preprint posted in the same arXiv window as this paper; that is not a fatal problem, but it means Theorem 1.3 rests on unverified and very recent literature.

The Fock theorem alone is worth a serious referee. The abstract extension is not ready in its current form, but the gap is identifiable and repairable. I would send it to review with a request for a revised Section 5 rather than desk-reject, and I would cite it for the Fock existence result even while the general theorem is still being cleaned up.

Referee Report

3 major / 4 minor

Summary. The paper proves an existence theorem for extremizers of the sum of the first k Toeplitz eigenvalues on the Fock space: for every s>0 and k≥1, the supremum S_k(s) over measurable sets Ω⊂C of measure s is attained (Theorems 1.1 and 2.1). The proof recasts the problem in terms of an orthonormal system F and the concentration functional J_s(F)=∫_{Ω_F(s)} u_F, where Ω_F(s) is the superlevel set of the joint Husimi function u_F. A maximizing sequence is translated so that u_F attains its maximum at 0; weak limits are re-orthogonalized with a geometric lemma (Lemma 3.1); the functional is decomposed with Lemma 2.6 into blocks of lower cardinality; and an induction with Lemma 2.8 forces the full normalized weak limit to be optimal. Section 5 extends the method to localization operators for abstract square-integrable projective representations under positivity assumption (5.6), with applications to STFT/Hermite windows, wavelet transforms, and the Paley-Wiener/Donoho-Stark setting.

Significance. If the main theorem holds, it settles an open existence problem for k>1 in the Fock space and, via the abstract extension, gives new existence results even for k=1 in several time-frequency settings. The Fock-space proof is a genuine direct method: it does not fit constants, the lemmas are explicit and checkable, and the induction mechanism is transparent. The main caveat is the abstract extension, which needs an additional hypothesis; once that is supplied, the result is a substantial contribution to eigenvalue optimization for Toeplitz and localization operators.

major comments (3)

[§5, proof of Theorem 5.5] The proof of Theorem 5.5 uses the assertion that Husimi functions vanish at infinity in the abstract setting: the paragraph after Remark 5.4 states that 'using the covariance property ... and the fact that Husimi functions vanish at infinity, one can translate each F^(n) so that u_{F^(n)} all achieve their maximum at the identity.' This property is never listed among the hypotheses of Theorem 5.5 and is not proved. For a general square-integrable projective representation, W_ψ f∈L^2(G) does not imply W_ψ f∈C_0(G); one would need an argument through the Mackey obstruction group or an extra assumption. The property is load-bearing in three places: Step I, to ensure the nonzero weak limit via (4.1); Step III, to obtain the compact set K used in (4.4)–(4.5); and Remark 2.2/Lemma 2.8, to guarantee that optimal sets are bounded so that disjoint translates can be chosen. Assumption (5.6) does not imply this property. Theorem 5.5 should be amended with a vanishing-at-infinity hypothesis or a lemma proving it.
[§5, Step III] The sentence 'also in the abstract setting weak convergence H^(n)⇀0 implies that u_{H^(n)} converges uniformly to 0 on compact sets' is asserted without proof. In the abstract setting this needs a short argument: weak convergence gives pointwise convergence of W_ψ H^(n) to 0, and strong continuity of π together with uniform boundedness gives equicontinuity on compact sets. This is likely fixable but should be included, since it is used in (4.5).
[§5, Lemma 2.8] The adaptation of Lemma 2.8 in the abstract setting is said to hold under (5.6), but the proof of Lemma 2.8 also uses the boundedness of the optimal sets Ω*_m(t) and Ω*_{k-m}(s-t) via Remark 2.2. In the abstract setting that boundedness is not available under (5.6) alone; it depends on the missing vanishing-at-infinity property. Thus the assumptions stated in Section 5 are insufficient as they stand.

minor comments (4)

[§2.1] The symbol k is used both for the number of eigenvalues and as a summation index, producing expressions such as 'for every k=1,...,k' and 'kX k=1'. Renaming the summation index (e.g., j) throughout Sections 2–4 would improve readability.
[§5.1.4] The Paley–Wiener representation x↦T_x on PW is not irreducible, yet §5.1.4 concludes 'Hence, Theorem 5.5 applies' and then states Theorem 5.6. Since Theorem 5.5 is phrased for irreducible representations, the statement should be softened to say that the proof, rather than the theorem as stated, carries over because the only needed property is the isometry of the wavelet transform.
[Abstract and Introduction] The abstract uses 'first K eigenvalues' with capital K while the main text uses k; the notation should be harmonized.
[Lemma 2.6] The normalization condition ∥f_k∥²+∥h_k∥²=1 for every k=1,...,k should be written with a second index to avoid the same k-clash; this would make the statement easier to read.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Fock-space existence proof is a self-contained direct method; self-citations are contextual and not load-bearing.

full rationale

The central claim is an existence statement proved by the direct method: the supremum S_k(s) is defined as the supremum of J_s(F) over orthonormal systems and is then shown to be attained by constructing a maximizing sequence, extracting a weak limit, repairing orthonormality with Lemma 3.1, and eliminating all mixed blocks via the induction step. No parameter is fitted to data, no quantity called a prediction is actually an input, and no external result is imported to define the functional. The use of lower-cardinality suprema in Lemma 2.8 is a legitimate induction on k, not an assumption of the target result; the strict inequality in that lemma is derived from the Ky Fan maximum principle and positivity, not from the theorem being proved. The self-citations [32] and [34] appear in the introduction and in the bathtub-principle discussion, but the proof itself relies on the standard bathtub principle [29] and on standard Fock-space facts; these citations are not load-bearing for Theorem 2.1. The Section 5 extension has a potential unproved premise, namely that Husimi functions vanish at infinity for arbitrary square-integrable projective representations, which is invoked to translate maximizing sequences; this is a correctness gap or missing hypothesis, not a circular reduction, because the property is not derived from the conclusion and no citation is used to smuggle it in. Accordingly, the derivation chain is not circular and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No quantities are fitted; s and k are variables of the theorem. The proof rests on standard measure-theoretic rearrangement and spectral principles, plus explicit structural hypotheses in the wavelet section. The main potential gap is that the abstract section uses vanishing at infinity of Husimi functions without listing it as a hypothesis; this is recorded as an ad hoc assumption. The Fock version is covered by known Fock-space facts.

assumptions (6)

standard math Bathtub principle: for a nonnegative integrable function u and prescribed measure s, the integral over any set of measure s is maximized by a superlevel set of u.
Used at (2.14) and (5.5) to reduce joint maximization over Omega and F to superlevel sets of the joint Husimi function; taken from Lieb-Loss [29, Theorem 1.14].
standard math Ky Fan maximum principle: the sum of the largest eigenvalues of a positive compact operator is monotone and strictly increases when a positive operator is added.
Used in Lemma 2.8 for the strict inequality when disjoint optimal sets are merged; cited to Bhatia [7, Problem I.6.15].
domain assumption In the Fock space, T_Omega is trace-class for finite measure Omega, and the Husimi function of a nonzero f vanishes at infinity.
Quoted from Zhu [46] and used throughout Steps I-IV; these facts make superlevel sets bounded and make evaluation functionals behave under weak convergence.
standard math The abstract wavelet transform W_psi is an isometry from H into L^2(G) for a square-integrable projective representation with admissible wavelet.
Used in Section 5 to identify localization-operator eigenvalues with concentration integrals; standard in [45, Theorem 7.2] and [9, Section 4].
ad hoc to paper Assumption (5.6): every subset of positive measure has strictly positive localization energy for every nonzero vector.
Explicitly imposed to make Lemma 2.8 strict in the abstract setting; it fails for atomic groups (Remark 5.4) and is not needed for k=1.
ad hoc to paper In the abstract setting, Husimi functions attain their maximum after a translation and vanish at infinity.
Invoked in Section 5 to translate maximizing sequences so the maximum is at the identity and to extract a nonzero weak limit; not listed among the hypotheses of Theorem 5.5 and not proved for arbitrary square-integrable projective representations.

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Pith. "Pith review of On the existence of extremizers for the sum of eigenvalues of Toeplitz operators." pith.science (2026). https://pith.science/paper/7DZ7SH6I

@misc{pith2026260720965,
  author       = {Pith},
  title        = {Pith review of: On the existence of extremizers for the sum of eigenvalues of Toeplitz operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DZ7SH6I}},
  note         = {Machine review of arXiv:2607.20965}
}

abstract

We prove that, among all measurable sets $\Omega\subset\mathbb{C}$ of prescribed Lebesgue measure, there exists a set maximizing the sum of the first $K$ eigenvalues ($K\geq 1$) of the associated Toeplitz operator on the Fock space. In the Fock setting, the case $K=1$ is well known, the optimal sets being balls of prescribed measure, whereas for $K>1$ the existence of optimal sets appears to be new (maximizers are not known explicitly, and the optimality of balls remains conjectural). Moreover, under mild assumptions, our proof extends to localization operators associated with abstract wavelet transforms. In this broader setting, the result is new even for $K=1$. As an application, we prove the existence of optimal sets for the Donoho--Stark concentration problem and its generalization to orthonormal systems.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere
quant-ph 2026-08 accept novelty 7.0 of 10

The passive rearrangement of a density operator maximizes convex Husimi functionals and minimizes Wehrl entropy among states with the same spectrum, and spherical caps maximize all Ky Fan sums of Toeplitz operators on...
The isoperimetric inequality for the Ky Fan norm
math.FA 2026-07 accept novelty 7.0 of 10

Among measurable sets of fixed area, the disc uniquely maximizes (up to translation) the Ky Fan norm of the Fock Toeplitz operator, confirming the Nicola–Riccardi–Tilli conjecture.

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[1] [4]

L. D. Abreu and M. Speckbacher. Optimal concentration in the Paley-Wiener space.arXiv preprint arXiv:2607.19192, 2026

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