Sharp spectral constants for scaled $q$-numerical ranges

Mohamed Amine Aouichaoui; Ryan O'Loughlin

REVIEW 2 major objections 4 minor 13 references

Sharp spectral constants for scaled $q$-numerical ranges

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that the optimal spectral constant for every scaled $q$-numerical range is $\max\{1, 2|q|/(1+\sqrt{1-|q|^2})\}$, with equality attained by a rank-one matrix.

desk verdict A clean, reusable transfer theorem for spectral constants that settles the scaled q-numerical range conjecture, with the caveat that the upper bound leans on the just-announced proof of Crouzeix's conjecture. read the letter →

arxiv 2608.09866 v1 pith:C2CZEYLS submitted 2026-08-10 math.FA

Mohamed Amine Aouichaoui , Ryan O'Loughlin This is my paper · ORCID

classification math.FA MSC 47A1247A2515A60

keywords q-numericalrangescalednumericalspectralsetoptimalconstantconditionnumberpolynomialfunctionalcalculusoperatornormbound

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

This paper settles the question of the optimal spectral constant for scaled $q$-numerical ranges. For every matrix $A$, every polynomial $p$, and every $0<|q|\leq 1$, the norm $\|p(A)\|$ is bounded by the maximum of $|p|$ on the scaled range $\Omega_q(A)=q^{-1}W_q(A)$, and the best possible constant is $\max\{1, 2|q|/(1+\sqrt{1-|q|^2})\}$. This confirms a conjecture from earlier work and is the first exact value for these sets. The proof is driven by a new geometric identity that writes $\Omega_{\eta(\gamma)}(A)$ as a union of ordinary numerical ranges of similarities of $A$; that identity transfers sharp spectral-set estimates from the classical numerical range to every scaled $q$-numerical range. Knowing the optimal constant makes the functional-calculus bound dimension-free and parameter-tight.

What carries the argument

The engine of the paper is Theorem 2.2, the exact similarity formula $\Omega_{\eta(\gamma)}(A)=\bigcup_{\kappa(S)\le\gamma} W(S^{-1}AS)$ with $\eta(\gamma)=2/(\gamma+\gamma^{-1})$. One inclusion uses the classical inequality $\langle Pu,u\rangle\langle P^{-1}u,u\rangle\le (M^2+m^2)^2/(4M^2m^2)$ for a positive matrix $P$ with extremal singular values $M,m$, which gives the condition-number bound; the reverse inclusion constructs an explicit positive matrix with eigenvalues $\gamma$ and $\gamma^{-1}$. The formula is used through the inverse relation $\chi(r)=(1+\sqrt{1-r^2})/r$: similarities with $\kappa(S)<\chi(r)$ send $W(S^{-1}AS)$ compactly inside $\Omega_r(A)$, which is precisely the containment hypothesis needed by the paper's abstract transfer theorem for spectral constants.

What would settle it

Compute, for a fixed nonnormal $A$ and fixed $|q|<1$, the supremum of $\|p(A)\|/\max_{z\in\Omega_q(A)}|p(z)|$ over polynomials; any value above $\max\{1, 2|q|/(1+\sqrt{1-|q|^2})\}$ would disprove the theorem. The paper's own sharpness example is the $2\times2$ matrix $N=\begin{pmatrix}0&2\\0&0\end{pmatrix}$, which attains the claimed constant, so a counterexample would have to be found elsewhere.

Watch

Extended reading notes

Core claim

The central discovery is an exact similarity formula and the sharp inequality that follows from it. For every $\gamma\ge 1$, $\Omega_{\eta(\gamma)}(A)=\bigcup_{\kappa(S)\le\gamma} W(S^{-1}AS)$, where $\eta(\gamma)=2/(\gamma+\gamma^{-1})$ and $\kappa(S)=\|S\|\,\|S^{-1}\|$. Using this formula, the paper proves that for every $0<|q|\le 1$, every $n\ge 2$, every $A\in M_n(\mathbb C)$, and every polynomial $p$, $\|p(A)\|\le \max\{1, 2|q|/(1+\sqrt{1-|q|^2})\}\max_{z\in\Omega_q(A)}|p(z)|$, and that the constant is optimal. The upper bound transfers the theorem that ordinary numerical ranges are $2$-spectral sets through the similarity formula, while the lower bound is a direct computation on a rank-one matrix.

Load-bearing premise

The upper bound in Theorem 4.1 is proved only on top of the external theorem, cited as [8] and [1], that every matrix's ordinary numerical range is a $2$-spectral set; if that theorem failed, the upper bound would fail, although the lower-bound example would still stand.

Editorial extensions

If this is right

For every $n\ge2$, $A\in M_n(\mathbb C)$, and polynomial $p$, the bound $\|p(A)\|\le \max\{1, 2|q|/(1+\sqrt{1-|q|^2})\}\max_{z\in\Omega_q(A)}|p(z)|$ holds, and no smaller universal constant exists.
As $|q|$ runs from $0$ to $1$, the optimal constant runs from $1$ to $2$, so the estimate interpolates between a trivial spectral-set bound and the classical $2$-spectral-set bound for ordinary numerical ranges.
The exact similarity formula gives a compact-containment criterion: whenever $\kappa(S)<\chi(|q|)$, the numerical range $W(S^{-1}AS)$ lies strictly inside $\Omega_q(A)$, so polynomial bounds on $\Omega_q(A)$ automatically control all sufficiently well-conditioned similarities.
The abstract transfer theorem applies to any family of $C$-spectral sets, so the same mechanism can turn a spectral-set proof for one class of sets into sharp constants for a parameterized family.

Reading between the lines

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

The lower-bound example is only $2\times2$, which suggests that the extremal matrices for this sharp constant may always be low-dimensional; a testable extension is to check whether every $q$ admits an extremizer of size at most $2$.
Because the constant collapses to $1$ as $|q|\to0$, one could investigate whether for small $q$ the scaled range behaves like a genuine spectral set with additional structure, such as a completely bounded version of the inequality.
The identity with condition-number-bounded similarities points to a computational route for the $q$-numerical radius: maximize $|z|$ over $W(S^{-1}AS)$ with $\kappa(S)\le\chi(r)$, which may be more tractable than directly solving the constrained bilinear form.

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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

Colleague,

The headline: this is a solid paper that resolves the sharp spectral constant for scaled q-numerical ranges, Conjecture 4.2 of the authors' own earlier paper, and does it with a genuinely reusable mechanism. The constant itself was also proved by Jin in a concurrent preprint, but the method here is the real contribution.

What's new: Theorem 2.2 is an exact similarity formula: Omega_{eta(gamma)}(A) = union_{kappa(S)<=gamma} W(S^{-1}AS) with eta(gamma)=2/(gamma+gamma^{-1}). It is a nice geometric fact, and the proof via the Kantorovich inequality is short and correct. Theorem 3.2 is an extremal-pair amplification: if a convex domain D has spectral constant M_D(A)>1, then you can find a similarity with condition number gamma that multiplies that constant by gamma. Theorem 3.3 turns that into an abstract transfer principle: if every matrix has a C-spectral set K(B), and all K(S^{-1}AS) sit compactly inside D for kappa(S)

The soft spot is exactly what the stress test flags: the upper bound in Theorem 4.1 assumes that W(B) is a 2-spectral set for every B, citing two 2026 preprints, [8] and [1]. The paper does not prove this. If those preprints are wrong, the upper bound falls to max{1, C/chi(|q|)} with an unknown C>2, and the lower bound would no longer match. That is a real dependency, but it is explicitly disclosed, and the authors also note Jin's independent verification of (1.3). A referee can check the cited preprints. The lower bound uses the rank-one formula from [6], another preprint, but that one is older and less likely to be problematic.

My take: the transfer mechanism is the intellectual content, and it is well executed. The proof of the sharp constant is conditional on the Crouzeix conjecture resolution, but that resolution now has two independent proofs, so the dependency is about as safe as it gets for a brand-new result. The paper deserves a serious referee and should be accepted, with a referee note to verify the external theorems. I would use Theorem 3.3 in my own work, and I would bring this to reading group.

Recommendation: send it to a good operator theory journal; accept after a standard check of [8] and [1].

Referee Report

2 major / 4 minor

Summary. The manuscript studies scaled q-numerical ranges Ω_q(A)=q^{-1}W_q(A). Its central results are: (i) Theorem 2.2, an exact similarity representation Ω_{η(γ)}(A)=union_{κ(S)≤γ} W(S^{-1}AS); (ii) Theorem 3.3, an abstract transfer principle that turns a C-spectral-set bound for W(B) into a spectral-set bound for Ω_q(A); and (iii) Theorem 4.1, the sharp spectral constant C_q = max{1, 2|q|/(1+√(1-|q|^2))} for every n≥2 and 0<|q|≤1. The upper bound is obtained by applying Theorem 3.3 with K(B)=W(B) and C=2, relying on the recently announced proof of Crouzeix's conjecture; the lower bound uses a rank-one nilpotent matrix and a cited formula for its q-numerical radius.

Significance. If the external 2-spectral-set theorem is accepted, the paper settles the conjecture from [11] and provides a clean mechanism that may be useful beyond q-numerical ranges. The similarity identity in Theorem 2.2 is elegant and proved in detail, and the internal proofs in Sections 2 and 3 are careful and correct. However, the advertised sharp constant is conditional on unproved external preprints ([8], [1]) for the upper bound and on [6] for the lower-bound formula; the unconditional contribution is the reduction, not the final number. This does not diminish the value of the mechanism, but the conditional nature should be stated precisely.

major comments (2)

[Section 4, Theorem 4.1 upper bound] The upper bound C_q ≤ max{1, 2|q|/(1+√(1-|q|^2))} is obtained by feeding the assertion 'W(B) is a 2-spectral set for every square matrix B' into Theorem 3.3, with that assertion cited only to the preprints [8] and [1]. No part of the manuscript proves or sketches this fact, and if the true universal constant for W(B) were C>2, the argument would only give C_q ≤ max{1, C/χ(|q|)}, which would not match the lower bound for |q| close to 1. The central theorem is therefore conditional. Please either state Theorem 4.1 explicitly as conditional on Crouzeix's conjecture, cite a peer-reviewed version of the 2-spectral-set theorem, or include a verification of the cited result.
[Section 4, Theorem 4.1 lower bound] The sharpness example N=[[0,2],[0,0]] uses the formula max_{z∈Ω_q(N)}|z|=(1+√(1-|q|^2))/|q| from [6, Theorem 2.1]. Since [6] is also an arXiv preprint, the lower bound is not fully proved within the manuscript. This is a smaller external dependency than the upper bound, but for a claimed sharp constant both sides should be justified either by a proof in the paper or by a reference that the reader can verify.

minor comments (4)

[Section 4, proof of Theorem 4.1] As written, the equality Ω_q(A)=Ω_{|q|}(A)=Ω_{|q|}(A)^\circ identifies the set with its interior. For a non-scalar A, Ω_q(A) is a closed convex set with nonempty interior and is not equal to its interior; the intended argument is that Theorem 3.3 is applied to D=Ω_{|q|}(A)^\circ and the spectral constant for the interior is the same as for its closure. Please rephrase.
[Introduction, final paragraph] The sentence about Jin's preprint [8] should be clarified: does [8] already prove (1.3) directly, and does it use the same similarity mechanism? The authors should state precisely how their result differs, especially since the abstract presents the sharp inequality as a main novelty.
[Section 2, Proposition 2.1(2)] The strict interior inclusion for non-scalar matrices is cited from [9, Theorem 2.5]; adding a few words about why that theorem applies to the scaled ranges would help the reader.
[Title/header] The full-text header renders the title with an extra space ('CONST ANTS'), which is presumably a typesetting artifact; the metadata title is correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the upper bound depends on external Crouzeix preprints, but that is a correctness risk, not a circular input.

full rationale

The claimed sharp constant is not used as an input anywhere in the derivation. Theorem 2.2, the exact similarity identity Omega_{eta(gamma)}(A) = union_{kappa(S)<=gamma} W(S^{-1}AS), is proved self-containedly via the Kantorovich inequality and an explicit construction of S, with no appeal to the target inequality. Theorem 3.2 and Theorem 3.3 are abstract transfer results proved from the quoted extremal-pair lemma and the spectral-set hypothesis; again the target inequality is not assumed. In Theorem 4.1, the upper bound applies Theorem 3.3 with K(B)=W(B), C=2, and gamma=chi(|q|); the value 2 enters solely through the external theorem that W(B) is a 2-spectral set for every matrix B, cited to [8] and [1]. That theorem is stronger than, and not identical to, the claimed conclusion, and its authors are not the present authors. The lower bound is a direct rank-one example with no fitted or imported constant. The self-citation to [11] supplies the conjecture and contextual inclusion statements, but is not load-bearing: the proof does not rely on [11] for any of its steps. The manuscript's own note that Jin's preprint also proves (1.3) is an acknowledgment of overlap rather than a circular dependency: the derivation would be valid if the external Crouzeix theorem is correct. Whether the external preprints are complete and correct is a verification risk, not a circularity. Therefore no circular step is exhibited, and the score is 0.

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

The central claim rests on established theorems and recent preprints; no fitted parameters appear. The main load-bearing external inputs are the 2-spectral set theorem for numerical ranges, the extremal-function machinery, and the rank-one formula. None is authored by the present authors except the conjecture from [11], and none is used as the target result.

assumptions (6)

domain assumption Crouzeix's theorem: the numerical range W(B) is a 2-spectral set for every square matrix B.
Invoked in Theorem 4.1 as C=2; source is preprint [8] with [1], not proved here.
domain assumption Extremal-pair lemma: if M_D(A)>1, there is f in A(D) with ||f||_D=1 and unit vectors x,y with x perpendicular to y and f(A)x=M_D(A)y.
Used to construct similarity S in Theorem 3.2; follows from [3, Theorem 2.1] and [2, Theorem 4.1], combined without proof.
domain assumption Geometry of q-numerical ranges: convexity, strict nesting Omega_t subset int Omega_r for r<t<1, and sigma(A) subset int Omega_r(A) for non-scalar A.
Cites [13] and [9]; used to ensure D is a bounded convex domain containing the spectrum and to get strict containment in Corollary 2.3(2).
domain assumption Rank-one q-numerical radius formula for N=[[0,2],[0,0]] giving max_{z in Omega_q(N)} |z| = (1+sqrt(1-|q|^2))/|q|.
Lower bound in Theorem 4.1; cited to [6, Theorem 2.1].
standard math Mergelyan's theorem and Runge's theorem: polynomials are dense in A(D) for bounded convex D and polynomial estimates determine spectral constants.
Justifies (3.1) and the reduction from rational functions to polynomials for convex sets.
standard math Kantorovich inequality for positive definite matrices.
Proved inside Theorem 2.2; bounds t>=eta(gamma).

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Cite this review

Pith. "Pith review of Sharp spectral constants for scaled $q$-numerical ranges." pith.science (2026). https://pith.science/paper/C2CZEYLS

@misc{pith2026260809866,
  author       = {Pith},
  title        = {Pith review of: Sharp spectral constants for scaled $q$-numerical ranges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C2CZEYLS}},
  note         = {Machine review of arXiv:2608.09866}
}

abstract

For $ n \geq 2$, $A\in M_n(\mathbb C)$ and $0<|q|\leq 1$, let $\Omega_q(A)=q^{-1}W_q(A)$ be the scaled $q$-numerical range. We prove that for every $\gamma \geq 1$, \[ \Omega_{\eta(\gamma)}(A) =\bigcup_{\kappa(S)\leq\gamma}W(S^{-1}AS), \qquad \eta(\gamma)=\frac{2}{\gamma+\gamma^{-1}}, \] where $\kappa(S)=\|S\|\,\|S^{-1}\|$. As a consequence, we prove the sharp inequality \[ \|p(A)\|\leq \max\!\left\{1,\frac{2|q|}{1+\sqrt{1-|q|^2}}\right\} \max_{z\in\Omega_q(A)}|p(z)|, \] for all polynomials $p$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages

[8]

Jin,The numerical range is a2-spectral set, Preprints (2026), preprint 202607.1919, version 4

S. Jin,The numerical range is a2-spectral set, Preprints (2026), preprint 202607.1919, version 4. doi:10.20944/preprints202607.1919.v4

arXiv 2026
[1]

A solution to Crouzeix's conjecture

E. Lorist and F. L. Schwenninger,A solution to Crouzeix’s conjecture, arXiv:2608.03841v1 (2026). doi:10.48550/arXiv.2608.03841

work page Pith review arXiv doi:10.48550/arxiv.2608.03841 2026
[6]

$q$-numerical radius of rank-one operators and the generalized Buzano inequality

D. Denčić, H. Stanković, M. Krstić, and I. Damnjanović,q-numerical radius of rank-one operators and the generalized Buzano inequality, arXiv:2503.05036v1 (2025). doi:10.48550/arXiv.2503.05036

work page Pith review arXiv doi:10.48550/arxiv.2503.05036 2025
[11]

O’Loughlin and J

R. O’Loughlin and J. Rani,q-numerical ranges and spectral sets, arXiv:2603.15536v1 (2026). doi:10.48550/arXiv.2603.15536

work page doi:10.48550/arxiv.2603.15536 2026
[2]

Bickel, P

K. Bickel, P. Gorkin, A. Greenbaum, T. Ransford, F. L. Schwenninger, and E. Wegert, Crouzeix’s conjecture and related problems, Comput. Methods Funct. Theory20(2020), no. 3–4, 701–728. doi:10.1007/s40315-020-00350-9

work page doi:10.1007/s40315-020-00350-9 2020
[3]

Crouzeix,Bounds for analytical functions of matrices, Integral Equations Operator Theory48(2004), no

M. Crouzeix,Bounds for analytical functions of matrices, Integral Equations Operator Theory48(2004), no. 4, 461–477. doi:10.1007/s00020-002-1188-6

work page doi:10.1007/s00020-002-1188-6 2004
[4]

Crouzeix,Numerical range and functional calculus in Hilbert space, J

M. Crouzeix,Numerical range and functional calculus in Hilbert space, J. Funct. Anal. 244(2007), no. 2, 668–690. doi:10.1016/j.jfa.2006.10.013

work page doi:10.1016/j.jfa.2006.10.013 2007
[5]

Crouzeix and C

M. Crouzeix and C. Palencia,The numerical range is a(1 + √ 2)-spectral set, SIAM J. Matrix Anal. Appl.38(2017), no. 2, 649–655. doi:10.1137/17M1116672

doi:10.1137/17m1116672 2017

Show all 13 references

[7]

Greub and W

W. Greub and W. Rheinboldt,On a generalization of an inequality of L. V. Kan- torovich, Proc. Amer. Math. Soc.10(1959), 407–415. doi:10.1090/S0002-9939-1959- 0105028-3

1959 doi
[9]

C.-K. Li, P. P. Mehta, and L. Rodman,A generalized numerical range: the range of a constrained sesquilinear form, Linear Multilinear Algebra37(1994), no. 1–3, 25–49. doi:10.1080/03081089408818311

1994 doi
[10]

Marcus and P

M. Marcus and P. Andresen,Constrained extrema of bilinear functionals, Monatsh. Math.84(1977), 219–235. doi:10.1007/BF01538033

1977 doi
[12]

Ransford and F

T. Ransford and F. L. Schwenninger,Remarks on the Crouzeix–Palencia proof that the numerical range is a(1 + √ 2)-spectral set, SIAM J. Matrix Anal. Appl.39(2018), no. 1, 342–345. doi:10.1137/17M1143757

2018 doi
[13]

Tsing,The constrained bilinear form and theC-numerical range, Linear Algebra Appl.56(1984), 195–206

N.-K. Tsing,The constrained bilinear form and theC-numerical range, Linear Algebra Appl.56(1984), 195–206. doi:10.1016/0024-3795(84)90125-3. Department of Mathematics, F aculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia Email address:aouichaoui.moh...

1984 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [8]

Jin,The numerical range is a2-spectral set, Preprints (2026), preprint 202607.1919, version 4

S. Jin,The numerical range is a2-spectral set, Preprints (2026), preprint 202607.1919, version 4. doi:10.20944/preprints202607.1919.v4

arXiv 2026

[2] [1]

A solution to Crouzeix's conjecture

E. Lorist and F. L. Schwenninger,A solution to Crouzeix’s conjecture, arXiv:2608.03841v1 (2026). doi:10.48550/arXiv.2608.03841

work page Pith review arXiv doi:10.48550/arxiv.2608.03841 2026

[3] [6]

$q$-numerical radius of rank-one operators and the generalized Buzano inequality

D. Denčić, H. Stanković, M. Krstić, and I. Damnjanović,q-numerical radius of rank-one operators and the generalized Buzano inequality, arXiv:2503.05036v1 (2025). doi:10.48550/arXiv.2503.05036

work page Pith review arXiv doi:10.48550/arxiv.2503.05036 2025

[4] [11]

O’Loughlin and J

R. O’Loughlin and J. Rani,q-numerical ranges and spectral sets, arXiv:2603.15536v1 (2026). doi:10.48550/arXiv.2603.15536

work page doi:10.48550/arxiv.2603.15536 2026

[5] [2]

Bickel, P

K. Bickel, P. Gorkin, A. Greenbaum, T. Ransford, F. L. Schwenninger, and E. Wegert, Crouzeix’s conjecture and related problems, Comput. Methods Funct. Theory20(2020), no. 3–4, 701–728. doi:10.1007/s40315-020-00350-9

work page doi:10.1007/s40315-020-00350-9 2020

[6] [3]

Crouzeix,Bounds for analytical functions of matrices, Integral Equations Operator Theory48(2004), no

M. Crouzeix,Bounds for analytical functions of matrices, Integral Equations Operator Theory48(2004), no. 4, 461–477. doi:10.1007/s00020-002-1188-6

work page doi:10.1007/s00020-002-1188-6 2004

[7] [4]

Crouzeix,Numerical range and functional calculus in Hilbert space, J

M. Crouzeix,Numerical range and functional calculus in Hilbert space, J. Funct. Anal. 244(2007), no. 2, 668–690. doi:10.1016/j.jfa.2006.10.013

work page doi:10.1016/j.jfa.2006.10.013 2007

[8] [5]

Crouzeix and C

M. Crouzeix and C. Palencia,The numerical range is a(1 + √ 2)-spectral set, SIAM J. Matrix Anal. Appl.38(2017), no. 2, 649–655. doi:10.1137/17M1116672

doi:10.1137/17m1116672 2017