Spectral constants for the quantum annulus

James E. Pascoe; Nitin Tomar; Sourav Pal

arxiv: 2601.17560 · v2 · pith:6647O5DNnew · submitted 2026-01-24 · 🧮 math.FA · math.CV· math.OA

Spectral constants for the quantum annulus

Sourav Pal , James E. Pascoe , Nitin Tomar This is my paper

Pith reviewed 2026-05-25 07:27 UTC · model grok-4.3

classification 🧮 math.FA math.CVmath.OA

keywords spectral setsquantum annulusdilation theorempolyannuluscommuting operatorsoperator theoryspectral constants

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The pith

New estimates are derived for the spectral constants making a closed annulus a K-spectral set for operators in the quantum annulus.

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

The paper derives several new estimates for the spectral constant K(A_r) such that the closed annulus or polyannulus is a K-spectral set for operators in the quantum annulus QA_r. It supplies two alternative proofs for an existing estimate, one using a dilation theorem and the other a variety in the Euclidean biball. For commuting and doubly commuting operators in QA_r the authors also obtain upper and lower bounds on the smallest such constants. These results refine the quantitative theory of spectral sets in the non-commutative setting.

Core claim

We find several new estimates for the spectral constants K(A_r) for which a closed annulus or closed polyannulus is a K-spectral set for operators in the quantum annulus QA_r. We give two alternative proofs to an existing estimate of spectral constant. The first proof capitalizes a dilation theorem due to McCullough and Pascoe, while the second proof involves a certain variety in the Euclidean biball. For commuting and doubly commuting operators in QA_r, we find upper and lower bounds for the smallest spectral constants.

What carries the argument

The quantum annulus QA_r of operators whose joint spectrum lies inside the annulus, together with the spectral constant K(A_r) that makes the closed annulus a K-spectral set for such operators.

If this is right

The closed annulus is a K-spectral set for QA_r operators with the new explicit bounds on K.
The same bounds hold for the closed polyannulus in several variables.
Commuting operators in QA_r satisfy tighter upper and lower bounds on the minimal spectral constant.
Doubly commuting operators in QA_r also admit explicit upper and lower bounds on the minimal spectral constant.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The two proof strategies suggest that dilation theorems and algebraic varieties can be combined to obtain spectral constants for other domains.
The commuting-case bounds may be used to test whether specific families of commuting operators achieve the minimal constant.

Load-bearing premise

The dilation theorem due to McCullough and Pascoe applies directly to operators in the quantum annulus QA_r.

What would settle it

An explicit operator in QA_r together with a polynomial whose norm on the operator exceeds K times its supremum norm on the closed annulus would show that the stated bound on K(A_r) fails.

read the original abstract

We find several new estimates for the spectral constants $K(\mathbb A_r)$ for which a closed annulus $\overline{\mathbb A}_r$ or closed polyannulus $\overline{\mathbb A}^n_r$ is a $K$-spectral set for operators in the quantum annulus $\mathbb Q \mathbb A_r$. We give two alternative proofs to an existing estimate of spectral constant. The first proof capitalizes a dilation theorem due to McCullough and Pascoe, while the second proof involves a certain variety in the Euclidean biball. For commuting and doubly commuting operators in $\mathbb Q \mathbb A_r$, we find upper and lower bounds for the smallest spectral constants.

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.

Desk Editor's Note private letter to a colleague

The paper supplies several new estimates for spectral constants on the quantum annulus plus two alternative proofs of a prior bound, with the commuting cases getting explicit upper and lower numbers.

read the letter

The main takeaway is that this work extends the program of K-spectral set estimates by producing fresh bounds for K(A_r) on the quantum annulus QA_r and by giving two different proofs of an existing estimate. One proof routes through the McCullough-Pascoe dilation theorem; the other works with a variety inside the Euclidean biball. It also separates the commuting and doubly commuting operator cases with concrete upper and lower bounds on the minimal spectral constant. That separation and the dual proofs are the concrete additions here. The biball-variety argument stands independently and looks like a useful alternative route. The dilation argument is the part that requires care. The abstract invokes the theorem directly for QA_r operators, so the paper must show that those operators satisfy the necessary hypotheses on the variety or contractivity. If that verification is present and correct in the full text, the proof goes through; if it is only asserted, the first proof rests on an unconfirmed step. The other results do not share that dependence. This sits in a narrow slice of multivariable operator theory. Readers already tracking spectral constants for annuli and polyannuli will get direct use from the new numbers and the proof options. Broader functional analysts will not need it. The claims are explicit extensions rather than circular or parameter-fitted, and the work shows clear engagement with the existing literature. It is worth sending to a referee who can check the dilation application and compare the tightness of the new bounds against prior literature.

Referee Report

1 major / 0 minor

Summary. The manuscript presents new estimates for the spectral constants K(A_r) such that the closed annulus or polyannulus is a K-spectral set for operators in the quantum annulus QA_r. It supplies two alternative proofs of an existing estimate (one via the McCullough-Pascoe dilation theorem, the other via a variety inside the Euclidean biball) and derives upper and lower bounds on the minimal spectral constant for commuting and doubly commuting elements of QA_r.

Significance. If the derivations are correct, the work supplies concrete new bounds on spectral constants for a noncommutative annulus domain and supplies independent proofs of a prior estimate; the commuting-case bounds are also potentially useful for applications in multivariable operator theory.

major comments (1)

[Section containing the first alternative proof (McCullough-Pascoe invocation)] The first alternative proof invokes the McCullough-Pascoe dilation theorem as applying directly to operators in QA_r. The manuscript must explicitly verify that elements of QA_r satisfy the theorem's hypotheses (contractivity on the relevant variety or domain); without this check the route is not self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on the McCullough-Pascoe application. We agree that an explicit verification of the theorem's hypotheses is needed to make the argument self-contained and will incorporate this in the revision.

read point-by-point responses

Referee: [Section containing the first alternative proof (McCullough-Pascoe invocation)] The first alternative proof invokes the McCullough-Pascoe dilation theorem as applying directly to operators in QA_r. The manuscript must explicitly verify that elements of QA_r satisfy the theorem's hypotheses (contractivity on the relevant variety or domain); without this check the route is not self-contained.

Authors: We agree that the manuscript should contain an explicit verification that operators T in QA_r are contractive with respect to the variety appearing in the McCullough-Pascoe theorem. In the revised version we will insert a short paragraph immediately preceding the invocation of the theorem that confirms ||p(T)|| ≤ sup_{z in V} |p(z)| for the relevant polynomials p and variety V, using the definition of QA_r and the fact that the spectral radius on the annulus is controlled by the maximum modulus on the boundary circles. This addition will render the proof self-contained without altering the overall argument. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to dilation theorem is not load-bearing

full rationale

The paper presents two independent alternative proofs for the spectral constant estimates, with the McCullough-Pascoe dilation theorem used only in the first. The second proof relies on a variety in the Euclidean biball and is independent of the cited theorem. No equations or derivations reduce by construction to fitted parameters, self-definitions, or a self-citation chain. The central claims retain independent mathematical content outside the self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.0 · 5640 in / 975 out tokens · 36768 ms · 2026-05-25T07:27:32.116494+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; Jcost_pos_of_ne_one matches

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matches
MATCHES: this paper passage directly uses, restates, or depends on the cited Recognition theorem or module.

An invertible operator T∈QA_r if and only if β(T*,T)≥0. … every operator in QA_r dilates to a quantum annulus unitary J … β(J*,J)=0
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean cost_alpha_one_eq_jcost; alpha_pin_under_high_calibration echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

T extends to ˆT = [T, T(T*T)^{-1/2}β^{1/2}; 0, T^{-*}] with β(ˆT*,ˆT)=0
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking; D3_admits_circle_linking echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

2^n ≤ K_dc(A^n_r) ≤ ((3r²-1)/(r²-1))^n … lim 2^n … 3^n

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

And ˆo,On a pair of commutative contractions, Acta Sci

T. And ˆo,On a pair of commutative contractions, Acta Sci. Math. (Szeged), 24 (1963), 88 – 90

work page 1963
[2]

Badea, B

C. Badea, B. Beckermann and M. Crouzeix,Intersections of several disks of the Riemann sphere as K-spectral sets, In: Commun. Pure Appl. Anal., 8 (2009), 37 – 54

work page 2009
[3]

Bercovici, C

H. Bercovici, C. Foias, L. Kerchy and B. Sz.-Nagy,Harmonic analysis of operators on Hilbert space, Universitext Springer, New York, 2010

work page 2010
[4]

Crouzeix,The annulus as a K-spectral set, Commun

M. Crouzeix,The annulus as a K-spectral set, Commun. Pure Appl. Anal., 11 (2012), 2291 – 2303

work page 2012
[5]

Crouzeix and A

M. Crouzeix and A. Greenbaum,Spectral sets: numerical range and beyond, SIAM J. Matrix Anal. Appl., 40 (2019), 1087 – 1101

work page 2019
[6]

Crouzeix,Spectral estimates in the quantum annulus and in the numerical annulus, arXiv: 2512.11813

M. Crouzeix,Spectral estimates in the quantum annulus and in the numerical annulus, arXiv: 2512.11813

work page arXiv
[7]

McCullough and J

S. McCullough and J. E. Pascoe,Geometric dilations and operator annuli, J. Funct. Anal., 285 (2023), Paper No. 110035, 20 pp

work page 2023
[8]

Pal and N

S. Pal and N. Tomar,Dilation on an annulus and von Neumann’s inequality on certain varieties in the biball, arXiv: 2505.08476

work page arXiv
[9]

Parrott,Unitary dilations for commuting contractions, Pacific J

S. Parrott,Unitary dilations for commuting contractions, Pacific J. Math., 34 (1970), 481 – 490

work page 1970
[10]

J. E. Pascoe,The spectral constant for the quantum cross and asymptotically sharp bounds for annuli, Acta Sci. Math. (Szeged) (2026), DOI: https://doi.org/10.1007/s44146-025-00222-5

work page doi:10.1007/s44146-025-00222-5 2026
[11]

Paulsen and D

V .I. Paulsen and D. Singh,Extensions of Bohr’s inequality, Bull. London Math. Soc., 38 (2006), 991 – 999

work page 2006
[12]

R. M. Range,Holomorphic functions and integral representations in several complex variables, Springer Science & Business Media, V ol. 108, 1998

work page 1998
[13]

A. L. Shields,Weighted shift operators and analytic function theory, Topics in operator theory, (13): 49 – 128, 1974

work page 1974
[14]

Tsikalas,A note on a spectral constant associated with an annulus, Oper

G. Tsikalas,A note on a spectral constant associated with an annulus, Oper. Matrices, 16 (2022), 95 – 99. (Sourav Pal) MATHEMATICSDEPARTMENT, INDIANINSTITUTE OFTECHNOLOGYBOMBAY, POWAI, MUMBAI - 400076, INDIA. Email address:sourav@math.iitb.ac.in (James E. Pascoe) DEPARTMENT OFMATHEMATICS, DREXELUNIVERSITY, PHILADELPHIA, USA Email address:jep362@drexel.edu...

work page 2022

[1] [1]

And ˆo,On a pair of commutative contractions, Acta Sci

T. And ˆo,On a pair of commutative contractions, Acta Sci. Math. (Szeged), 24 (1963), 88 – 90

work page 1963

[2] [2]

Badea, B

C. Badea, B. Beckermann and M. Crouzeix,Intersections of several disks of the Riemann sphere as K-spectral sets, In: Commun. Pure Appl. Anal., 8 (2009), 37 – 54

work page 2009

[3] [3]

Bercovici, C

H. Bercovici, C. Foias, L. Kerchy and B. Sz.-Nagy,Harmonic analysis of operators on Hilbert space, Universitext Springer, New York, 2010

work page 2010

[4] [4]

Crouzeix,The annulus as a K-spectral set, Commun

M. Crouzeix,The annulus as a K-spectral set, Commun. Pure Appl. Anal., 11 (2012), 2291 – 2303

work page 2012

[5] [5]

Crouzeix and A

M. Crouzeix and A. Greenbaum,Spectral sets: numerical range and beyond, SIAM J. Matrix Anal. Appl., 40 (2019), 1087 – 1101

work page 2019

[6] [6]

Crouzeix,Spectral estimates in the quantum annulus and in the numerical annulus, arXiv: 2512.11813

M. Crouzeix,Spectral estimates in the quantum annulus and in the numerical annulus, arXiv: 2512.11813

work page arXiv

[7] [7]

McCullough and J

S. McCullough and J. E. Pascoe,Geometric dilations and operator annuli, J. Funct. Anal., 285 (2023), Paper No. 110035, 20 pp

work page 2023

[8] [8]

Pal and N

S. Pal and N. Tomar,Dilation on an annulus and von Neumann’s inequality on certain varieties in the biball, arXiv: 2505.08476

work page arXiv

[9] [9]

Parrott,Unitary dilations for commuting contractions, Pacific J

S. Parrott,Unitary dilations for commuting contractions, Pacific J. Math., 34 (1970), 481 – 490

work page 1970

[10] [10]

J. E. Pascoe,The spectral constant for the quantum cross and asymptotically sharp bounds for annuli, Acta Sci. Math. (Szeged) (2026), DOI: https://doi.org/10.1007/s44146-025-00222-5

work page doi:10.1007/s44146-025-00222-5 2026

[11] [11]

Paulsen and D

V .I. Paulsen and D. Singh,Extensions of Bohr’s inequality, Bull. London Math. Soc., 38 (2006), 991 – 999

work page 2006

[12] [12]

R. M. Range,Holomorphic functions and integral representations in several complex variables, Springer Science & Business Media, V ol. 108, 1998

work page 1998

[13] [13]

A. L. Shields,Weighted shift operators and analytic function theory, Topics in operator theory, (13): 49 – 128, 1974

work page 1974

[14] [14]

Tsikalas,A note on a spectral constant associated with an annulus, Oper

G. Tsikalas,A note on a spectral constant associated with an annulus, Oper. Matrices, 16 (2022), 95 – 99. (Sourav Pal) MATHEMATICSDEPARTMENT, INDIANINSTITUTE OFTECHNOLOGYBOMBAY, POWAI, MUMBAI - 400076, INDIA. Email address:sourav@math.iitb.ac.in (James E. Pascoe) DEPARTMENT OFMATHEMATICS, DREXELUNIVERSITY, PHILADELPHIA, USA Email address:jep362@drexel.edu...

work page 2022