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arxiv: 2604.22289 · v1 · submitted 2026-04-24 · 🧮 math.FA

On numerical invariants for submodules [(z-w)²] in H²(mathbb{D}²)

Yin Liu , Yufeng Lu , Chao Zu This is my paper

Pith reviewed 2026-05-08 09:29 UTC · model grok-4.3

classification 🧮 math.FA
keywords numerical invariantshomogeneous submodulesHardy modulebidiskmonotonicity property(z-w)^2functional analysis
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The pith

Explicit formulas are derived for numerical invariants of the submodule generated by (z-w)^2 in the bidisk Hardy space, confirming monotonicity.

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

The paper examines numerical invariants associated with the homogeneous submodule generated by the polynomial (z-w)^2 inside the Hardy module over the bidisk. The authors obtain explicit formulas for these invariants in this quadratic setting. They apply the formulas to verify that the monotonicity property holds for this submodule. This provides a concrete example of the invariants' behavior outside the simpler linear case.

Core claim

For the submodule M = [(z-w)^2] in H^2(D^2), explicit formulas are obtained for the associated numerical invariants. These formulas are then used to verify the monotonicity property in this concrete setting, providing a detailed example beyond the linear case.

What carries the argument

The homogeneous submodule generated by the polynomial (z-w)^2, with its numerical invariants computed via the standard machinery of the bidisk Hardy module.

Load-bearing premise

The submodule generated by (z-w)^2 is homogeneous and the numerical invariants are well-defined and computable via the standard machinery of the bidisk Hardy module.

What would settle it

A computation of the invariants by an independent method, such as direct series expansion or numerical approximation on the bidisk, that produces values differing from the derived explicit formulas would disprove the results.

read the original abstract

In this paper, we study numerical invariants associated with a homogeneous submodule of the Hardy module over the bidisk. We focus on the submodule generated by the polynomial $(z-w)^2$ and obtain explicit formulas for the corresponding invariants. As an application, we verify the monotonicity property in this concrete setting. Our results provide a detailed example illustrating the behavior of these invariants beyond the linear case.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript examines numerical invariants for the homogeneous submodule M = [(z-w)^2] inside the bidisk Hardy module H²(𝔻²). It derives explicit formulas for the invariants and applies them to verify the monotonicity property, providing a concrete example beyond the linear case.

Significance. This work supplies an explicit, computable case study for numerical invariants of homogeneous submodules of degree 2 in the bidisk setting. Such formulas enable direct verification of monotonicity and serve as a benchmark for general theories of Hardy modules and their submodules. The explicit nature of the results strengthens the literature on multivariable operator theory by moving beyond abstract existence statements.

minor comments (3)
  1. [Introduction] The introduction would benefit from a short paragraph recalling the general definition of the numerical invariants (e.g., via curvature or reproducing-kernel formulas) before specializing to the submodule generated by (z-w)^2.
  2. [Section 3 or 4 (formulas)] Explicit formulas are stated in the abstract and presumably derived in the body; adding a brief check against the known linear case (z-w) would strengthen the presentation and allow readers to see the degree-2 correction terms immediately.
  3. [Section 2] Notation for the submodule [(z-w)^2] is standard but could be accompanied by a one-sentence reminder that it denotes the closed submodule generated by the polynomial in H²(𝔻²).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript. The work provides explicit formulas for the numerical invariants of the homogeneous submodule generated by (z-w)^2 in H²(𝔻²) and uses them to verify monotonicity. The referee recommends minor revision, but the report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity in explicit formulas for invariants

full rationale

The paper derives explicit formulas for numerical invariants of the homogeneous submodule M = [(z-w)^2] inside H²(𝔻²) by applying standard Hardy-module constructions (orthogonal complements, reproducing kernels, curvature-type invariants) to this low-degree homogeneous polynomial case. These steps consist of direct algebraic computations that do not reduce to fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. The subsequent verification of the monotonicity property follows immediately from the obtained closed-form expressions without importing uniqueness theorems or ansatzes from prior author work. The derivation remains self-contained against the external benchmarks of bidisk module theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work rests on the standard definition of the bidisk Hardy module and the notion of homogeneous submodules, both drawn from prior literature.

axioms (2)
  • domain assumption The Hardy module H^2(D^2) is the standard reproducing-kernel Hilbert space of analytic functions on the bidisk with the usual inner product.
    Invoked implicitly when defining the submodule and its invariants.
  • domain assumption Numerical invariants for homogeneous submodules are well-defined via quotient dimensions or trace formulas.
    Required for the explicit formulas to make sense.

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discussion (0)

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Reference graph

Works this paper leans on

19 extracted references

  1. [1]

    Beurling, On two problems concerning linear transformations in Hilbert space

    A. Beurling, On two problems concerning linear transformations in Hilbert space. Acta Math. 81 (1948), 239- 255

    1948

  2. [2]

    B ¨ottcher and B

    A. B ¨ottcher and B. Silbermann, Toeplitz matrices and determinants with Fisher-Hartwig symbols. J. Funct. Anal. 63 (1985), no. 2, 178-214

    1985

  3. [3]

    X. Chen, K. Guo, Analytic Hilbert modules. Chapman and Hall/CRC Res. Notes Math., vol. 433, Chapman and Hall/CRC, Boca Raton, FL 2003

    2003

  4. [4]

    R. G. Douglas, V . I. Paulsen, Hilbert modules over function algebras. Pitman Res. Notes Math. Ser., vol. 217, Longman Sci. Tech., Harlow; copublished in the United States with John Wiley and Sons, Inc., New York 1989

    1989

  5. [5]

    M. E. Fischer and R. E. Hartwig, Toeplitz determinants: Some applications, theorems, and conjectures, Advan. Chem. Phys. 15 (1968), 333-353

    1968

  6. [6]

    Guo and R

    K. Guo and R. Yang, The core function of submodules over the bidisk. Indiana Univ. Math. J. 53 (2004), no. 1, 205-222

    2004

  7. [7]

    Azari Key, Y

    F. Azari Key, Y . Lu and R. Yang, On numerical invariants for homogeneous submodules inH2(D2). New York J. Math. 23 (2017), 505-526

    2017

  8. [8]

    S. Luo, K. Izuchi and R. Yang, Hilbert-Schmidtness of some finitely generated submodules inH 2(D2). J. Math. Anal. Appl. 465 (2018), 531-546. 18

    2018

  9. [9]

    Rudin, Function theory in polydiscs

    W. Rudin, Function theory in polydiscs. Benjamin, New York 1969

    1969

  10. [10]

    Yang, Operator theory in the Hardy space over the bidisk

    R. Yang, Operator theory in the Hardy space over the bidisk. III. J. Funct. Anal. 186 (2001), 521-545

    2001

  11. [11]

    Yang, Operator theory in the Hardy space over the bidisk

    R. Yang, Operator theory in the Hardy space over the bidisk. II. Integral Equ. Oper. Theory 42 (2002), 99-124

    2002

  12. [12]

    Yang, Beurling’s phenomenon in two variables

    R. Yang, Beurling’s phenomenon in two variables. Integral Equ. Oper. Theory 48 (2004), 411-423

    2004

  13. [13]

    Yang, The core operator and congruent submodules, J

    R. Yang, The core operator and congruent submodules, J. Funct. Anal. 228 (2005), 469-489

    2005

  14. [14]

    Yang, Hilbert-Schmidt submodules and issues of unitary equivalence, J

    R. Yang, Hilbert-Schmidt submodules and issues of unitary equivalence, J. Oper. Theory 53 (2005), 169-184

    2005

  15. [15]

    Yang, A brief survey of operator theory inH 2(D2)

    R. Yang, A brief survey of operator theory inH 2(D2). Handbook of analytic operator theory, 223-258, CRC Press/Chapman Hall Handb. Math. Ser., CRC Press, Boca Raton, FL, 2019

    2019

  16. [16]

    A Course of Modern Analysis,

    E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis,” 4th ed., Cambridge Univ. Press, London/ New York, 1952

    1952

  17. [17]

    H. Zou, T. Yu and Y . Yang, The core operator and the compressed multipliers onMψ,ϕ-type submodules, Banach J. Math. Anal. 26 (2022)

    2022

  18. [18]

    C. Zu, Y . Yang and Y . Lu, Hilbert-Schmidtness of submodules inH2(D2) containingθ(z)−ϕ(w), Commun. Math. Res. 39 (2023), 331-341

    2023

  19. [19]

    Zu and Y

    C. Zu and Y . Lu, Hilbert-Schmidtness of theM θ,ϕ-type submodules, J. Math. Anal. Appl. 548 (2025), 129405. School ofMathematics andStatistics, Nany angNormalUniversity, Nany ang, Henan, 473061, P. R. China Email address:lylight@mail.bnu.edu.cn School ofMathematicsSciences, DalianUniversity ofTechnology, Dalian, Liaoning, 116024, P. R. China Email address...

    2025