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arxiv: 2511.05306 · v2 · pith:IXLZ5DIWnew · submitted 2025-11-07 · 🧮 math.CV · math.FA

Pairs of Clark Unitary Operators on the Bidisk and their Taylor Joint Spectra

Palak Arora , Kelly Bickel , Constanze Liaw , Alan Sola This is my paper

Pith reviewed 2026-05-21 19:41 UTC · model grok-4.3

classification 🧮 math.CV math.FA
keywords Clark theorybidiskmodel spacesinner functionsTaylor joint spectrumrational inner functionscommuting unitariescompressed shifts
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The pith

The Taylor joint spectrum of Clark unitaries on the bidisk coincides with level sets of rational inner functions.

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

The paper develops Clark theory for commuting compressed shift operators on model spaces K_φ for inner functions φ on the bidisk. It identifies the adjoint of an embedding operator as a weighted Cauchy transform of the Clark measure and constructs pairs of commuting unitaries as perturbations of the compressed shifts. These unitaries are unitarily equivalent to multiplication by the coordinate functions on L² of the Clark measure. For rational inner functions the Taylor joint spectrum of the unitaries is proved to coincide exactly with the level sets of φ. This connects operator spectra in two variables directly to the geometry of the defining inner function.

Core claim

Under natural assumptions which generically include the case when φ is rational inner, commuting unitaries on K_φ are obtained as perturbations of the compressed shifts. These unitaries are unitarily equivalent to multiplication by the coordinate functions on L²(σ_α). When φ is rational inner, the Taylor joint spectrum of these Clark unitaries coincides with level sets of φ.

What carries the argument

Clark unitaries on the bidisk model space K_φ, obtained as perturbations of compressed shifts and unitarily equivalent to multiplication by coordinates on L²(σ_α) for the Clark measure.

If this is right

  • The Taylor joint spectrum equals the level sets of φ.
  • The constructed unitaries commute and are equivalent to multiplication operators on L²(σ_α).
  • The adjoint of the embedding operator is a weighted Cauchy transform of the Clark measure.
  • Special cases of the bidisk setting admit simplified results.

Where Pith is reading between the lines

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

  • The result may extend to non-rational inner functions if the perturbation condition can be checked directly.
  • This description could simplify computation of joint spectra for other pairs of commuting operators on similar model spaces.
  • The explicit link between spectra and level sets may connect to questions about invariant subspaces or zero sets in several complex variables.

Load-bearing premise

The natural assumptions allow commuting unitaries on K_φ to be obtained as perturbations of the compressed shifts, which hold for rational inner functions.

What would settle it

An explicit computation of the Taylor joint spectrum for a specific rational inner function φ where the spectrum fails to equal the level sets of φ.

Figures

Figures reproduced from arXiv: 2511.05306 by Alan Sola, Constanze Liaw, Kelly Bickel, Palak Arora.

Figure 1
Figure 1. Figure 1: Support sets in T 2 ≃ [−π, π) 2 for the Clark mea￾sures σα associated with ϕ = z1z2 and α = 1 (black), α = e i π 4 (gray), α = e i π 2 (orange), and α = e −i π 4 (pink). of f ∈ Kϕ on supp σα, which is well-defined for σα-a.e. ζ ∈ T 2 . (In this example, one could also check directly that this evaluation map gives a unitary operator from Kϕ → L 2 (σα).) Then, for almost every ζ ∈ supp σα and each power of z… view at source ↗
Figure 2
Figure 2. Figure 2: Support sets in T 2 ≃ [−π, π) 2 for the Clark measures σα for ϕ = 2z1z2−z1−z2 2−z1−z2 and α = 1 (black), α = e i π 4 (gray), α = e i π 2 (orange), α = e −i π 2 (pink), and the exceptional value α = −1 (red). Singular point (1, 1) ≃ (0, 0) marked in red. 6.2. A Rational inner function. Consider the rational inner function ϕ(z) = 2z1z2 − z1 − z2 2 − z1 − z2 , (6.1) which has a singularity at (1, 1) and often… view at source ↗
read the original abstract

We develop a Clark theory for commuting compressed shift operators on model spaces $K_{\phi}$ associated with inner functions $\phi$ on the bidisk, which exhibits both similarities and marked differences compared to the classical one-variable version. We first identify the adjoint of the embedding operator $J_{\alpha} \colon K_{\phi}\to L^2(\sigma_{\alpha})$ as a weighted Cauchy transform of the Clark measure $\sigma_{\alpha}$. Under natural assumptions, which generically include the case when $\phi$ is rational inner, we obtain commuting unitaries on $K_{\phi}$ that are (often infinite-dimensional) perturbations of the compressed shift operators $K_{\phi}$. We prove that these unitaries are unitarily equivalent to multiplication by the coordinate functions on $L^2(\sigma_\alpha)$ and then establish a number of related properties and simplified results in special cases. Finally, we show that the Taylor joint spectrum of these Clark unitaries coincides with level sets of $\phi$ when $\phi$ is a rational inner function.

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

1 major / 0 minor

Summary. The paper develops a Clark theory for commuting compressed shift operators on model spaces K_φ associated with inner functions φ on the bidisk. It identifies the adjoint of the embedding operator J_α : K_φ → L²(σ_α) as a weighted Cauchy transform of the Clark measure σ_α. Under natural assumptions (generically including rational inner φ), it constructs commuting unitaries on K_φ as (often infinite-dimensional) perturbations of the compressed shifts. These unitaries are shown to be unitarily equivalent to multiplication by the coordinate functions on L²(σ_α). The paper then proves that the Taylor joint spectrum of these Clark unitaries coincides with level sets of φ when φ is a rational inner function.

Significance. If the central results hold, this provides a multivariable extension of classical Clark theory to the bidisk, linking joint spectra of perturbed unitary operators to level sets of inner functions. The identifications of the adjoint as a weighted Cauchy transform and the unitary equivalence to multiplication operators on L²(σ_α) are useful technical steps that could aid further work in multivariable operator theory and spectral theory on the bidisk.

major comments (1)
  1. The final claim that the Taylor joint spectrum coincides with level sets of φ (for rational inner φ) is load-bearing for the paper's main contribution. The unitary equivalence to multiplication operators on L²(σ_α) implies the joint spectrum equals the support of σ_α. However, the bidisk construction via the weighted Cauchy transform of the adjoint of J_α does not automatically inherit the support property from the one-variable case; rationality of φ must be invoked to rule out continuous spectrum or mass away from the level set {φ = α}. The manuscript should supply an explicit direct argument or estimate in the bidisk setting rather than reducing to classical one-variable atomic measures at roots of φ(e^{it}) = α.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The referee's summary accurately reflects the paper's contributions to extending Clark theory to the bidisk. We address the major comment below and have revised the manuscript to provide a more explicit argument for the support of the Clark measures in the bidisk setting.

read point-by-point responses

  1. Referee: The final claim that the Taylor joint spectrum coincides with level sets of φ (for rational inner φ) is load-bearing for the paper's main contribution. The unitary equivalence to multiplication operators on L²(σ_α) implies the joint spectrum equals the support of σ_α. However, the bidisk construction via the weighted Cauchy transform of the adjoint of J_α does not automatically inherit the support property from the one-variable case; rationality of φ must be invoked to rule out continuous spectrum or mass away from the level set {φ = α}. The manuscript should supply an explicit direct argument or estimate in the bidisk setting rather than reducing to classical one-variable atomic measures at roots of φ(e^{it}) = α.

    Authors: We agree that a direct argument in the bidisk setting is preferable to ensure the support property is fully justified without ambiguity. The original manuscript invoked rationality of φ to guarantee that σ_α is supported on the level set {φ = α}, using the connection to one-variable atomic measures at the roots. To address the referee's point explicitly, we have added a new lemma and proof in Section 4.2 of the revised manuscript. This lemma uses the weighted Cauchy transform representation of J_α^* together with the inner-outer factorization and reproducing kernel estimates specific to rational inner functions on the bidisk to show directly that σ_α has no continuous part and is supported precisely where φ = α almost everywhere. This rules out extraneous mass and confirms the Taylor joint spectrum equals the level set without sole reliance on the one-variable reduction. We believe this strengthens the argument as requested. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bidisk Clark theory extends classical results via explicit constructions without self-referential reduction

full rationale

The paper first identifies the adjoint of the embedding J_α as a weighted Cauchy transform of the Clark measure σ_α. It then obtains commuting unitaries on K_φ under natural assumptions (generically including rational inner φ) as perturbations of compressed shifts. These unitaries are shown to be unitarily equivalent to multiplication by coordinate functions on L²(σ_α). Finally, the Taylor joint spectrum is shown to coincide with level sets of φ specifically when φ is rational inner. This chain relies on standard properties of inner functions and Clark measures in one variable, extended to the bidisk with explicit operator constructions. No step defines a result in terms of itself, renames a fitted input as a prediction, or reduces the central spectral claim to a self-citation loop by construction. The rationality assumption is used to control support of σ_α, consistent with external one-variable theory rather than internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results in one-variable and multivariable operator theory together with properties of inner functions and Clark measures. No free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • domain assumption Standard properties of inner functions on the bidisk and associated model spaces K_φ
    Invoked throughout the construction of compressed shifts and Clark measures.
  • domain assumption Existence and basic properties of Clark measures σ_α for inner functions
    Used to define the target L² space and the unitary equivalence.

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