Composition operators and Rational Inner Functions on the bidisc

Athanasios Beslikas

REVIEW 3 major objections 4 minor 22 references

Composition operators and Rational Inner Functions on the bidisc

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read One boundary singularity forces an unbounded composition operator

desk verdict Good question and a solid boundedness result for stable polynomials, but the main unboundedness theorem is not proven—the volume estimates only cover δ>1/2 and the key lemma misidentifies distances. read the letter →

arxiv 2412.16593 v3 pith:23OPK4XQ submitted 2024-12-21 math.CV

Athanasios Beslikas This is my paper · ORCID

classification math.CV MSC 32A3732A4030J10

keywords compositionoperatorsrationalinnerfunctionsbidiscweightedBergmanspacesboundarysingularitiesalgebraicdistanceestimatesSchur-Aglerclasspolydisc

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 asks whether a rational inner self-map of the bidisc can induce an unbounded composition operator on the Bergman space when the map has a boundary singularity. Its main theorem answers yes: for a symbol whose two coordinate functions are the same rational inner function $\phi=\widetilde p/p$ with a denominator having exactly one zero on the distinguished boundary and satisfying a mild lower-bound condition, $C_\Phi$ is unbounded on $A^2(\mathbb{D}^2)$. A contrasting theorem shows that when the denominator is stable on $\mathbb{D}^2$, the same kind of symbol induces a bounded operator between appropriate weighted Bergman spaces. Together the results say that boundary singularities of rational inner functions are not automatically harmless, and the transition between boundedness and unboundedness is controlled by the zero set of the denominator.

What carries the argument

The load-bearing object is a rational inner function $\phi=\widetilde p/p$ on the bidisc, where $\widetilde p$ is the reflection of $p$; such functions have modulus one on the distinguished boundary but may have singularities where $p$ vanishes there. Near a single boundary zero $\tau$, Lemma 4.1 converts the assumed lower bound on $|p(z)|$ into an upper bound on $|\phi(z)-1|$, showing that the sublevel set $\{z:|\phi(z)-1|<\delta\}$ contains a bi-annulus of Euclidean width comparable to $\delta^{1/q}$. Its Bergman volume is at least $C\delta^{4/q}$. The boundedness criterion on weighted Bergman spaces says $C_\Phi$ is bounded only if pulled-back volumes of boundary boxes grow no faster than the boxes themselves, which on $A^2(\mathbb{D}^2)$ is $\delta^4$; the $4/q<4$ growth rate is the contradiction that yields unboundedness.

What would settle it

Test the comparison in Lemma 4.1 for $p=2-z_1-z_2$: near $(1,1)$, $Z_p$ is the line $z_1+z_2=2$, so the distance from $z$ to $Z_p$ is comparable to $|p(z)|$, whereas the distance from $z$ to the point $(1,1)$ is not; the claimed bound on $|\phi(z)-1|$ in terms of the point distance therefore does not follow from the hypothesis. A direct computation of the volume $V(\{z:|\phi(z)+1|<\delta\})$ in Example 5.2 would show which exponent of $\delta$ is actually correct.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 2.1: if $\Phi=(\phi,\phi)$ is a holomorphic self-map of the bidisc and $\phi=\widetilde p/p$ is a rational inner function whose denominator polynomial $p$ is stable on $\mathbb{D}^2$, has exactly one zero $\tau$ on $\mathbb{T}^2$, and obeys $|p(z)|\ge C\,\mathrm{dist}^{q}(z,Z_p)$ near $\tau$ for some $q>1$, then $C_\Phi$ is unbounded on the unweighted Bergman space $A^2(\mathbb{D}^2)$. The proof shows that the inverse image of a boundary box around the singular value contains a thin bi-annulus whose volume is bounded below by $C\delta^{4/q}$, while boundedness on $A^2(\mathbb{D}^2)$ would force that volume to be at most $C\delta^4$; since $q>1$, the lower bound is the larger one as $\delta\to0$. The paper also proves Theorem 2.2, that for stable $p$ the operator $C_\Phi$ is bounded from $A^2_{\beta/2-2}(\mathbb{D}^2)$ to $A^2_\beta(\mathbb{D}^2)$ for all $\beta>4$, and Theorem 2.3, a sufficient volume condition in the Schur-Agler class on the polydisc.

Load-bearing premise

The unboundedness conclusion rests on the claim that, near the single boundary zero, the denominator shrinks like a power of the distance from $z$ to that point; if the zero set is a curve rather than a point, that comparison can fail and the exponent used in the volume estimate changes.

Editorial extensions

If this is right

Every rational inner function satisfying the single-zero lower-bound hypothesis with $q>1$ induces a composition operator that is unbounded on $A^2(\mathbb{D}^2)$.
For a stable denominator $p$ on $\mathbb{D}^2$, the same diagonal symbol is bounded from $A^2_{\beta/2-2}(\mathbb{D}^2)$ into $A^2_\beta(\mathbb{D}^2)$ for every $\beta>4$, with a polydisc analogue holding for $\beta>2n$.
For Schur-Agler rational inner functions on the polydisc, a volume decay condition of the form $V_\beta(\{z:\sum_j(1-|z_j|^2)\mathrm{SOS}_{ij}\le\delta_i M\})\le C\delta_i^{n(\beta+2)}$ is sufficient for boundedness of $C_\Phi$ on $A^2_\beta(\mathbb{D}^n)$.
The examples show that the same denominator $2-z_1-z_2$ can produce a bounded operator between weighted spaces for $\beta\ge8$ and an unbounded operator on the unweighted space, so the boundedness range is sensitive to the weights and to the numerator.

Reading between the lines

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

A direct comparison of the distance to the zero set with the distance to the singular point would decide whether the exponent in Theorem 2.1 is optimal; for $p=2-z_1-z_2$, whose zero set near $(1,1)$ is a line, $\mathrm{dist}(z,Z_p)$ is comparable to $|p(z)|$, so the natural sharp exponent there is $1$ rather than $2$.
The proof treats only diagonal symbols $\Phi=(\phi,\phi)$; a natural next step is to test whether one singular coordinate is already enough when paired with a smooth second coordinate, since the volume mechanism would then see only one shrinking direction.
If the volume bound in the singular example is sharp at $\delta^2$, the corresponding operator should remain unbounded on every weighted space $A^2_\beta$ with $\beta<4$; computing this threshold would turn the paper's boundedness/unboundedness gap into a sharp phase transition.

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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

Dear [Colleague],

You should know two things about this paper. First, it's the first attempt I've seen to treat composition operators on the bidisc whose symbols are rational inner functions with boundary singularities. That's a natural gap in the Bayart–Kosinski program, and Theorem 2.2 (boundedness for stable polynomials between different Bergman spaces) looks like a real, citable result. Second, the advertised unboundedness result, Theorem 2.1, is not proved. The paper deserves a serious referee, but the core idea is worth keeping on the radar.

What's new: the question itself, and the observation that Lojasiewicz-type control on the denominator can in principle be fed into Carleson-measure estimates. Theorem 2.2 seems to work; the proof via Bickel's inequality gives the right box-counting. Examples 5.1 and 5.3 are useful sanity checks.

The soft spots are at the center. The proof of Theorem 2.1 estimates the volume of Φ^{-1}(S(e,(δ,δ))) by a bi-annulus that exists only for δ∈(1/2,1). The Carleson criterion requires control for all δ arbitrarily small; a finite interval of scales can always be absorbed into the constant. So the argument never touches the scales that matter. Lemma 4.1, which is supposed to provide the local bound |φ−1| ≤ Cε/dist(z,τ)^q, replaces dist(z,Zp) by dist(z,τ) even though Zp is a hypersurface through τ. Those two distances are not comparable for points near the hypersurface; inside D^2 the ratio can be arbitrarily small. This is the central estimate, and it is not justified. Example 5.2 inherits both problems and adds a sign error: it estimates |φ+1| where the relevant quantity is |φ−1|. So the paper's main claim, as written, is unsupported.

Who is this for? Specialists in composition operators on polydiscs or rational inner functions. If you work in that area, you'd want to know about the question and the partial result, but you cannot rely on the unboundedness theorem. A serious referee could probably help the author fix the technical steps, but the revision would be substantial. I'd send it to review rather than desk-reject, because the question is genuine and Theorem 2.2 is solid. Just don't expect the central theorem to survive in its current form.

Best, [Your name]

Referee Report

3 major / 4 minor

Summary. The paper studies composition operators C_Φ on weighted Bergman spaces A^2_β(D^2) for symbols Φ=(φ,φ) whose coordinate function φ is a rational inner function (RIF) on the bidisc. The main result, Theorem 2.1, claims that if φ=p̃/p with p stable on D^2, p has a single zero τ on T^2, and |p(z)| ≥ C dist^q(z,Z_p) near τ for some q>1, then C_Φ is unbounded on the unweighted space A^2(D^2). Theorem 2.2 claims boundedness between two different weighted Bergman spaces, A^2_{β/2-2} → A^2_β, for β>4 when p is stable. Theorem 2.3 gives a sufficient volume condition for boundedness on A^2_β(D^n) for Schur-Agler RIFs. The paper also contains three examples: one for the boundedness theorem, one purportedly showing unboundedness, and one for the stable-polynomial boundedness result.

Significance. If Theorem 2.1 were correct, it would be a significant first step: it would give the first unboundedness result for composition operators on A^2(D^2) induced by rational inner symbols with a boundary singularity. The proof strategy—using Carleson-box volume estimates and local estimates of |φ(z)-1| near the singularity—is natural and connects with the existing literature of Bayart and Kosiński. The paper also contains a potentially valid positive result in Theorem 2.2, and the use of established tools (Carleson measure criterion, Lojasiewicz inequality, Rudin's representation, Bickel's stability criterion) is appropriate. However, the main theorem is not established: the proof has a load-bearing gap and the key lemma contains an unjustified geometric comparison. The unconditional significance of the paper therefore rests on a repair that the current argument does not provide.

major comments (3)

[§4, proof of Theorem 2.1, Eqs. (4.6)–(4.11)] The volume lower bound V(Φ^{-1}(S(e,δ_1))) > C'(ε)δ_1^{4/q} is proved only for δ_1 ∈ (1/2,1). Indeed, the chosen bi-annulus (ε/(2δ_1))^{1/q} < |z_i-1| < ε^{1/q} is nonempty exactly when δ_1 > 1/2, and the text explicitly restricts to 1 > δ_1 > 1/2. Unboundedness of C_Φ on A^2(D^2) requires a violation of the Carleson estimate for boxes of arbitrarily small side length δ→0; a lower bound on a bounded interval away from zero is compatible with boundedness and gives no contradiction. Moreover, as δ_1→0, the lower radius (ε/(2δ_1))^{1/q} grows without bound, so the bi-annulus eventually leaves the neighborhood U provided by Lemma 4.1; the construction cannot be rescaled to small δ. Thus Theorem 2.1 is not proven by the argument presented.
[§4, Lemma 4.1, Eqs. (4.4)–(4.5)] The proof replaces dist(z, Z_p ∩ T^2) by the Euclidean distance to the point τ, i.e. (|z_1-1|^2+|z_2-1|^2)^{1/2}, and writes dist^q(z, Z_p∩T^2) = (|z_1-1|^2+|z_2-1|^2)^{q/2}. This is unjustified: Z_p is a complex hypersurface through τ, and the distance from z to that hypersurface is generally strictly smaller than the distance from z to the single point τ. Since inequality (4.4) is used as an upper bound on |φ(z)-1|, replacing the larger quantity dist(z,Z_p) in the denominator by the smaller quantity |z-τ| makes the fraction larger, not smaller; the needed comparison is the opposite. This error changes the exponent q in the subsequent volume estimate and is load-bearing for the claimed δ^{4/q} lower bound.
[§5, Example 5.2, Eqs. (5.7)–(5.13)] This example repeats the same defect as Theorem 2.1. The annulus constructed after inequality (5.9) requires (1/√2)√(ε/(2δ)) < |z_i-1| < √ε, which is nonempty only when δ > ε/2, and the proof concludes with a 'suitable subinterval of (1/2,1)'. Therefore the displayed lower bound V > C(ε)δ^2 is obtained only for δ in (1/2,1), not for δ→0. Since unboundedness on A^2(D^2) requires boxes of arbitrarily small radius, this example does not demonstrate non-boundedness.

minor comments (4)

[§4, proof of Theorem 2.1] The text says 'by Lemma 4.3' where the reference should be to Lemma 4.1.
[§5, Example 5.2] The phrase 'Inequality (5.10) lets us conclude' appears to refer to inequality (5.9), not (5.10).
[Throughout, esp. §1.3, §3.3, §4] There are stray '/suppress' commands in the text (e.g., '/suppress Lukasz Kosiński', '/suppress Lojasiewicz') that should be removed; they appear to be typographical artifacts.
[Lemma 4.1] The lemma is stated for a general RIF with one singularity, but the proof tacitly uses that p has a zero at τ and that p̃(τ)=p(τ)=0; this should be stated explicitly for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain uses external theorems and estimates without reducing its conclusions to its own inputs.

full rationale

The paper's derivation chain is not circular. The central unboundedness result (Theorem 2.1) is proved by combining the Carleson measure criterion for weighted Bergman spaces on the bidisc (Lemma 3.2, attributed to Bayart and Kosiński), the Lojasiewicz inequality applied to the denominator polynomial p, and elementary containment/volume estimates. Lemma 4.1 is a local upper bound on |φ−1| in terms of distance to a singular point, obtained from continuity and Lojasiewicz; it is an auxiliary estimate, not an assumption equivalent to the conclusion. Theorem 2.2 uses Bickel's stability criterion (Theorem 3.6) to control sublevel sets, again an external parameter-free theorem. Theorem 2.3 is conditional: it assumes a volume bound for explicit SOS sublevel sets and derives boundedness by containment and the Carleson criterion; this is a straightforward implication rather than a disguised restatement. The proof does contain a quantitative gap in Theorem 2.1, since the displayed lower bound is established only for δ1 ∈ (1/2,1) and therefore does not violate the Carleson condition for arbitrarily small boxes, but that is a correctness/estimate concern, not circularity. No load-bearing argument reduces to a self-citation or to a fitted parameter renamed as a prediction.

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

The central claims rest on standard tools from operator theory, real analysis, and rational inner function theory. No free parameters are fitted to data. The main concern is the ad hoc geometric assumption about distance to the zero set in Lemma 4.1, plus the incomplete Carleson box argument in Theorem 2.1.

assumptions (6)

standard math Rudin's representation theorem: every rational inner function on the bidisc has the form lambda z1^M z2^N ptilde/p.
Invoked in Section 3.2 to justify the representation phi=ptilde/p used throughout the paper.
standard math Lojasiewicz inequality: for a real analytic function f on a compact set, |f(x)| >= C dist(x,Zf)^q.
Used in Lemma 4.1 and Theorem 2.1 to convert lower bounds on |p| into distance lower bounds.
domain assumption Carleson measure criterion for composition operators on weighted Bergman spaces, Lemma 3.2: boundedness is equivalent to V_beta(Phi^{-1}(S)) <= C V_alpha(S) for all Carleson boxes.
This is the foundational criterion behind all volume estimates; it is taken from Bayart and Kosinski.
domain assumption Bickel stability criterion: p stable on the closed bidisc iff |ptilde|^2 - |p|^2 >= C(1-|z1|^2)(1-|z2|^2).
Used in Theorem 2.2 and Example 5.3 to bound sublevel sets of |phi-zeta|.
domain assumption Non-tangential limits of rational inner functions exist at singular points and lie on the unit circle.
Used in Theorem 2.1 and Example 5.2 to normalize the behavior at the singularity, following Knese's results.
ad hoc to paper For a stable polynomial with one boundary zero tau, distance to the zero set Zp is effectively comparable to the Euclidean distance to tau near tau.
This geometric identification is asserted implicitly in Lemma 4.1 and is load-bearing for the volume lower bound, but it is not proven or referenced. It is the weakest step in the main proof.

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Cite this review

Pith. "Pith review of Composition operators and Rational Inner Functions on the bidisc." pith.science (2026). https://pith.science/paper/23OPK4XQ

@misc{pith2026241216593,
  author       = {Pith},
  title        = {Pith review of: Composition operators and Rational Inner Functions on the bidisc},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/23OPK4XQ}},
  note         = {Machine review of arXiv:2412.16593}
}

abstract

In the present article, composition operators induced by Rational Inner Functions on the bidisc $\mathbb{D}^2$ are studied, acting on the weighted Bergman space $A^2_{\beta}(\mathbb{D}^2).$ We prove that under mild conditions that Rational Inner Functions with one singularity on $\mathbb{T}^2$ induce unbounded composition operator on $A^2(\mathbb{D}^2).$ We also prove that under the condition of stability of the polynomial inducing the Rational Inner Function, the composition operator is bounded between two different Bergman spaces.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 21 canonical work pages

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Pith tools

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

[1] [1]

Agler, J.E

J. Agler, J.E. M cCarthy, and N.J. Young,A Caratheodory theorem for the bidisk via Hilbert space methods, Math. Ann. 352 (2012), 581-624

work page 2012

[2] [2]

F.Bayart, Composition operators on the polydisk induced by aﬃne maps Journal of Func- tional Analysis Volume 260, Issue 7, 1 April 2011, Pages 1969-2003

work page 2011

[3] [3]

F.Bayart, Composition operators on the Hardy space of the tridisc , to appear in Indiana University Mathematics Journal

work page

[4] [4]

L.Bergqvist, Rational Inner Functions and their Dirichlet Type Norms ,Computational Methods and Function Theory, Volume 23, pages 563–587, (2023)

work page 2023

[5] [5]

K.Bickel, Fundamental Agler Decompositions , Integral Equations and Operator Theory, Published: 11 September 2012 Volume 74, pages 233–257, (2012)

work page 2012

[6] [6]

Knese, Inner functions on the bidisk and associated Hilbert spaces , Journal of Functional Analysis Volume 265, Issue 11 , 1 December 2013, Pages 2753-2790

K.Bickel, G. Knese, Inner functions on the bidisk and associated Hilbert spaces , Journal of Functional Analysis Volume 265, Issue 11 , 1 December 2013, Pages 2753-2790

work page 2013

[7] [7]

K.Bickel, J.E.Pascoe, A.Sola, Singularities of Rational Functions in higher dimensions, American Journal of Mathematics Johns Hopkins University Press V olume 144, Number 4, August 2022

work page 2022

[8] [8]

Pascoe, A.Sola, Derivatives of Rational Inner Functions and integrability at the boundary, Proceedings of the London Mathematical Society, Vol

K.Bickel, J.E. Pascoe, A.Sola, Derivatives of Rational Inner Functions and integrability at the boundary, Proceedings of the London Mathematical Society, Vol. 116, Issue 2, pp.281- 329

work page