REVIEW 2 major objections 3 minor 12 references
Projecting onto Helson matrices in Schatten classes
T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The orthogonal projection from Hilbert–Schmidt operators onto Helson matrices, and every non-negative multiplicative weighted averaging projection onto them, is unbounded on the Schatten class $S_q$ for every $q \neq 2$.
desk verdict The paper gives a clean negative answer to a natural analogue of Peller's theorem, but the proof of the main theorem has a repairable gap in the tensor-product step. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument runs on two rails. First, $\ell^2(\mathbb{N})$ is identified with the infinite tensor product over primes $\bigotimes_p \ell^2(\langle p \rangle)$, so a multiplicative matrix factors as $A = \bigotimes_p A_p$ and a multiplicative Helson matrix corresponds to a tensor product of Hankel matrices. Second, for each prime $p$ the weight $\Phi$ induces an additive weight $\varphi_p(i,j) = \Phi(p^i, p^j)$ satisfying $\sum_{i+j=k} \varphi_p(i,j) = 1$, and the projection factors as $P_\Phi(\bigotimes_p A_p) = \bigotimes_p P_{\varphi_p}(A_p)$. Lemma 4 is the quantitative heart: for every $q \neq 2$ there is a universal $\delta_q > 0$ such that every weighted Hankel averaging projection $P_\varphi$ has norm at least $1 + \delta_q$ on $S_q$, proved by testing on the four $3 \times 3$ matrices $A(t), B(t), C(t), D(t)$ and optimizing the resulting lower bounds. Feeding the lemma into the tensor-product factorization and using the norm identity $\|\bigotimes_p A_p\|_{S_q} = \prod_p \|A_p\|_{S_q}$ yields $\|P_\Phi\|_{S_q \to S_q} \geq (1+\delta_q)^N$ for every $N$, forcing unboundedness.
What would settle it
Take $q=4$ and a multiplicative weight $\Phi$ from the paper's family $\Phi_{\alpha,\beta}$, choose $N$ primes, set $A_p$ to the $3 \times 3$ matrix $C(t)$ from Lemma 4 for $p \leq p_N$ and $A_p = H_{e_0}$ otherwise, and compute the ratio $\|P_\Phi(\bigotimes_{p \leq p_N} A_p)\|_{S_4} / \|\bigotimes_{p \leq p_N} A_p\|_{S_4}$. If this ratio stays bounded as $N$ grows, Theorem 2 is false; the proof predicts it grows like $(1+\delta_4)^N$.
Extended reading notes
Core claim
The paper's central claim is Theorem 2: for any non-negative multiplicative function $\Phi : \mathbb{N} \times \mathbb{N} \to \mathbb{R}$ with $\sum_{mn=k} \Phi(m,n) = 1$ for every $k \geq 1$, the weighted projection $P_\Phi$ defined by $(P_\Phi A)_{m,n} = \varrho_{mn}$ with $\varrho_k = \sum_{mn=k} \Phi(m,n) a_{m,n}$ is unbounded on the Schatten class $S_q$ for every $1 \leq q \neq 2 < \infty$. In particular, the uniform averaging projection $P$ of Theorem 1—the orthogonal projection from $S_2$ onto Hilbert–Schmidt Helson matrices—does not extend to a bounded operator on any other Schatten class. The authors view this as the multiplicative failure of the classical boundedness theorem for Hankel projections, and they also show that there are no bounded projections onto the spaces of compact or bounded Helson matrices, and that the natural duality pairing between Helson matrices in $S_q$ and $S_r$ (with $1/q + 1/r = 1$) is not surjective for $q \neq 2$.
Load-bearing premise
The proof depends on the identity that the Schatten $q$-norm of an infinite tensor product of operators is the product of their individual Schatten $q$-norms, together with the convergence of the specific infinite tensor products constructed from the $3 \times 3$ test matrices; if that norm identity or convergence fails for these operators, the exponential lower bound collapses.
Editorial extensions
If this is right
- Theorem 1 settles the multiplicative analogue of the classical Hankel projection theorem: the Hilbert–Schmidt orthogonal projection onto Helson matrices has no bounded extension to $S_q$ for any $q \neq 2$.
- Theorem 2 rules out a whole family of repairs: no non-negative multiplicative weight satisfying the projection condition can make a weighted averaging projection onto Helson matrices bounded on $S_q$ for $q \neq 2$.
- There are no bounded projections from the compact operators onto the compact Helson matrices, nor from the bounded operators onto the bounded Helson matrices, even without assuming multiplicativity of the weight.
- For $1 < q \neq 2 < \infty$ and $1/q + 1/r = 1$, the natural embedding of the Helson matrices in $S_r$ into the dual of the Helson matrices in $S_q$ is not surjective; the duality structure of Helson matrices in Schatten classes differs from that of Hankel matrices.
- Because the lower bound $\|P_\Phi\| \geq (1+\delta)^N$ holds for every finite set of primes, finite-dimensional tensor-product examples exhibit the unboundedness quantitatively.
Reading between the lines
- The method suggests that any bounded projection onto Helson matrices on $S_q$ (if one exists for $q \neq 2$) would have to be non-multiplicative or take negative values; the paper leaves that possibility open.
- The proof's lower bound is constructive: for any finite set of primes it builds explicit operators, tensoring the $3 \times 3$ matrices from Lemma 4, where the projection gains a factor $(1+\delta)^N$, so the unboundedness is witnessed by finite-dimensional data and could be checked numerically.
- Because the obstruction comes from the infinite product structure over primes, analogous projection problems for matrices indexed by other multiplicative semigroups might be attacked with the same two-step argument of tensor factorization plus a uniform additive lower bound.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the orthogonal projection from the Hilbert-Schmidt class S_2 onto the subspace of Hilbert-Schmidt Helson matrices, and more generally the weighted averaging projections P_Phi defined in (9) with a non-negative multiplicative weight Phi satisfying the normalization condition (8). The main theorem states that for every 1 <= q != 2 < infinity, every such P_Phi is unbounded on the Schatten class S_q; in particular, the uniform averaging projection (5) is unbounded on S_q for all q != 2. The proof introduces an explicit finite-dimensional lower bound (Lemma 4) showing that any Hankel-type weighted averaging projection has S_q norm at least 1+delta, and then transfers this bound to infinitely many tensor factors via an infinite tensor product representation of ell^2(N). Additional results assert the nonexistence of bounded projections onto Helson matrices in the spaces of compact and bounded operators, and a duality failure for Helson matrices in Schatten classes.
Significance. If the results are correct, the paper settles a natural multiplicative analogue of Peller's theorem on Hankel projections: unlike the Hankel case, the natural projection onto Helson matrices is unbounded for every q != 2, and no non-negative multiplicative reweighting of the averaging projection can repair this. The finite-dimensional computations in Lemma 4 are explicit and self-contained, and the uniform lower bound over all admissible weights is a notable strength. The tensor-product reduction is elegant and connects the problem to known results on multiplicative matrices. The paper is concise and the main ideas are transparent, though one step in the proof of Theorem 2 requires repair, as detailed below.
major comments (2)
- [Section 3, proof of Theorem 2] The proof chooses, for each prime p <= p_N, an operator A_p with ||A_p||_{S_q}=1 and ||P_{phi_p} A_p||_{S_q} >= 1+delta, and then states that A = tensor_p A_p is a multiplicative matrix. This is inconsistent with the definition in Section 2.3: if ||A_p||_{S_q}=1 and <A_p e_1, e_1>=1 (the condition for A to be multiplicative), then e_1 is a singular vector with singular value 1 and all other singular values vanish, so A_p is a rank-one projection. For such A_p one has P_{phi_p} A_p = A_p and hence ||P_{phi_p} A_p||_{S_q}=1, contradicting the required bound >= 1+delta. Consequently, the inference that P_Phi A = tensor_p P_{phi_p} A_p, which in the written proof uses the converse for multiplicative matrices, is not justified for the chosen A_p. The factorization is in fact valid for arbitrary tensor products by an entrywise computation, so the proof can be repaired by removing the multiplicativity assumption and proving the factorization directly, or by choosing A_p with <A_p e_1, e_1>=1 and the ratio property without unit normalization. As written, the key estimate rests on an unproved step.
- [Section 3, use of identity (15)] The paper applies the norm identity (15) to the infinite tensor products A = tensor_p A_p and tensor_p P_{phi_p} A_p, but it does not verify the hypotheses of [6, Thm. 2.4] quoted in Section 2.3: convergence of product_p ||A_p||, convergence of sum_p ||A_p e_1||-1 and sum_p <A_p e_1,e_1>-1, and membership of the tensor products in S_q. For the specific construction (finitely many finite-rank factors and A_p = H_{e_0} for p > p_N) these conditions hold, so the gap is easily fixed, but the verification should be included because (15) is the mechanism by which the finite-factor lower bound (1+delta)^N is converted into a lower bound for ||P_Phi||.
minor comments (3)
- [Section 4.1, Theorem 5] The statement of Theorem 5 claims that there are no bounded projections from the compact (resp. bounded) operators onto the compact (resp. bounded) Helson matrices, but the proof only addresses projections of the weighted form (9). The sentence 'Clearly, a bounded projection P_Phi must satisfy (8)' is justified only for projections of that form, not for arbitrary idempotent maps onto the Helson subspace. Please either restrict the theorem to weighted averaging projections or provide an argument showing that an arbitrary projection can be reduced to this case.
- [Lemma 4, optimization over x] In the final estimates of the two cases in Lemma 4, the interval for x in the infimum is written as 0 <= x <= 1, but the bound involving 1-2x is only meaningful for x <= 1/2 since phi_1 >= 0. It would clarify the argument to state explicitly that x ranges over [0, 1/2] and that the intersection point x_q lies in this interval.
- [Section 2.3, definition of multiplicative matrix] The condition <A_p e_1, e_1> = 1 for every p is described as making the tensor product a multiplicative matrix, but the reason is not stated: this condition ensures that the associated function f satisfies f(1,1)=1, which is part of the definition of multiplicativity. A short explanatory sentence would help the reader.
Circularity Check
No significant circularity: the lower bounds are proved directly and the only imported norm identity is from external, non-overlapping work.
full rationale
The derivation is self-contained rather than circular. Lemma 4 explicitly constructs 3x3 test matrices and computes their singular values, yielding the uniform lower bound ||P_phi|| >= 1+delta for every weighted Hankel projection satisfying (6); this is an unconditional computation, not an assumed prediction. Theorem 2 then uses multiplicativity of Phi and the tensor-product factorization P_Phi(⊗_p A_p) = ⊗_p P_{phi_p}(A_p), together with the norm identity (15). Identity (15) is quoted from Hilberdink [6], an external source whose author does not overlap with the present authors, so it is independent support rather than self-citation. The self-references [2], [7], [8], and [11] are cited for context and inspiration and are not load-bearing for the unboundedness proof. The reviewer-identified normalization concern about the hypotheses of the tensor-product factorization is a possible gap in verifying the setup, not a case in which a claimed output is identical to an input by construction; it is a correctness issue orthogonal to circularity. No parameter is fitted to the target result, and no equation is defined in terms of the conclusion.
Assumptions & free parameters
assumptions (4)
- standard math Identity (15): ||⊗_p A_p||_{S_q} = Π_p ||A_p||_{S_q} for tensor products of Schatten class operators, cited from [6, Thm. 2.4].
- standard math The infinite tensor product A = ⊗_p A_p built in the proof of Theorem 2 is a well-defined bounded Schatten class operator satisfying the convergence conditions of Section 2.3.
- standard math Peller's theory of Hankel operators in Schatten classes, including boundedness of the averaging projection for 1 < q < ∞ and the absence of bounded projections onto compact/bounded Hankel operators ([9], [10, Ch. 6.5]).
- standard math Standard Schatten duality (S_q)^* ≃ S_r for 1/q + 1/r = 1, annihilator duality, the open mapping theorem, and the link between topological complements and bounded projections ([12, Thm. 5.16]).
Cite this review
Pith. "Pith review of Projecting onto Helson matrices in Schatten classes." pith.science (2026). https://pith.science/paper/6ETNTK7U
@misc{pith2026190804521,
author = {Pith},
title = {Pith review of: Projecting onto Helson matrices in Schatten classes},
year = {2026},
howpublished = {\url{https://pith.science/paper/6ETNTK7U}},
note = {Machine review of arXiv:1908.04521}
}
abstract
A Helson matrix is an infinite matrix $A = (a_{m,n})_{m,n\geq1}$ such that the entry $a_{m,n}$ depends only on the product $mn$. We demonstrate that the orthogonal projection from the Hilbert--Schmidt class $\mathcal{S}_2$ onto the subspace of Hilbert--Schmidt Helson matrices does not extend to a bounded operator on the Schatten class $\mathcal{S}_q$ for $1 \leq q \neq 2 < \infty$. In fact, we prove a more general result showing that a large class of natural projections onto Helson matrices are unbounded in the $\mathcal{S}_q$-norm for $1 \leq q \neq 2 < \infty$. Two additional results are also presented.
Reference graph
Works this paper leans on
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[6]
Titus Hilberdink, Matrices with multiplicative entries are tensor products , Linear Algebra Appl. 532 (2017), 179–197
work page 2017
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[1]
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Henry Helson, Hankel forms and sums of random variables , Studia Math. 176 (2006), no. 1, 85–92
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Nazar Miheisi and Alexander Pushnitski, A Helson matrix with explicit eigenvalue asymp- totics, J. Funct. Anal. 275 (2018), no. 4, 967–987
work page 2018
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Joaquim Ortega-Cerdà and Kristian Seip, A lower bound in Nehari’s theorem on the polydisc , J. Anal. Math. 118 (2012), no. 1, 339–342
work page 2012
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[9]
V. V. Peller, Hankel operators of class Sp and their applications (rational approximation, Gaussian processes, the problem of majorization of operato rs), Mat. Sb. (N.S.) 113(155) (1980), no. 4(12), 538–581
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[10]
Peller, Hankel operators and their applications , Springer Monographs in Mathe- matics, Springer-Verlag, New York, 2003
Vladimir V. Peller, Hankel operators and their applications , Springer Monographs in Mathe- matics, Springer-Verlag, New York, 2003
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[11]
Karl-Mikael Perfekt and Alexander Pushnitski, On Helson matrices: moment problems, non- negativity, boundedness, and finite rank , Proc. Lond. Math. Soc. (3) 116 (2018), no. 1, 101– 134
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[12]
W alter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. Department of Mathematical Sciences, Nor wegian University of Science and Tech- nology (NTNU), NO-7491 Trondheim, Nor w ay E-mail address...
1973
Reviewed August 14, 2026 · model on record in the stance chip above.
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