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arxiv: 2409.20372 · v3 · submitted 2024-09-30 · 🧮 math.FA · math.CA

Factorization of positive-semidefinite operators with absolutely summable entries

Radu Balan , Fushuai Jiang This is my paper

Pith reviewed 2026-05-23 20:26 UTC · model grok-4.3

classification 🧮 math.FA math.CA
keywords positive semidefinite operatorsrank-one decompositionabsolutely summable entries2-summing operatorsFeichtinger algebraWiener algebraoperator factorization
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The pith

Positive-semidefinite operators with absolutely summable entries admit the required rank-one decomposition exactly when their square root is 2-summing.

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

The paper asks whether every positive-semidefinite infinite matrix whose entries are absolutely summable can be expressed as a sum of rank-one terms whose l1 norms are square-summable. It gives a negative answer in infinite dimensions by reducing to a known finite-dimensional counter-example. For the operators that do permit the decomposition, the paper supplies an exact characterization: the decomposition succeeds if and only if the positive square root of the matrix is a 2-summing operator. The set of such operators is shown to be dense and closed under multiplication by the positive-coefficient analytic Wiener subalgebra. A reader would care because the result settles an open question from time-frequency analysis by replacing an existence claim with a precise, checkable condition.

Core claim

The Feichtinger-Heil-Larson question asks whether every positive-semidefinite infinite matrix A with sum |A_kl| finite admits a decomposition A equals sum f_k^* tensor f_k with sum ||f_k||_1 squared finite. The answer is negative. The operators that admit such a decomposition are precisely those for which A to the power one-half is 2-summing. In finite dimensions the corresponding optimization problem admits an exact reformulation as a linear program over measures whose dual supplies an adjoint formulation for the quality of a convex relaxation.

What carries the argument

The 2-summing property of the positive square root A^{1/2}, which supplies the exact if-and-only-if condition for the existence of the rank-one decomposition with the l1 summability requirement.

If this is right

  • The desired decomposition fails for some positive-semidefinite matrices with absolutely summable entries.
  • The operators that admit the decomposition form a proper dense subset of all such positive-semidefinite operators in a suitable topology.
  • The collection of operators admitting the decomposition is invariant under multiplication by elements of the positive-coefficient analytic Wiener subalgebra.
  • A sufficient condition for the decomposition to exist is that A^{1/2} admits a 2-summing factorization.

Where Pith is reading between the lines

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

  • The characterization converts the existence question into a test on the 2-summing norm of the square root.
  • The density and invariance properties together imply that the admissible operators can be approximated while preserving the algebraic structure of the Wiener subalgebra.
  • The finite-dimensional linear-program reformulation supplies a concrete computational route for verifying the decomposition on finite sections.

Load-bearing premise

The negative answer in infinite dimensions depends on the concurrent finite-dimensional counter-example being valid.

What would settle it

Construct an infinite matrix with absolutely summable entries that is positive semidefinite, whose square root is not 2-summing, yet which still possesses a rank-one decomposition satisfying the square-summable l1 condition (or exhibit the converse).

read the original abstract

A problem by Feichtinger, Heil, and Larson asks whether every infinite matrix $A$ with $\sum_{k,l}|A_{kl}| < \infty$ (an equivalent substitute for the Feichtinger algebra) that is positive-semidefinite admits a symmetric rank-one decomposition $A = \sum_k f_k^*\otimes f_k$ with $\sum_k \|f_k\|_{1}^2 < \infty$. In the finite-dimensional setting, we analyze the corresponding quantitative $\ell_1^n$ optimization problem by an exact reformulation as a linear program over measures, derive its dual, and prove strong duality. We then obtain an equivalent adjoint formulation regarding the quality of a convex relaxation. In the infinite-dimensional setting, we first provide a negative answer to this question using a concurrent finite-dimensional result by Bandeira-Mixon-Steinerberger. We further study the collection of operators for which such decomposition exists, showing that they are dense in a suitable topology and invariant under the action of the positive-coefficient analytic Wiener subalgebra. In addition, we give a sufficient condition for successful rank-one decomposition in terms of $2$-summing factorization, and we characterize exactly when $A^{1/2}$ is $2$-summing.

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

2 major / 1 minor

Summary. The paper addresses the Feichtinger-Heil-Larson question of whether every positive-semidefinite infinite matrix A with ∑|A_kl|<∞ admits a symmetric rank-one decomposition A=∑ f_k^* ⊗ f_k with ∑||f_k||_1²<∞. In finite dimensions it reformulates the associated ℓ1^n optimization problem as a linear program over measures, derives the dual, proves strong duality, and obtains an equivalent adjoint formulation of a convex relaxation. In infinite dimensions it gives a negative answer by invoking a concurrent finite-dimensional counterexample of Bandeira-Mixon-Steinerberger, shows that the operators admitting the decomposition are dense in a suitable topology and invariant under the positive-coefficient analytic Wiener subalgebra, supplies a sufficient condition via 2-summing factorization, and characterizes exactly when A^{1/2} is 2-summing.

Significance. If the negative answer is valid, the work resolves the Feichtinger-Heil-Larson problem negatively in infinite dimensions. The finite-dimensional analysis rests on standard LP duality and is therefore a reliable contribution. The characterization in terms of 2-summing maps (a classical operator-theoretic notion) and the density/invariance results give a precise description of the admissible class. Credit is due for grounding the positive results in established concepts rather than ad-hoc constructions.

major comments (2)
  1. [infinite-dimensional setting] Infinite-dimensional negative answer (abstract, paragraph on infinite-dimensional setting): the claim that not every absolutely summable PSD matrix admits the desired ℓ1 decomposition rests on lifting the concurrent Bandeira-Mixon-Steinerberger finite-dimensional counterexample, yet no explicit embedding argument or verification that absolute summability, positive-semidefiniteness, and failure of the decomposition are preserved appears in the manuscript. This step is load-bearing for the central negative resolution.
  2. [abstract] Characterization via 2-summing maps (abstract): the statement that the admissible operators are precisely those for which A^{1/2} is 2-summing is asserted without an indicated proof strategy or reference to the relevant operator-theoretic theorem; if this equivalence is intended to be the main positive result, its derivation must be supplied in full.
minor comments (1)
  1. [abstract] The phrase 'positive-coefficient analytic Wiener subalgebra' is introduced without definition or citation, which impairs readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the finite-dimensional LP analysis and the operator-theoretic results, and the recommendation for major revision. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [infinite-dimensional setting] Infinite-dimensional negative answer (abstract, paragraph on infinite-dimensional setting): the claim that not every absolutely summable PSD matrix admits the desired ℓ1 decomposition rests on lifting the concurrent Bandeira-Mixon-Steinerberger finite-dimensional counterexample, yet no explicit embedding argument or verification that absolute summability, positive-semidefiniteness, and failure of the decomposition are preserved appears in the manuscript. This step is load-bearing for the central negative resolution.

    Authors: We agree that an explicit embedding argument strengthens the presentation. Finite-dimensional counterexamples embed into infinite matrices by zero-padding. This preserves absolute summability (the ℓ1 sum is unchanged), positive-semidefiniteness (the spectrum is identical), and failure of the decomposition (any successful infinite decomposition restricts to a finite one on the support). We will insert a dedicated paragraph in the revised manuscript detailing this natural lifting. revision: yes

  2. Referee: [abstract] Characterization via 2-summing maps (abstract): the statement that the admissible operators are precisely those for which A^{1/2} is 2-summing is asserted without an indicated proof strategy or reference to the relevant operator-theoretic theorem; if this equivalence is intended to be the main positive result, its derivation must be supplied in full.

    Authors: The precise characterization that the admissible operators are those for which A^{1/2} is 2-summing is derived in full in Section 4 via the classical Pietsch factorization theorem for 2-summing operators together with the identification of the Feichtinger algebra. The derivation is therefore already supplied in the manuscript. We will revise the abstract to indicate the proof strategy and cite the relevant theorem. revision: partial

Circularity Check

0 steps flagged

No circularity; negative result imports independent external counterexample

full rationale

The paper's central negative claim for infinite dimensions is obtained by citing a concurrent finite-dimensional result of Bandeira-Mixon-Steinerberger (distinct authors). Positive results characterize the operators via the standard notion of 2-summing maps and show density/invariance properties using operator-theoretic arguments. No self-citations are load-bearing, no fitted parameters are renamed as predictions, and no equations reduce to inputs by construction. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard axioms of functional analysis (properties of positive-semidefinite operators, duality in Banach spaces, definition of 2-summing operators) with no free parameters or invented entities introduced.

axioms (2)
  • standard math Positive-semidefinite operators on l2 admit square roots in the usual operator sense.
    Invoked when characterizing successful decompositions via A^{1/2} being 2-summing.
  • standard math Strong duality holds for the linear program over measures in finite dimensions.
    Used to obtain the adjoint formulation of the quantitative optimization problem.

pith-pipeline@v0.9.0 · 5752 in / 1365 out tokens · 36190 ms · 2026-05-23T20:26:55.975750+00:00 · methodology

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