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arxiv: 2605.21232 · v1 · pith:5XDML4NZnew · submitted 2026-05-20 · 🧮 math.FA

On the nonnegative rank of positive operators

Roman Drnov\v{s}ek , Marko Kandi\'c This is my paper

Pith reviewed 2026-05-21 01:19 UTC · model grok-4.3

classification 🧮 math.FA
keywords nonnegative rankpositive operatorsordered vector spacesBanach latticesoperator ranknonnegative matricespositive cones
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The pith

Under mild assumptions on the target cone, nonnegative rank equals ordinary rank for positive operators of rank at most two.

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

The paper defines the nonnegative rank of a positive operator between ordered vector spaces as the smallest number of positive rank-one operators that sum to it, which recovers the usual nonnegative rank in the matrix case. It proves that this nonnegative rank coincides with the operator's ordinary rank whenever the rank is one or two, provided the target positive cone satisfies certain natural and mild assumptions. This supplies an infinite-dimensional counterpart to a known result for nonnegative matrices. The authors also exhibit a positive operator of rank three on the space C[0,1] whose nonnegative rank is infinite, showing that the equality can fail for higher ranks.

Core claim

We introduce the nonnegative rank of a positive operator T colon X to Y between ordered vector spaces. In the case of nonnegative matrices, our definition agrees with the standard definition of a nonnegative rank. Under some natural and mild assumptions on the cone Y_+, we prove that the nonnegative rank and the rank agree whenever the rank is at most two. This can be considered as the infinite-dimensional version of Theorem 4.1 in CR93. We also provide an example of a positive rank-three operator on the Banach lattice C[0,1] with an infinite nonnegative rank.

What carries the argument

The nonnegative rank of a positive operator, defined as the smallest k such that the operator equals the sum of k positive operators each having rank at most one.

If this is right

  • Nonnegative rank supplies no extra information beyond ordinary rank for rank-one and rank-two positive operators under the cone assumptions.
  • The equality supplies an infinite-dimensional extension of the corresponding matrix result.
  • Rank-three positive operators can have infinite nonnegative rank, as shown by the explicit example on C[0,1].

Where Pith is reading between the lines

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

  • The new rank may distinguish positive operators that share the same ordinary rank but differ in how many positive rank-one pieces are needed.
  • Characterizing the precise cone conditions under which nonnegative rank always equals rank remains open.
  • The construction on C[0,1] suggests that infinite nonnegative rank can occur even for compact positive operators of small ordinary rank.

Load-bearing premise

The positive cone in the codomain satisfies certain natural and mild assumptions that force nonnegative rank to equal ordinary rank for operators of rank at most two.

What would settle it

A positive operator of rank two, acting between spaces whose target cone meets the stated assumptions, whose nonnegative rank is strictly greater than two would disprove the main theorem.

read the original abstract

In this paper we introduce the concept of a nonnegative rank of a positive operator $T\colon X\to Y$ between ordered vector spaces. In the case of nonnegative matrices, our definition agrees with the standard definition of a nonnegative rank. Under some natural and mild assumptions on the cone $Y_+$, we prove that the nonnegative rank and the rank agree whenever the rank is at most two. This can be considered as the infinite-dimensional version of \cite[Theorem 4.1]{CR93}. We also provide an example of a positive rank-three operator on the Banach lattice $C[0,1]$ with an infinite nonnegative rank.exceed $\lceil 6\min\{m,n\}/7\rceil$.

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 / 2 minor

Summary. The paper introduces the nonnegative rank of a positive operator T: X → Y between ordered vector spaces (agreeing with the standard definition for nonnegative matrices). Under some natural and mild assumptions on the cone Y_+, it proves that nonnegative rank equals ordinary rank whenever the rank is at most 2; this is framed as the infinite-dimensional version of Theorem 4.1 from CR93. It also constructs an example of a positive rank-3 operator on the Banach lattice C[0,1] whose nonnegative rank is infinite.

Significance. If the assumptions on Y_+ can be made fully explicit and the infinite-dimensional adaptation verified, the work provides a useful extension of nonnegative-rank ideas into functional analysis and ordered vector spaces, with the rank-3 counterexample serving as a clear demarcation of where the equality fails. The new definition and the explicit infinite-dimensional example are the primary contributions.

major comments (2)
  1. [Abstract and §1] Abstract and §1 (or wherever the main theorem is stated): the phrase 'some natural and mild assumptions on the cone Y_+' is invoked to guarantee that a rank-2 positive operator factors through a nonnegative rank-2 operator, yet the assumptions are never listed explicitly (e.g., whether Y_+ is generating, pointed, has nonempty interior, or satisfies a lattice property). This list is load-bearing for the central claim and must be supplied before the result can be checked against the finite-dimensional argument of CR93.
  2. [Proof of the rank-≤2 theorem (likely §3)] Proof of the rank-≤2 theorem (likely §3): the adaptation from the finite-dimensional setting of CR93 to general ordered vector spaces is not yet verified in detail. In particular, it is unclear whether the argument relies on finite bases, compactness, or interior-point arguments that may fail in infinite dimensions; an explicit check that the factorization construction survives without these tools is required.
minor comments (2)
  1. [Counterexample section] The Banach-lattice counterexample on C[0,1] would be strengthened by an explicit formula or diagram showing why the nonnegative rank is forced to be infinite.
  2. [Notation throughout] Notation for cones and positive operators should be introduced once and used uniformly; minor inconsistencies in the use of Y_+ versus Y_+^o appear in the early sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below and will revise the paper to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1 (or wherever the main theorem is stated): the phrase 'some natural and mild assumptions on the cone Y_+' is invoked to guarantee that a rank-2 positive operator factors through a nonnegative rank-2 operator, yet the assumptions are never listed explicitly (e.g., whether Y_+ is generating, pointed, has nonempty interior, or satisfies a lattice property). This list is load-bearing for the central claim and must be supplied before the result can be checked against the finite-dimensional argument of CR93.

    Authors: We agree that the assumptions on Y_+ need to be stated explicitly for the main result to be verifiable. In the revised manuscript we will replace the phrase with a precise list at the statement of the theorem (both in the abstract and in Section 1). The assumptions we use are that Y is a Banach space and Y_+ is a closed, pointed, generating cone; these are the minimal conditions under which the factorization construction proceeds exactly as in the finite-dimensional case of CR93. revision: yes

  2. Referee: [Proof of the rank-≤2 theorem (likely §3)] Proof of the rank-≤2 theorem (likely §3): the adaptation from the finite-dimensional setting of CR93 to general ordered vector spaces is not yet verified in detail. In particular, it is unclear whether the argument relies on finite bases, compactness, or interior-point arguments that may fail in infinite dimensions; an explicit check that the factorization construction survives without these tools is required.

    Authors: We have reviewed the proof in Section 3 and confirm that it does not rely on finite bases, compactness, or interior-point arguments. The factorization is obtained directly from the definition of rank (dimension of the image) together with positivity and the cone assumptions; every step is algebraic or uses only the ordered-vector-space structure. In the revision we will insert a short paragraph immediately after the proof that explicitly maps each step of the CR93 argument to the infinite-dimensional setting and notes the absence of any finite-dimensional tools. revision: yes

Circularity Check

0 steps flagged

No significant circularity: new definition and adaptation of external finite-dimensional result

full rationale

The paper introduces the nonnegative rank directly for positive operators between ordered vector spaces and states that it agrees with the matrix case by definition. The central theorem asserts agreement of nonnegative rank with ordinary rank for rank at most two under explicitly invoked (though mild) assumptions on the cone Y_+, presented as an infinite-dimensional extension of the cited external result Theorem 4.1 in CR93. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the cited theorem is from unrelated prior work and the proof is claimed to adapt it under the stated cone assumptions. The rank-three counterexample on C[0,1] is independent. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the newly introduced definition of nonnegative rank together with standard background properties of ordered vector spaces and positive cones; no free parameters or invented physical entities are present.

axioms (1)
  • domain assumption Ordered vector spaces and positive cones satisfy the usual compatibility axioms (closed under addition and positive scalar multiplication).
    Invoked throughout the definition and the low-rank proof.
invented entities (1)
  • Nonnegative rank of a positive operator no independent evidence
    purpose: To measure the minimal number of positive rank-one operators needed to express a given positive operator.
    Newly defined concept; no independent evidence outside the paper is supplied in the abstract.

pith-pipeline@v0.9.0 · 5645 in / 1301 out tokens · 54478 ms · 2026-05-21T01:19:42.869221+00:00 · methodology

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Reference graph

Works this paper leans on

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