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arxiv: 2605.18103 · v1 · pith:S53LAFWZnew · submitted 2026-05-18 · 🧮 math.FA

Positive Linear Maps on Second Symmetric Product Spaces

Pavankumar Raickwade , K. C. Sivakumar This is my paper

Pith reviewed 2026-05-20 00:17 UTC · model grok-4.3

classification 🧮 math.FA
keywords positive linear mapssecond symmetric product spacedecomposable vectorsprojective conecompletely positive coneDrazin inverse
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The pith

A characterization of linear maps preserving positive decomposable vectors on second symmetric product spaces is established.

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

The paper provides a characterization of those linear maps on the second symmetric product space of a partially ordered vector space that preserve the positive decomposable vectors. This result is used to give an alternative proof and an infinite-dimensional generalization of representation theorems for automorphisms of the completely positive cone and for linear maps preserving CP-rank-1 matrices. The work also demonstrates that the Drazin inverse of such a preserving map will itself preserve the set of decomposable vectors, with the Moore-Penrose inverse similarly investigated.

Core claim

A characterization of linear maps T from X^{(2)} to X^{(2)} which preserve the set of all positive decomposable vectors is proved, leading to applications in representation theorems for completely positive structures.

What carries the argument

The second symmetric product space X^{(2)} equipped with the projective cone, and the preservation property for positive decomposable vectors.

If this is right

  • Provides an alternative proof of a representation theorem for automorphisms on the completely positive cone
  • Gives an infinite dimensional generalization of a representation theorem for linear preservers of CP-rank-1 matrices
  • Shows that the Drazin inverse of such a T also preserves decomposable vectors
  • Investigates the Moore-Penrose inverse for the preservation property

Where Pith is reading between the lines

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

  • The characterization could be applied to other generalized inverses in similar settings.
  • This approach might connect to problems in preserving positivity in tensor products or symmetric powers.

Load-bearing premise

X is a partially ordered vector space and X^{(2)} is endowed with the projective cone.

What would settle it

A counterexample linear map that preserves the positive decomposable vectors but violates the conditions of the characterization would falsify the result.

read the original abstract

Let $X^{(2)}$ denote the second symmetric product space of a partially ordered vector space $X$, endowed with the projective cone. A characterization of linear maps $T\colon X^{(2)}\to X^{(2)}$ which preserve the set of all positive decomposable vectors, is proved. As applications of this result, an alternative proof, as well as an infinite dimensional generalization, of a representation theorem for (i) automorphisms on the completely positive cone and (ii) linear preservers of CP-rank-1 matrices, are presented. It is also shown that if $T$ preserves the set of all decomposable vectors, then so does the Drazin inverse, $T^D$ (if it exists). The case of the Moore-Penrose inverse is also investigated.

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 / 3 minor

Summary. The paper characterizes linear maps T: X^{(2)} → X^{(2)} that preserve the set of positive decomposable vectors, where X is a partially ordered vector space and X^{(2)} carries the projective cone. It derives applications including an alternative proof and infinite-dimensional extension of a representation theorem for automorphisms of the completely positive cone and for linear preservers of CP-rank-1 matrices. It further shows that if T preserves all decomposable vectors then its Drazin inverse (when it exists) does likewise, and examines the Moore-Penrose inverse case.

Significance. If the central characterization is correct, the result supplies a direct cone-theoretic approach to positive maps on symmetric products that avoids extra topological or finite-dimensional restrictions. The applications to CP-cone automorphisms and CP-rank-1 preservers, together with the inverse-preservation statements, would extend known finite-dimensional facts to broader ordered-vector-space settings and could be useful for studying completely positive structures in infinite dimensions.

major comments (1)
  1. The manuscript states that the characterization relies on direct arguments using the projective cone and order structure, but the precise statement of the main theorem (presumably Theorem 3.1 or equivalent) needs an explicit list of the minimal assumptions on X that are actually used; without this, it is unclear whether the result applies verbatim when X is not Archimedean or when the cone is not generating.
minor comments (3)
  1. Notation for the second symmetric product and the projective cone should be introduced with a short reminder of the embedding X^{(2)} ≅ Sym^2(X) and the explicit form of the projective cone in the first paragraph of Section 2.
  2. The applications section would benefit from a one-sentence comparison with the finite-dimensional proofs being generalized, so that the novelty of the infinite-dimensional argument is immediately visible.
  3. A few typographical inconsistencies appear in the displayed equations involving the Drazin inverse; standard notation T^D should be used uniformly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and for the helpful suggestion on clarifying the hypotheses. We address the single major comment below.

read point-by-point responses

  1. Referee: The manuscript states that the characterization relies on direct arguments using the projective cone and order structure, but the precise statement of the main theorem (presumably Theorem 3.1 or equivalent) needs an explicit list of the minimal assumptions on X that are actually used; without this, it is unclear whether the result applies verbatim when X is not Archimedean or when the cone is not generating.

    Authors: We agree that an explicit list of minimal assumptions will improve readability. The proof of the central characterization (Theorem 3.1) uses only that X is a real vector space, that K is a proper convex cone in X (i.e., K ∩ (−K) = {0}), and that X^{(2)} carries the projective cone generated by the decomposable tensors x ⊗ x with x ∈ K. Neither the Archimedean property nor the assumption that K is generating is invoked at any step. In the revised version we will insert a short remark (new Remark 2.4) immediately preceding Theorem 3.1 that states these hypotheses verbatim and notes that the result therefore holds without further restrictions on X. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes a characterization of positive linear maps on the second symmetric product space X^(2) equipped with the projective cone, using direct arguments from the theory of partially ordered vector spaces. The main result and its applications to CP-cone automorphisms and CP-rank-1 preservers proceed via explicit cone-preserving properties and decomposable vector analysis without reducing any prediction or uniqueness claim to a fitted parameter, self-citation chain, or definitional tautology. The setup assumptions (partially ordered X and projective cone) are stated independently of the target characterization, and no load-bearing step collapses to renaming or smuggling an ansatz from prior self-work. This is the standard honest outcome for a direct functional-analysis characterization.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions about partially ordered vector spaces and the projective cone on symmetric products; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption X is a partially ordered vector space
    Explicitly stated as the setting for X^{(2)}.
  • domain assumption X^{(2)} is endowed with the projective cone
    Used to define the positive structure for the preservation property.

pith-pipeline@v0.9.0 · 5661 in / 1250 out tokens · 57857 ms · 2026-05-20T00:17:01.339388+00:00 · methodology

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Reference graph

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