bibkey: lin2016gproperty
authors: Minghua Lin
year: 2016
title: "A singular value inequality related to a linear map"
doi: 10.13001/1081-3810.3108
url: https://arxiv.org/abs/1609.08231
claim: "Conjecture 3.1 asks whether trace augmentation gives the g-property for every positive semidefinite two-by-two block matrix."
strata_touched:
  - D5/S3/Quantum/Matrix/LinGProperty/ArithmeticBound
  - D5/S3/Quantum/Matrix/LinGProperty/TraceDistanceBound
  - D5/S3/Quantum/Matrix/LinGProperty/GeometricMeanBound
  - D5/S3/Quantum/Matrix/LinGProperty/ScalarPerturbation
  - D5/S3/Quantum/Matrix/LinGProperty/Result
license: citation-only
triage: anchor

Trace augmentation and the geometric-mean singular-value inequality

Locator

The original question is Conjecture 3.1, printed page 124, in Electronic Journal of Linear Algebra 31 (2016), 120–124, DOI 10.13001/1081-3810.3108. The fixed restatement is New properties for certain positive semidefinite matrices, arXiv:1609.08231v1, §3, Example 3.5; its g-property is Definition 1.1(iv). The restatement appeared in Linear Algebra and its Applications 520 (2017), 32–43.

The supplied text of Definition 1.1(iv) is:

M is said to have g-property if sj (M12 ) ≤ λj (M11 ♯ M22 ), j = 1, . . . , n.

The supplied text of Example 3.5 is:

[Φ(A) Φ(X); Φ(X∗) Φ(B)] where Φ : X ↦ X+(trX)In , is PPT … Numerical experiments suggest that the matrix [Φ(A) Φ(X); Φ(X∗) Φ(B)] has g-property, which we haven’t been able to prove yet.

The setup states:

we assume that [A X; X∗ B], where A, X, B are n × n, is positive semidefinite.

Mathematical scope

For every dimension and every complex positive semidefinite block matrix $[AX;X^{\ast }B]$, the question is whether

$$s_j(X+(\mathrm{Tr}X)I)\le {\lambda }_j((A+(\mathrm{Tr}A)I)\#(B+(\mathrm{Tr}B)I))$$

holds at every index in decreasing order. On positive definite inputs, $P\#Q=P^{1/2}(P^{-1/2}Q{P}^{-1/2}{)}^{1/2}{P}^{1/2}$. Zero diagonal blocks force the off-diagonal block to vanish.

Literature-check scope and result

The supplied preregistration records searches of the title and DOI with conjecture/proof/solved; “g-property” with “X + tr(X) I”; and “Lin’s conjecture” geometric mean singular value, 2024–2026. The named references checked were Yang–Lu–Chen, Operators and Matrices 14 (2020), for dimension two; Abbas–Ghabries, Linear and Multilinear Algebra 70 (2022), for $AX=XA$; Fu–Lau–Tam, Mathematical Inequalities & Applications 23 (2020), for ${\Phi }_c$ with $c\ge (n+3)/2$; Rani–Kapil–Singh, Advances in Operator Theory 8 (2023), which restates the problem as Conjecture 1.1; and a Filomat 34 (2020) restatement.

No settlement was found within that scope. Citation indexes were not exhaustively checked. This is the preregistration’s concrete reading, not an independent exhaustive literature certification.

Attribution boundary

The quotations, bibliographic metadata and literature-check reading are supplied source information, ASSUMED-UNVERIFIED against original papers under the offline constraint. The trace-corrected arithmetic bound, the trace-overlap difference estimate and the quantitative harmonic comparison are the supplied proof route; no proof of the general conjecture is attributed to the cited literature. This citation-only note incorporates no source software.