Lin: both spectral properties of the map Ψ

The fixed source is M. Lin, New properties for certain positive semidefinite matrices, arXiv:1609.08231v1, §3, Example 3.6; the journal version is Linear Algebra and its Applications 520 (2017), 32–43. The setup states:

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

Definition 1.1(iii) states:

M is said to have a-property if 2sj (M12 ) ≤ λj (M11 + M22 ), j = 1, . . . , n.

Definition 1.1(iv) states:

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

Example 3.6 states:

If we consider the map Ψ : X ↦ 2(trX)In − X, then using the approach in [14] we can show that the matrix [Ψ(A) Ψ(X); Ψ(X∗) Ψ(B)] is PPT. Though there are strong numerical evidence suggesting that this block matrix also has g-property, yet we have not even been able to show that it has a-property.

The concluding §5 states:

Besides the challenging problems described in Example 3.5 and Example 3.6, …

For every natural n, complex n-by-n blocks A,X,B with [A X; X* B] positive semidefinite, and every decreasing index j, prove both inequalities

$$ 2s_j(\Psi(X))\le\lambda_j(\Psi(A)+\Psi(B)), \qquad s_j(\Psi(X))\le\lambda_j(\Psi(A)#\Psi(B)), \quad \Psi(T)=2(\operatorname{tr}T)I-T. $$

These are comparisons at each spectral index, with no commutation, invertibility or rank assumption on the original blocks. The PPT property alone does not give the matching-index comparisons. The statement uses the same decreasing singular values, Hermitian eigenvalues and total geometric mean as the trace-augmentation comparison.

With both diagonal traces normalized to one, set t=|tr X| and R(T)=(tr T)I−T. The decisive additional positive matrix is [R(A) R(X*); R(X) R(B)]. Its Gram discrepancy is at most 2(1−t)I after a unit phase makes tr X real. A general discrepancy estimate gives an arithmetic spectral bound for this reduced block. This estimate must retain the same t as the diagonal difference-square estimate.

For basis indices k<l put F_kl=e_k e_l−e_l e_k. Entrywise calculation gives Σ F_kl T F_kl*=(tr T)I−Tᵀ. Apply the corresponding block-diagonal congruences to the entrywise conjugate of the original positive block. Their sum is the required positive reduction block.

For any positive block [P Y;Y* Q] whose discrepancy P+Q−Y−Y* is at most δI, Gram factorization and Loewner eigenvalue monotonicity give 2s_j(Y)≤λ_j(P+Q)+δ/2. Applying this to the reduction block gives 2s_j(tI−X)≤λ_j(2I−A−B)+1−t. Scalar perturbation then gives s_j(Ψ(X))≤λ_j(H)−(1−t)/2 for H=(Ψ(A)+Ψ(B))/2.

The existing normalized difference-square bound is (A−B)²≤(1−t²)I. The existing geometric-mean comparison bounds Ψ(A)#Ψ(B) below by H−(1−t²)I/4. The retained arithmetic deficit absorbs this correction and gives the normalized g-property.

If either diagonal block is zero, positivity forces X=0. Otherwise a block congruence normalizes both diagonal traces. Independent positive homogeneity of the geometric mean restores the g-property for the original blocks. The matrix arithmetic-geometric mean comparison then implies the a-property for arbitrary traces; the arithmetic sum itself is not homogeneous under independent scaling of its two summands.

A positive semidefinite complex block matrix and an index violating either displayed inequality would refute the conjunction. A failure of the reduction-block identity or of the general discrepancy estimate would invalidate this route without itself refuting the target.

The exact universal conjunction is conjecture, and its settlement theorem is D5.S3.Quantum.Matrix.LinPsiMap.result in the same file. The new intermediate statements are reduction_block_posSemidef and gram_discrepancy_bound. The difference-square, scalar perturbation, phase rotation, block normalization and geometric-mean estimates are reused from D5/S3/Quantum/Matrix/LinGProperty.

What the settlement shows

PROVED — both properties. D5/S3/Quantum/Matrix/LinPsiMap/Result.result states the a-property and g-property together for every natural dimension, every positive semidefinite complex block matrix and every decreasing index. Dimension zero is vacuous, and zero diagonal blocks are included.

PROVED — the new positivity input. D5/S3/Quantum/Matrix/LinPsiMap/ReductionBlock.reduction_block_posSemidef provides the positive reduction block without a commutation assumption.

PROVED — the general discrepancy estimate. D5/S3/Quantum/Matrix/LinPsiMap/GramDiscrepancy.gram_discrepancy_bound controls each singular value using the corresponding diagonal-sum eigenvalue and one half of the discrepancy bound.

OPEN — separate quantitative statement. The normalized route gives the margin (1−|tr X|)²/4 in the g-property. No separate public theorem for that strengthened conclusion is asserted here.

OPEN — equality classification. A rank-one trace-one projection with A=B=X gives equality in both properties. A complete equality classification is not part of the settlement.

The verbatim quotations, source locators and literature reading are supplied information, not independently checked against original papers under the offline constraint. Yang–Lu–Chen, Filomat 34 (2020), treats Ψ₁(T)=(2n+1)(tr T)I−T. Alakhrass, arXiv:2305.10461 (2023), gives a PPT-block geometric-mean estimate with index floor((j+1)/2), weaker than the matching-index g-property. Rani–Kapil–Singh, Advances in Operator Theory 8 (2023), is also included in the supplied reading. None settles Example 3.6 within that reading. Citation indexes were not exhaustively checked, and no literature-wide priority claim is certified. See the Library note.