bibkey: dumitrufranco2017powermeans
authors: Raluca Dumitru; Jose Franco
year: 2017
title: On the monotonicity of weighted power means of matrices
doi: null
url: https://arxiv.org/abs/1701.07023v1
claim: The withdrawn first version conjectures decreasing distance to A for weighted power means with every unitarily invariant norm and every exponent 0 < p <= 2; an exact positive definite pair refutes its p=2 Euclidean operator-norm specialization.
strata_touched:
  - D5/S3/Quantum/Matrix/PowerMeanOperatorRefutation
license: citation-only
triage: anchor

Weighted power means and a version-specific counterexample

Verified locator

Raluca Dumitru and Jose Franco, On the monotonicity of weighted power means of matrices, arXiv:1701.07023v1, submitted 24 January 2017: https://arxiv.org/abs/1701.07023v1. The official source is https://arxiv.org/src/1701.07023v1. The final conjecture is numbered Conjecture 9 in the first-version PDF.

Source assertion and version boundary

Conjecture 9 reads:

Let $A,B\ge 0$ and let $\Vert \cdot \Vert$ be a unitarily invariant norm. Let $0<p\le 2$. Then, $t\mapsto \Vert A-{\mu }_p(A,B,t)\Vert$ is monotonically decreasing for $t\in [0,1]$.

Here ${\mu }_p(A,B,t)=(t{A}^p+(1-t)B^p{)}^{1/p}$, with positive spectral powers. Version 2, dated 30 January 2017, withdraws the preprint because Proposition 4 is false and major revisions are necessary. Withdrawal alone does not prove Conjecture 9 false. The counterexample below is attributed only to the displayed assertion of version 1.

The published successor is Trung Hoa Dinh, Raluca Dumitru, and Jose A. Franco, On the monotonicity of weighted power means for matrices, Linear Algebra and its Applications 527 (2017), 128–140, https://doi.org/10.1016/j.laa.2017.04.003. Its author manuscript is https://trunghoamath.wordpress.com/wp-content/uploads/2017/03/main-v-31.pdf. It proves the $p=2$ Frobenius branch and the $p=1/2$ branch for every unitarily invariant norm; those conclusions are compatible with this counterexample. This comparison uses the public three-author author manuscript. Publisher full version-of-record text was not obtained, so no complete comparison with it is claimed.

No exact prior refutation was found in the bounded later primary literature inspected through 2026-11-01. The directly relevant The matrix power means and interpolations (2018), https://doi.org/10.15352/aot.1801-1288, remained text-inaccessible; other candidates were metadata-only. Current-open status, novelty, priority, and uniqueness remain unverified. No comprehensive later-resolution claim is made, and this component earns zero verified-original or unique credit.

Exact formal scope

The closed Lean proposition claim quantifies over all natural dimensions, complex positive semidefinite matrices $A,B$, and real $0\le s\le t\le 1$. It asserts

$$\Vert A-{\mu }_2(A,B,t){\Vert }_{2\to 2}\le \Vert A-{\mu }_2(A,B,s){\Vert }_{2\to 2}.$$

The norm is the operator norm induced by the Euclidean norm on complex vectors (Matrix.Norms.L2Operator). The power mean uses the positive spectral square root through continuous functional calculus. The theorem result has the literal closed type Not claim. This specialization follows from the full version-1 assertion, so its negation refutes that universal assertion. The formal proposition does not quantify over all norms, and the result does not refute every norm separately.

Exact matrices and the strict inequality

Take $A=Y$, $B=X$, $s=0$, and $t=1/100$, where

$$X=\left(\begin{array}{cc}1/10& 0\\ 0& 4\end{array}\right),Y=\left(\begin{array}{cc}19/50& -24/25\\ -24/25& 93/25\end{array}\right).$$

Both matrices are positive definite. Their determinants are $2/5$ and $123/250$, respectively. The difference $D=X-Y$ satisfies $D^{\ast }D=I$, giving the endpoint distance $1$. At the positive parameter put

$$S=\frac{99X^2+Y^2}{100}=\left(\begin{array}{cc}257/12500& -123/3125\\ -123/3125& 39969/2500\end{array}\right),d=\frac{51118101}{156250000},T=\frac{100051}{6250}.$$

For $r=\sqrt{d}$ and $w=\sqrt{T+2r}$, the matrix $Q=(S+rI)/w$ is positive semidefinite and satisfies $Q^2=S$, hence is the positive spectral root. The Euclidean unit vector $v=(3/5,4/5)$ has

$$a=v^{\ast }Sv=\frac{127521}{12500},c=v^{\ast }Xv=\frac{649}{250},v^{\ast }Yv=\frac{399}{250}=c-1.$$

The exact certificates

$$d-(57/100{)}^2=\frac{88119}{39062500}>0,$$

$$(a+r{)}^2-c^2{w}^2=-\frac{339943099}{97656250}+\frac{108202}{15625}r>\frac{22763263}{48828125}>0$$

give $(a+r)/w>c$, and therefore $v^{\ast }(Q-Y)v>1$. Cauchy–Schwarz and the Euclidean operator-norm bound yield $\Vert Y-Q{\Vert }_{2\to 2}>1$. No floating approximation is a premise.