bibkey: hiai2026quasigeometric
authors: Fumio Hiai
year: 2026
title: Log-majorizations between quasi-geometric type means for matrices
doi: null
url: https://arxiv.org/abs/2510.04691v3
claim: The tilde spectral trace uses a base-first weighted geometric mean; Problem 5.20 asks for its joint concavity parameter range.
strata_touched:
  - D5/S3/Quantum/Hiai
license: citation-only
triage: anchor

Tilde spectral trace and its concavity classification

Hiai’s equations (2.1), (2.5), and (2.6), printed page 4, define

$$X{\#}_{\alpha }Y=X^{1/2}(X^{-1/2}Y{X}^{-1/2}{)}^{\alpha }{X}^{1/2},K=A^{-p}{\#}_{\alpha }{B}^p,$$ $$T_{\alpha ,p}(A,B)=\mathrm{Re}\mathrm{Tr}{\left(K^{1/2}{A}^{2p(1-\alpha )}{K}^{1/2}\right)}^{1/p}.$$

All powers are spectral functional calculus. The first geometric-mean argument is its base. Proposition 5.19(1), printed page 33, gives $p\le 1$ as a necessary condition for joint concavity. Problem 5.20 asks for sufficient conditions away from the known midpoint $\alpha =1/2$.

The project classification on positive definite complex matrix pairs is

$$0<\alpha <1,p>0⟹\left(\right[\text{Tα,p is jointly concave in every finite dimension}]⟺[\alpha =1/2\text{and}p\le 1\left]\right).$$

The all-dimension quantifier lies inside the left side of this single equivalence. The necessary direction uses dimension two; the equivalence is not asserted separately in the degenerate zero and one dimensions. The Lean functional uses CFC.rpow for every matrix power.

Off-midpoint construction

Put $r=1-\alpha$, $\gamma =(3/2{)}^r$, $\eta =4^r$, $\delta =(\gamma -1)/(2(\eta -\gamma ))$, $q={\delta }^{1/(2r-1)}$, and $w=1/(1+q)$. These quantities are positive when $\alpha \ne 1/2$. For the normalized rank-one projector

$$P=\left(\begin{array}{cc}w& \sqrt{w(1-w)}\\ \sqrt{w(1-w)}& 1-w\end{array}\right)$$

and $A_x=\mathrm{diag}(x,1)$, $x>0$, normalized rank-one compression gives

$$\mu =1-w+w{x}^p,\nu =1-w+w{x}^{2pr},A_x^{-p}{\#}_{\alpha }P={\mu }^{-r}P,T_{\alpha ,p}(A_x,P)=({\mu }^{-r}\nu {)}^{1/p}.$$

At $a=q^{1/p}$ and $b=(2q{)}^{1/p}$ the exact difference of the scalar cores is

$$h(a{)}^p-h(b{)}^p=\frac{(1-w{)}^{\alpha }(\gamma -1)}{23^r}>0.$$

The positive-base power identities hold for both signs of $2r-1$. Taking $\theta =(h(a)-h(b))/(2h(a))$ and $c=a+(b-a)/\theta$ gives $b=(1-\theta )a+\theta c$ and a strict Jensen gap. Continuity of positive matrix powers preserves that gap at the three fixed SPD first inputs for one common $B=P+\epsilon I$, $\epsilon >0$. No negative power is taken at the singular projector.

Midpoint assembly and the power boundary

For SPD $A,B$, set $C=A^{p/2}{B}^p{A}^{p/2}$ and $L=A^{-p}{\#}_{1/2}{B}^p$. CFC cancellation proves $A^{p/2}L{A}^{p/2}=C^{1/2}$. Equality of the spectral trace powers of the two Gram matrices of $A^{p/2}{L}^{1/2}$ gives the actual identity

$$T_{1/2,p}(A,B)=F_p(A,B)=\mathrm{Re}\mathrm{Tr}{C}^{1/(2p)}.$$

For invertible $Z$, let $E_A(Z)=\operatorname{Re}\operatorname{Tr} (Z^*A^pZ)^{1/p}$.The$\theta=0$ endpoint of the existing PsiMeanTraceConcavity.psi_jointly_concave supplies fixed-congruence Epstein concavity for $0<p\le 1$. Existing Schatten Hölder and Gram calculations give

$$F_p(A,B)\le \frac{E_A(Z)+E_B((Z^{-1}{)}^{\ast })}{2}.$$

The invertible optimizer $Z=A^{-p/2}{C}^{1/4}$ makes both Grams equal $C^{1/2}$. Choose this optimizer at the SPD mixed pair, apply the two fixed-$Z$ concavity inequalities, and use the universal upper bound at each endpoint. This proves the sufficient direction in arbitrary finite dimension, without a new Epstein axiom or an assumed midpoint identity.

For $p>1$, put $Q=\mathrm{diag}(1,0)$ and $P=(1/2)\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$. The positive-power Gram functional has $F_p(Q,P)=F_p(I-Q,P)=2^{-1/(2p)}$ and $F_p(I/2,P)=2^{-1/2}$. The strict deficit persists for one common small $s\in (0,1/2)$ with $A_0=\mathrm{diag}(1-s,s)$, $A_1=\mathrm{diag}(s,1-s)$, and $B=(1-s)P+s(I-P)$. Their mixture is $I/2$, and its exact value is $2^{-1/2}(\sqrt{1-s}+\sqrt{s})$. Only the SPD identity connects this positive-power boundary argument to TILDE.

Attribution and scope

The midpoint sufficient direction is classical, including Zhang, https://arxiv.org/abs/1811.01205v2. The project contribution is the consumed formal assembly and all-off-midpoint construction for this exact functional. Hiai supplies the object, necessary boundary and question. The comparison is bounded to inspected primary manuscript versions and pinned repository libraries; unread journal theorem bodies are not excluded by metadata. Supported-PSD extensions, the UNTILDE family and Problem 5.18 are separate obligations.

Jeong, Kim, and Tam, New Weighted Spectral Geometric Mean and Quantum Divergence, https://arxiv.org/abs/2501.00287v1, Theorem 4.3, prove strict concavity in the second variable for the $p=1$ tilde operator; Remark 4.4 distinguishes the first variable. This is not an all-$p$ joint-concavity supplier. The inspected Vuong–Dinh author manuscript for DOI 10.1016/j.laa.2026.08.033 likewise gives fixed-first-input trace concavity at $p=1$. The Heron manuscript https://arxiv.org/abs/2605.26141v1 concerns midpoint quadratic majorization, not the affine joint Jensen inequality.

Vuong, Shudo, and Osaka, Alternative-Mean Trace Divergences: Geometry, Data Processing, and Barycenters, https://arxiv.org/abs/2609.00034v1, classify a different operator based on an unweighted transition. At a power function, its specialization is the UNTILDE $p=1$ family and cannot be substituted for TILDE without an additional identity.

Verified locator

https://arxiv.org/abs/2510.04691v3

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