Log-majorizations between quasi-geometric type means for matrices

• URL: http://arxiv.org/abs/2510.04691v2
• Date: Mon, 20 Oct 2025 02:05:27 GMT
• Title: Log-majorizations between quasi-geometric type means for matrices
• Authors: Fumio Hiai,
• Abstract summary: The log-majorization $mathcalM_alpha,p(A,B)prec_logmathcalN_alpha,q(A,B)$ is examined for pairs $(mathcalM,calN)$ those $alpha$-weighted geometric type means.<n>The joint concavity/ity of the trace functions $mathrmTr,mathcalM_alpha,p$ is also discussed based on theory of quantum divergences.
• Score: 4.56877715768796
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, for $\alpha\in(0,\infty)\setminus\{1\}$, $p>0$ and positive semidefinite matrices $A$ and $B$, we consider the quasi-extension $\mathcal{M}_{\alpha,p}(A,B):=\mathcal{M}_\alpha(A^p,B^p)^{1/p}$ of several $\alpha$-weighted geometric type matrix means $\mathcal{M}_\alpha(A,B)$ such as the $\alpha$-weighted geometric mean in Kubo--Ando's sense, the R\'enyi mean, etc. The log-majorization $\mathcal{M}_{\alpha,p}(A,B)\prec_{\log}\mathcal{N}_{\alpha,q}(A,B)$ is examined for pairs $(\mathcal{M},\mathcal{N})$ of those $\alpha$-weighted geometric type means. The joint concavity/convexity of the trace functions $\mathrm{Tr}\,\mathcal{M}_{\alpha,p}$ is also discussed based on theory of quantum divergences.

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