Log-majorizations between quasi-geometric type means for matrices

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 Rényi 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.

44 pages, this includes the second half of the author's plenary talk in ILAS2025. Final print version

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