ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Geometry and Order Structure in Matrices and Operators

Fri 11:30–12:00, Room AV 03.12
A generalization of Araki's log-majorization
Fumio Hiai (Tohoku University)

For $n\times n$ positive semidefinite matrices $A$ and $B$, the log-majorization $A\prec_{(\log)}B$ means that $\prod_{i=1}^k\lambda_i(A)\le\prod_{i=1}^k\lambda_i(B)$ for $k=1,\dots,n$ (with quality for $k=n$), where $\lambda_i(A)$'s are the eigenvalues of $A$ in decreasing order. We generalize Araki's log-majorization $(A^{1/2}BA^{1/2})^r\prec_{(\log)}A^{r/2}B^r A^{r/2}$, $r\ge1$, in such a way that the function $$ p\ge0\longmapsto \prod_{i=1}^k\lambda_i\bigl(\Phi(A^p)^{1/2}\Psi(B^p)\Phi(A^p)^{1/2}\bigr) $$ is log-convex for every $k=1,\dots,n$, where $\Phi$ and $\Psi$ are positive linear maps between matrix algebras. We further show, in a similar vein, some generalizations of Ando-Hiai's log-majorization $A^r\,\#_\alpha\,B^r\prec_{(\log)}(A\,\#_\alpha\,B)^r$, $r\ge1$, for the weighted geometric mean $\#_\alpha$.