# MS-10 Matrix Analysis: Inequalities, Means, and Majorization (Matrix Inequalities)

### Organizers: Fumio Hiai (Tohoku University, Japan) and Yongdo Lim (Sungkyunkwan University, Korea)

A core concept in Matrix Analysis is order structure constituting different notions of positive semidefiniteness order (or Loewner order), spectral and majorization order for eigenvalues, etc. Correspondingly, different types of matrix/operator inequalities have become a big research field with wide applications. In particular, various majorization results have played vital roles in matrix norm inequalities for unitarily invariant norms. A subject strongly connected to these order relations is theory of matrix/operator means, whose multivariate extensions have recently been developed extensively. In particular, the extension of the geometric mean in Riemannian geometry approach is remarkable. This mini-symposium is aimed to exchange new developments and ideas in topics including matrix (norm) inequalities, matrix/operator means, and majorizations for matrices.

### Speakers

**Man-Duen Choi**, *Some assorted inequalities for positive linear maps*

**Douglas Farenick**, *Isometries and contractions of density operators relative to the Bures metric*

**Frank Hansen**, *Peierls-Bogolyubov's inequality for deformed exponentials*

**Fumio Hiai**, *Log-majorization and Lie-Trotter formula for the Cartan barycenter*

**Masatoshi Ito**, *Estimations of power difference mean by Heron mean*

**Sejong Kim**, *An order inequality characterizing Cartan barycenters of positive definite matrices*

**Fuad Kittaneh**, *A generalization of the Ando-Hiai-Okubo trace inequalities*

**Seung-Hyeok Kye**, *Separability of three qubit X-states*

**Jimmie Lawson**, *The Karcher Barycentric Map for Positive Operator Probability Measures on Hilbert Space*

**Yongdo Lim**, *Multiplicative Geometric Means*

**Lajos Molnar**, *Order automorphisms in matrix algebras and in operator algebras and their applications*

**Takashi Sano**, *On classes of non-normal matrices/operators*

**Takeaki Yamazaki**, *Properties of weighted operator means via generalized relative operator entropy*

**Pingping Zhang**, *Remarks on two determinantal inequalities*