Inverse Problems for Matrix Means and Trace characterization

Thời gian: 14:30 chiều thứ Hai, 11/5/2026

Địa điểm: Phòng seminar tầng 5- nhà Ạ,6

Tóm tắt: We present a unified mechanism connecting three problems in matrix analysis: inverse problems for matrix means, characterizations of operator monotone functions, and characterizations of the trace among positive linear functionals. Let $m$ and $M$ be two matrix means satisfying $m(A,B) \leq M(A,B), A,B > 0$. If the ordered pair $(m,M)$ has a suitable inverse-realization property, then two parallel characterizations follow. First, a continuous function $f$ is matrix monotone precisely when $f(m(A,B)) \leq f(M(A,B))$ for all positive definite $A,B$. Second, a positive linear functional $\phi$ is a scalar multiple of the trace precisely when $\phi(M(A,B)^2 - m(A,B)^2) \geq 0$ for all positive definite $A,B$.