math.FA — Functional Analysis

On Lin's Arithmetic–Geometric Means Conjecture for Singular Values

Contributed by Suvrit Sra

Let $A$ and $B$ be positive semidefinite, and let $0\le t\le 1$. Lin (2017) conjectured that $\lambda_j((1-t)A+tB)\ge\sigma_j^{1/2}(A^{2(1-t)}B^{2t})$ for $j=1,\ldots,n$. The case $t=1/2$ is Drury's theorem, which answered a long-standing question of Bhatia and Kittaneh. Lin remarked that Drury's method does not seem to reach other values of $t$. We prove his conjecture for every $t\in[0,1]$. The proof builds on Drury's argument, but replaces his block-matrix step (unique to the midpoint $t=1/2$) by an eigenvalue count based on an inertia identity and a one-variable inequality.