An arithmetic-geometric mean inequality for products of three matrices

Abstract

Consider the following noncommutative arithmetic-geometric mean inequality: given positive-semidefinite matrices ${\mathbf{A}}_1,\dots ,{\mathbf{A}}_n$, the following holds for each integer $m\le n$: $$\frac{1}{n^m}\sum _{j_1,j_2,\dots ,j_m=1}^n|||{\mathbf{A}}_{j_1}{\mathbf{A}}_{j_2}\dots {\mathbf{A}}_{j_m}|||\ge \frac{(n-m)!}{n!}\sum _{\begin{array}{c}{j}_1,j_2,\dots ,j_m=1\\ \text{all distinct}\end{array}}^n|||{\mathbf{A}}_{j_1}{\mathbf{A}}_{j_2}\dots {\mathbf{A}}_{j_m}|||,$$ where $|||\cdot |||$ denotes a unitarily invariant norm, including the operator norm and Schatten p-norms as special cases. While this inequality in full generality remains a conjecture, we prove that the inequality holds for products of up to three matrices, $m\le 3$. The proofs for $m=1,2$ are straightforward; to derive the proof for $m=3$, we appeal to a variant of the classic Araki-Lieb-Thirring inequality for permutations of matrix products.

11 pages, no figures