A note on the non-commutative arithmetic-geometric mean inequality

Abstract

This note proves the following inequality: if $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $A_1,A_2,\cdots ,A_n$, \begin{equation} \frac{1}{n^3}\Big\|\sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\| \geq \frac{(n-3)!}{n!} \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{$j_1$, $j_2$, $j_3$ all distinct}}}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\|, \end{equation} where $\Vert \cdot \Vert$ represents the operator norm. This inequality is a special case of a recent conjecture by Recht and Ré.

References in corpus (1)

• Beneath the valley of the noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences

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