Noncommutative versions of the arithmetic-geometric mean inequality

TL;DR

This paper extends the noncommutative arithmetic-geometric mean inequality to general operators with dimension-free constants, introduces an order version under certain conditions, and demonstrates its near validity for various random matrices.

Contribution

It provides a dimension-free AGM inequality for operators, an order version under specific assumptions, and shows its applicability to random matrices.

Findings

Dimension-free AGM inequality for general operators

Order version of AGM inequality under additional hypotheses

Approximate AGM inequality for many random matrix examples

Abstract

Recht and R\'{e} introduced the noncommutative arithmetic geometric mean inequality (NC-AGM) for matrices with a constant depending on the degree $d$ and the dimension $m$. In this paper we prove AGM inequalities with a dimension-free constant for general operators. We also prove an order version of the AGM inequality under additional hypothesis. Moreover, we show that our AGM inequality almost holds for many examples of random matrices .