Noncommutative versions of the arithmetic-geometric mean inequality

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 .