Applications of geometric means on symmetric cones

Abstract.

Let V be a Euclidean Jordan algebra, and let $\Omega$ be the corresponding symmetric cone. The geometric mean $a\#b$ of two elements a and b in $\Omega$ is defined as a unique solution, which belongs to $\Omega,$ of the quadratic equation $P(x)a^{-1}=b,$ where P is the quadratic representation of V. In this paper, we show that for any a in $\Omega$ the sequence of $n^{\mathrm{th}}$ iterate $f_a^n(x)$ of the function $f_a:\Omega\to \Omega$ defined by $f_a(x)={1\over 2}(x+P(a)x^{-1})$ converges to a. As applications, we obtain that the geometric mean $a\#b$ of $\Omega$ can be represented as a limit of successive iteration of arithmetic means and harmonic means, and we derive the Löwner-Heinz inequality on the symmetric cone $\Omega: 0\leq a\leq b {\mathrm{ implies}} a^p\leq b^p{\mathrm{ for}} 0\leq p\leq 1.$ Furthermore, we obtain a formula ${\mathrm exp} (x+y)=\lim_{n\to \infty}({\mathrm exp}{\frac{2}{n}{x}}\#\exp{\frac{2}{n}}y)^n$ which leads a Golden-Thompson type inequality $||{mathrm exp} (x+y)||\leq ||P({\mathrm exp}{x\over 2}){\mathrm exp} y||$ for the spectral norm on V.