Lie-Trotter means of positive definite operators

As extension of the Lie-Trotter product formula, we define the two-variable and multivariate Lie-Trotter means with several examples including the Sagae-Tanabe and Hansen inductive means. We show that the weighted means satisfying the arithmetic-geometric-harmonic mean inequalities are the multivariate Lie-Trotter means. We finally give a few interesting versions of the extended Lie-Trotter formula derived from multivariate Lie-Trotter means.