exp(log + log) for positive semidefinite matrices

Question

Let $A$ and $B$ be positive definite matrices. What is known about $f(A,B)=\exp(\log A + \log B)$? Does this function have a name? This is interesting because $f(A,B) = AB$ for commuting matrices and $f(A,B)=f(B,A)$ even for non-commuting matrices.

Answer 1

Since $\log(A)$ and $\log(B)$ are symmetric matrices, by the Golden-Thompson theorem we have $$ \text{tr}\big(f(A,B)\big) \le \text{tr}\big(\exp\big(\log(A)\big)\exp\big(\log(B)\big)\big) =\text{tr}(AB). $$

Answer 2

Petz (1994) A survey of certain trace inequalities, Lemma 4 leads to

$f(A, B) = \lim_{s \to \infty} (A^{1/s} B^{1/s})^s$

This gives a nice Lie algebra geometric interpretation: suppose Alice is trying to rotate in the $\log A$ direction and Bob is trying to rotate in the $\log B$ direction. They compromise and rotate in the $\log A + \log B$ direction, or take turns rotating an infinitesimal amount.