I'm working on a certain problem that involves the following question:
Let $A,B$ be two self-adjoint operators, and define $C=e^{-A}e^{-B}e^{-A}$. Is there a "convenient" way to express $log(C)$? I'm not entirely sure what "convenient" is, looking for anything that could be useful.
I tried two things - the first was using the Campbell-Baker-Hausdorff formula, but I got very complicated expressions that I do not know how to deal with (since the two operators do not necessarily commute). The second was assuming that one of the operators is diagonal (I simply calculate everything up to a conjugation), but still, I couldn't find a fairly simple way to express what I'm looking for.
Does anyone know of some useful identities/methods that could be useful for this? In the problem I'm working on, $A,B$ depend on two parameters which vary in a certain region, and I want to find a "simple" expression for $log(C)$, so that I can change the parameters and immediately see what happens.
Thanks in advance.
We can't say much. We assume that the considered marices are real.
For example, let $R=e^{-A}e^{-B/2}$. Then $C=e^{-A}e^{-B}e^{-A}=RR^T$ is symmetric $>0$.
We consider the $R$'s SVD: $R=U\Sigma V^T$. Then $C=U\Sigma ^2 U^T$ where $U$ is orthogonal. Finally
$\log(C)=2Udiag(\log(\sigma_i)_i)U^T$ where the $(\sigma_i)_i$ are the singular values of $R$.