Suppose $A$ and $B$ are (non-commuting) hermitian $n\times n$ matrices and $k$ is a large positive number. Suppose we write the product of matrix exponentials as

$e^{kA + B} e^{-kA} = e^{C(k)}$

for some matrix $C(k)$. I am interested in how large $C(k)$ is (with respect to some appropriate norm, lets say the operator norm of matrices).

In particular is $\| C(k)\| = O(k)$? Some numerical experiments suggest a bound of the form $||C(k)|| \le O(k) ||[ A,B]||$ but I cannot prove this.

I know the Baker-Campbell-Hausdorff formula gives us an (explicit) equation for $C(k)$. But bounding this expression term-by-term using the triangle inequality gives a terrible bound. I am hoping somebody knows either another argument (or a trick) that gives a more sensible bound.

Thanks a lot!