Consider an operator or matrix $O$ the adjoint action then is $ad_O M=[O,M]$, where $M$ is an arbitrary test matrix.
My question as follows: Is there a norm that has been defined for $ad_O$ and if so how does it relate to the operator norm of $O$? Take the Frobenius norm specificially $norm(O)=\sqrt{OO^\dagger}$. What norm for $ad_O$ is there that relates the easiest to this norm if any?
Pleas note that specificically I am looking for a relation of the form $$\left\lVert ad_O\right\rVert_{1}=f(\left\lVert g_1(O)\right\rVert,...,\left\lVert g_n(O)\right\rVert)$$, where $f$ is a function and $g_n$ with $n\in N$ are also functions.
Also I would like to reiterate that $ad$ is the adjoint action in this case and not the adjoint representation