TITLE: Characterizing when matrices are 'dissipative'
QUESTION [7 upvotes]: An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly dissipative if they are each dissipative with respect to the same norm.
Here $e^{At}$ is defined by
$$e^{At}:=\sum_{n=0}^{n=\infty}\frac{1}{n!}A^nt^n.$$ 
My question is the following: what are necessary and sufficient conditions (on $A$ and $B$) for the existence a norm for which $A$ and $B$ are jointly dissipative? 
Of course depending on what we take $n$ to be, this question can be quite hard. If anyone has any particularly clever suggestions for an approach, or solutions to special cases ($n\geq 2$), I would love to hear from you.

REPLY [2 votes]: The problem has been addressed in the infinite dimensional setting in the following paper: 
Máté Matolcsi, On the relation of closed forms and Trotter’s product formula, J. Funct. Anal. 205 (2003), no. 2, 401--413.
Theorem 2 is of special interest here. If there are any practical consequences, I do not know.