Characterizing when matrices are 'dissipative'

Question

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.

Answer 1

The discrete-time analogue -- there exists a norm in which $|A_1^nx|\leq |x|$, $|A_2^nx|\leq |x|$ for all $n\geq 1$ and $x \in\mathbb{R}^d$ -- is equivalent to the property that the semigroup generated by $A_1$ and $A_2$ is bounded. To see this we just define $$|x|:=\sup_{n\geq 0} \|A_{i_n}\cdots A_{i_1}x\|$$ for all $x$. Determining whether or not this property holds is known to be an algorithmically undecidable problem by work of Blondel and Tsitsiklis.

There could easily be continuous-time analogues of these statements, but I'm not very familiar with that part of the literature.

Answer 2

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.