I am interested in the explicit calculation of the following norm $\vert \cdot \vert$. Let $X$ a Banach space with norm $\Vert \cdot \Vert$ and $(T(t))_{t \geq 0}$ a strongly continuous one-parameter semigroup on X (i.e.: $T(t): X \to X$ for all $t >0$), lets define $\vert \cdot \vert$ as $$ |x| = \sup_{t \geq 0} \Vert T(t)x \Vert $$ for all $x \in X.$ We assume that the semigroup is bounded so that the norm is well-defined. As we know the new norm $\vert \cdot \vert$ is equivalent to the original norm $\Vert \cdot \Vert$ on $X$. I am interested in that norm in context of ODEs/PDEs (especially non-symmetric evolution equations and long-time behaviour/asymptotics). Does this norm have a special name or does anyone know some helpful references for the calculation or useful properties of the norm? Would be very grateful for any references.

I tried to compute this norm for the matrix exponential of a (n by n)-matrix A with constant coefficients, where $T(t)x = \operatorname{e}^{-At}x$ can be seen as the solution of the following linear ODE with constant coefficients:

$x(t)=(x_1(t),\cdots,x_n(t))^\intercal \in \mathbb{R}^n:$ $$ \begin{cases} \frac{d}{dt}x=-Ax,t \geq 0.\\ f(0)=x_0 \in \mathbb{R}^n, \end{cases} $$ with a real in general non-symmetric matrix $A \in \mathbb{R}^{n\times n}$.

So I would like to compute $$ |x| = \sup_{t \geq 0} \Vert \operatorname{e}^{-At}x \Vert $$ for a given constant matrix $A \in \mathbb{R}^{n \times n}$ and $x \in \mathbb{R}^{n}.$ Is there an easy way to do this, or if not, are there references? Would be very grateful for any help!

Best regards