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 and A should be positive stable, 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, positive stable (every Eigenvalue of A has positive real part) 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? Maybe with additional assumptions? Would be very grateful for any help!
Best regards
This norm is well-known for its theoretical advantage in order to restrict the study to bounded semigroups. From practical point of view, it is the generator that is known whereas the associated semigroup is not explicit in general. I expect that this norm would not be explicit, except for a few special examples that probably cannot be interesting for the study of evolution equations.