Definition:Infinitesimal Generator of Semigroup

Definition
Let $\GF \in \set {\R, \C}$.

Let $X$ be a Banach space over $\GF$.

Let $\family {\map T t}_{t \ge 0}$ be a $\hointr 0 \infty$-indexed family of bounded linear transformations $\map T t : X \to X$.

Define:


 * $\ds \map D A = \set {x \in X : \lim_{t \mathop \to 0^+} \frac {\map T t x - x} t \text { exists} }$

Define $A : \map D A \to X$ by:


 * $\ds A x = \lim_{t \mathop \to 0^+} \frac {\map T t x - x} t$

for $x \in \map D A$.

We say that $\struct {\map D A, A}$ is the infinitesimal generator of $\family {\map T t}_{t \ge 0}$.