Definition:Infinitesimal Generator of Semigroup
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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 $\family {\map T t}_{t \ge 0}$.
Sources
- 1983: Amnon Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations ... (previous) ... (next): $1.1$: Uniformly Continuous Semigroups of Bounded Linear Operators