Definition:Semigroup of Bounded Linear Operators

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 $\map T t : X \to X$.

We say that $\family {\map T t}_{t \ge 0}$ is a semigroup of bounded linear operators if and only if:

$(1): \quad$ $\map T 0 = I$

$(2): \quad$ for $t, s \ge 0$ we have $\map T {t + s} = \map T t \map T s$

Sources

• 1983: Amnon Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations ... (next): $1.1$: Uniformly Continuous Semigroups of Bounded Linear Operators