Definition:C0 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 semigroup of bounded linear operators.

We say that $\family {\map T t}_{t \ge 0}$ is a $C_0$ semigroup if and only if:

$\ds \lim_{t \mathop \to 0^+} \map T t x = x$ for each $x \in X$.

Also known as

A $C_0$ semigroup may also be called strongly continuous, in contrast to uniformly continuous.

Also see

• Semigroup of Bounded Linear Operators is C0 iff Point Evaluations Continuous

• Results about $C_0$ semigroups can be found here.

Sources

• 1983: Amnon Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations ... (previous) ... (next): $1.2$: Strongly Continuous Semigroups of Bounded Linear Operators