Contraction semigroup

Remarks

Example: on $L^p(\R)$ with $1 \le p < \infty$, let $(T(t)f)(s) = f(s + t)$. Each $T(t)$ preserves the $L^p$ norm, so $\norm{T(t)} = 1$, and $t \mapsto T(t)f$ is continuous for every $f$, so this is a contraction semigroup.

Non-example: on $\C$, the semigroup $T(t)z = e^{t}z$ is strongly continuous, but $\norm{T(t)} = e^{t} > 1$ for $t > 0$.

The generator of a contraction semigroup is dissipative. Some authors write $C_0$ contraction semigroup to stress strong continuity.