C0-semigroup

Remarks

Conditions 1 and 2 say that $T$ is a monoid homomorphism from $([0, \infty), +)$ into the multiplicative semigroup $\mathcal{L}(X)$.

Example: for a bounded operator $A$, $T(t) = e^{tA}$, the exponential series, is a $C_0$-semigroup with generator $A$; here even $\norm{T(t) - I} \to 0$ in operator norm.

Example: on $X = L^p(\R)$ with $1 \le p < \infty$, the left shifts $(T(t) f)(s) = f(s + t)$ form a $C_0$-semigroup. Condition 3 holds because translation is continuous in $L^p$ norm for $p < \infty$, which follows by approximation with continuous functions of compact support. The generator is $A f = f'$ on the Sobolev space $W^{1,p}(\R)$.

Non-example: the same shifts on $L^\infty(\R)$ fail condition 3. For $f$ the indicator function of $[0, 1]$ and $0 < t < 1$, $T(t) f - f$ equals $1$ in absolute value on $[-t, 0)$, so $\norm{T(t) f - f}_\infty = 1$.

Every $C_0$-semigroup satisfies $\norm{T(t)} \le M e^{\omega t}$ for some constants $M \ge 1$ and $\omega \in \R$, and its generator is closed and densely defined.