# Strongly-continuous semi-group

A family $T ( t)$, $t > 0$, of bounded linear operators on a Banach space $X$ with the following properties:

1) $T ( t + \tau ) x = T ( t) T ( \tau ) x$, $t, \tau > 0$, $x \in X$;

2) the function $t \mapsto T ( t) x$ is continuous on $( 0, \infty )$ for any $x \in X$.

When 1) holds, the measurability of all functions $t \mapsto T ( t) x$, $x \in X$, and, in particular, their one-sided (right or left) weak continuity, implies strong continuity of $T ( t)$. For a strongly-continuous semi-group the finite number

$$\omega = \ \inf _ {t > 0 } \ t ^ {-} 1 \mathop{\rm ln} \| T ( t) \| = \ \lim\limits _ {t \rightarrow \infty } \ t ^ {-} 1 \mathop{\rm ln} \| T ( t) \|$$

is called the type of the semi-group. Thus, the norms of the functions $t \mapsto T ( t) x$ grow at $\infty$ no faster than the exponential $e ^ {\omega t }$. The classification of strongly-continuous semi-groups is based on their behaviour as $t \rightarrow 0$. If there is a bounded operator $J$ such that $\| T ( t) - J \| \rightarrow 0$ as $t \rightarrow 0$, then $J$ is a projection operator and $T ( t) = Je ^ {tA}$, where $A$ is a bounded linear operator commuting with $J$. In this case $T ( t)$ is continuous with respect to the operator norm. If $J = I$, then $T ( t) = e ^ {tA}$, $- \infty < t < \infty$, is a uniformly-continuous group of operators.

If $T ( t) x \rightarrow Jx$ for each $x \in X$, then $J$ is also a projection operator, projecting $X$ onto the subspace $X _ {0}$ that is the closure of the union of all $T ( t) x$, $t > 0$, $x \in X$.

For $J$ to exist and to be equal to $I$ it is necessary and sufficient that $\| T ( t) \|$ be bounded on $( 0, 1)$ and that $X _ {0} = X$. In this case the semi-group $T ( t)$ can be extended by the equality $T ( 0) = I$ and is strongly continuous for $t \geq 0$( it satisfies the $C _ {0}$- condition). For broader classes of semi-groups the limit relation $T ( t) \rightarrow I$ is satisfied in a generalized sense:

$$\lim\limits _ {t \rightarrow 0 } \ { \frac{1}{t} } \int\limits _ { 0 } ^ { t } T ( \tau ) x d \tau = x,\ \ x \in X$$

(Cesáro summability, the $C _ {1}$- condition), or

$$\lim\limits _ {\lambda \rightarrow \infty } \ \lambda \int\limits _ { 0 } ^ \infty e ^ {- \lambda \tau } T ( \tau ) x d \tau = x,\ \ x \in X$$

(Abel summability, the $A$- condition). Here it is assumed that the function $\| T ( t) x \|$, $x \in X$, is integrable on $[ 0, 1]$( and, hence, on any finite interval).

The behaviour of a strongly-continuous semi-group as $t \rightarrow 0$ can be completely irregular. For example, the function $t \mapsto \| T ( t) x \|$ may have a power singularity at $t = 0$.

For a dense set of $x$ in $X _ {0}$ the function $t \mapsto T ( t) x$ is differentiable on $[ 0, \infty )$. An important role is played by strongly-continuous semi-groups for which the function $t\mapsto T ( t) x$ is differentiable for all $x$ for $t > 0$. In this case the operator $T ^ \prime ( t)$ is bounded for each $t$ and its behaviour as $t \rightarrow 0$ gives new opportunities for classifying semi-groups. The classes of strongly-continuous semi-groups for which $T ( t)$ admits a holomorphic extension in a sector of the complex plane containing the semi-axis $( 0, \infty )$ have been characterized.

#### References

 [1] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)