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.

See Semi-group of operators; Generating operator of a semi-group.

References

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

References

[a1]	A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983)
[a2]	W. Arendt, A. Grabosch, G. Greiner, U. Groh, H.P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, U. Schlotterbeck, "One parameter semigroups of positive operators" , Lect. notes in math. , 1184 , Springer (1986)
[a3]	Yu.I. [Yu.I. Daletskii] Daleckii, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)