Strongly-continuous semi-group
A family ,
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) |
Comments
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) |
Strongly-continuous semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strongly-continuous_semi-group&oldid=48879