View source for Strongly-continuous semi-group

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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|Semi-group of operators]]; [[Generating operator of a semi-group|Generating operator of a semi-group]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,   R.S. Phillips,   "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pazy,   "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Yu.I. [Yu.I. Daletskii] Daleckii,   "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR></table>