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A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s0906201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s0906202.png" />, of bounded linear operators on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s0906203.png" /> with the following properties:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s0906204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s0906205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s0906206.png" />;
+
{{TEX|auto}}
 +
{{TEX|done}}
  
2) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s0906207.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s0906208.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s0906209.png" />.
+
A family  $  T ( t) $,
 +
$  t > 0 $,
 +
of bounded linear operators on a Banach space  $  X $
 +
with the following properties:
  
When 1) holds, the measurability of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062011.png" />, and, in particular, their one-sided (right or left) weak continuity, implies strong continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062012.png" />. For a strongly-continuous semi-group the finite number
+
1) $  T ( t + \tau ) x = T ( t) T ( \tau ) x $,
 +
$  t, \tau > 0 $,  
 +
$  x \in X $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062013.png" /></td> </tr></table>
+
2) the function  $  t \mapsto T ( t) x $
 +
is continuous on  $  ( 0, \infty ) $
 +
for any  $  x \in X $.
  
is called the type of the semi-group. Thus, the norms of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062014.png" /> grow at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062015.png" /> no faster than the exponential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062016.png" />. The classification of strongly-continuous semi-groups is based on their behaviour as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062017.png" />. If there is a bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062019.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062021.png" /> is a projection operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062023.png" /> is a bounded linear operator commuting with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062024.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062025.png" /> is continuous with respect to the operator norm. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062028.png" />, is a uniformly-continuous group of operators.
+
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
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062029.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062031.png" /> is also a projection operator, projecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062032.png" /> onto the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062033.png" /> that is the closure of the union of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062036.png" />.
+
$$
 +
\omega  = \
 +
\inf _ {t > 0 } \
 +
t  ^ {-} 1  \mathop{\rm ln}  \| T ( t) \|  = \
 +
\lim\limits _ {t \rightarrow \infty } \
 +
t  ^ {-} 1  \mathop{\rm ln}  \| T ( t) \|
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062037.png" /> to exist and to be equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062038.png" /> it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062039.png" /> be bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062040.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062041.png" />. In this case the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062042.png" /> can be extended by the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062043.png" /> and is strongly continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062044.png" /> (it satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062046.png" />-condition). For broader classes of semi-groups the limit relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062047.png" /> is satisfied in a generalized sense:
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062048.png" /></td> </tr></table>
+
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 $.
  
(Cesáro summability, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062050.png" />-condition), or
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062051.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow 0 } \
 +
{
 +
\frac{1}{t}
 +
}
 +
\int\limits _ { 0 } ^ { t }
 +
T ( \tau ) x  d \tau  = x,\ \
 +
x \in X
 +
$$
  
(Abel summability, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062053.png" />-condition). Here it is assumed that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062055.png" />, is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062056.png" /> (and, hence, on any finite interval).
+
(Cesáro summability, the $  C _ {1} $-
 +
condition), or
  
The behaviour of a strongly-continuous semi-group as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062057.png" /> can be completely irregular. For example, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062058.png" /> may have a power singularity at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062059.png" />.
+
$$
 +
\lim\limits _ {\lambda \rightarrow \infty } \
 +
\lambda \int\limits _ { 0 } ^  \infty 
 +
e ^ {- \lambda \tau }
 +
T ( \tau ) x  d \tau  = x,\ \
 +
x \in X
 +
$$
  
For a dense set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062061.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062062.png" /> is differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062063.png" />. An important role is played by strongly-continuous semi-groups for which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062064.png" /> is differentiable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062065.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062066.png" />. In this case the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062067.png" /> is bounded for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062068.png" /> and its behaviour as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062069.png" /> gives new opportunities for classifying semi-groups. The classes of strongly-continuous semi-groups for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062070.png" /> admits a holomorphic extension in a sector of the complex plane containing the semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090620/s09062071.png" /> have been characterized.
+
(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]].
 
See [[Semi-group of operators|Semi-group of operators]]; [[Generating operator of a semi-group|Generating operator of a semi-group]].
Line 31: Line 124:
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:24, 6 June 2020


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)

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)
How to Cite This Entry:
Strongly-continuous semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strongly-continuous_semi-group&oldid=48879
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article