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A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841401.png" /> of operators on a [[Banach space|Banach space]] or [[Topological vector space|topological vector space]] with the property that the composite of any two operators in the family is again a member of the family. If the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841402.png" /> are "indexed" by elements of some abstract [[Semi-group|semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841403.png" /> and the binary operation of the latter is compatible with the composition of operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841404.png" /> is known as a representation of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841405.png" />. The most detailed attention has been given to one-parameter semi-groups (cf. [[One-parameter semi-group|One-parameter semi-group]]) of bounded linear operators on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841406.png" />, which yield a representation of the additive semi-group of all positive real numbers, i.e. families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841407.png" /> with the property
+
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<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/s084/s084140/s0841408.png" /></td> </tr></table>
+
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 +
{{TEX|done}}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841409.png" /> is strongly measurable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414011.png" /> is a [[Strongly-continuous semi-group|strongly-continuous semi-group]]; this will be assumed in the sequel.
+
A family  $  \{ T \} $
 +
of operators on a [[Banach space|Banach space]] or [[Topological vector space|topological vector space]] with the property that the composite of any two operators in the family is again a member of the family. If the operators  $  T $
 +
are "indexed" by elements of some abstract [[Semi-group|semi-group]]  $  \mathfrak A $
 +
and the binary operation of the latter is compatible with the composition of operators,  $  \{ T \} $
 +
is known as a representation of the semi-group  $  \mathfrak A $.
 +
The most detailed attention has been given to one-parameter semi-groups (cf. [[One-parameter semi-group|One-parameter semi-group]]) of bounded linear operators on a Banach space  $  X $,
 +
which yield a representation of the additive semi-group of all positive real numbers, i.e. families  $  T ( t) $
 +
with the property
 +
 
 +
$$
 +
T ( t + \tau ) x  =  T ( t) T ( \tau ) x ,\  t , \tau > 0 ,\  x \in X .
 +
$$
 +
 
 +
If  $  T ( t) $
 +
is strongly measurable, $  t > 0 $,  
 +
then $  T ( t) $
 +
is a [[Strongly-continuous semi-group|strongly-continuous semi-group]]; this will be assumed in the sequel.
  
 
The limit
 
The limit
  
<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/s084/s084140/s08414012.png" /></td> </tr></table>
+
$$
 +
\omega  = \lim\limits _ {t \rightarrow \infty } \
 +
t  ^ {-1}  \mathop{\rm ln}  \| T ( t) \|
 +
$$
  
exists; it is known as the type of the semi-group. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414013.png" /> increase at most exponentially.
+
exists; it is known as the type of the semi-group. The functions $  T ( t) x $
 +
increase at most exponentially.
  
 
An important characteristic is the infinitesimal operator (infinitesimal generator) of the semi-group:
 
An important characteristic is the infinitesimal operator (infinitesimal generator) of the semi-group:
  
<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/s084/s084140/s08414014.png" /></td> </tr></table>
+
$$
 +
A _ {0} x  = \lim\limits _ {t \rightarrow 0 }  t  ^ {-1} [ T ( t) x - x ] ,
 +
$$
  
defined on the linear set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414015.png" /> of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414016.png" /> for which the limit exists; the closure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414017.png" />, of this operator (if it exists) is known as the generating operator, or generator, of the semi-group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414018.png" /> be the subspace defined as the closure of the union of all values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414019.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414020.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414021.png" />. If there are no non-zero elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414023.png" />, then the generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414024.png" /> exists. In the sequel it will be assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414025.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414026.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414027.png" />.
+
defined on the linear set $  D ( A _ {0} ) $
 +
of all elements $  x $
 +
for which the limit exists; the closure, $  A $,  
 +
of this operator (if it exists) is known as the generating operator, or generator, of the semi-group. Let $  X _ {0} $
 +
be the subspace defined as the closure of the union of all values $  T ( t) x $;  
 +
then $  D ( A _ {0} ) $
 +
is dense in $  X _ {0} $.  
 +
If there are no non-zero elements in $  X _ {0} $
 +
such that $  T ( t) x \equiv 0 $,  
 +
then the generating operator $  A $
 +
exists. In the sequel it will be assumed that $  X _ {0} = X $
 +
and that $  T ( t) x \equiv 0 $
 +
implies $  x = 0 $.
  
The simplest class of semi-groups, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414030.png" />, is defined by the condition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414031.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414032.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414033.png" />. This is equivalent to the condition: The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414034.png" /> is bounded on any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414035.png" />. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414036.png" /> has a generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414037.png" /> whose [[Resolvent|resolvent]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414038.png" /> satisfies the inequalities
+
The simplest class of semi-groups, denoted by $  C _ {0} $,  
 +
is defined by the condition: $  T ( t) x \rightarrow x $
 +
as $  t \rightarrow 0 $
 +
for any $  x \in X $.  
 +
This is equivalent to the condition: The function $  \| T ( t) \| $
 +
is bounded on any interval $  ( 0 , a ] $.  
 +
In that case $  T ( t) $
 +
has a generating operator $  A = A _ {0} $
 +
whose [[Resolvent|resolvent]] $  R ( \lambda , A ) = ( A - \lambda I )  ^ {-1} $
 +
satisfies the inequalities
  
<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/s084/s084140/s08414039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\| R  ^ {n} ( \lambda , A ) \|  \leq  M ( \lambda - \omega )  ^ {-} n ,\ \
 +
n = 1 , 2 , . . . ; \  \lambda > \omega ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414040.png" /> is the type of the semi-group. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414041.png" /> is a [[Closed operator|closed operator]] with domain of definition dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414042.png" /> and with a resolvent satisfying (1), then it is the generating operator of some semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414043.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414045.png" />. Condition (1) is satisfied if
+
where $  \omega $
 +
is the type of the semi-group. Conversely, if $  A $
 +
is a [[Closed operator|closed operator]] with domain of definition dense in $  X $
 +
and with a resolvent satisfying (1), then it is the generating operator of some semi-group $  T ( t) $
 +
of class $  C _ {0} $
 +
such that $  \| T ( t) \| \leq  M e  ^ {\omega t } $.  
 +
Condition (1) is satisfied if
  
<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/s084/s084140/s08414046.png" /></td> </tr></table>
+
$$
 +
\| R ( \lambda , A ) \|  \leq  ( \lambda - \omega )  ^ {-1}
 +
$$
  
(the Hill–Yosida condition). If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414047.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414048.png" /> is a [[Contraction semi-group|contraction semi-group]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414049.png" />.
+
(the Hill–Yosida condition). If, moreover, $  \omega = 0 $,  
 +
then $  T ( t) $
 +
is a [[Contraction semi-group|contraction semi-group]]: $  \| T ( t) \| \leq  1 $.
  
A summable semi-group is a semi-group for which the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414050.png" /> are summable on any finite interval for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414051.png" />. A summable semi-group has a generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414052.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414053.png" /> is closed if and only if, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414054.png" />,
+
A summable semi-group is a semi-group for which the functions $  \| T ( t) x \| $
 +
are summable on any finite interval for all $  x \in X $.  
 +
A summable semi-group has a generating operator $  A = \overline{ {A _ {0} }}\; $.  
 +
The operator $  A _ {0} $
 +
is closed if and only if, for every $  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/s084/s084140/s08414055.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow 0 } 
 +
\frac{1}{t}
 +
\int\limits _ { 0 } ^ { t }  T ( s) x  d s  = x .
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414056.png" /> one can define the Laplace transform of a summable semi-group,
+
For $  \mathop{\rm Re}  \lambda > \omega $
 +
one can define the Laplace transform of a summable semi-group,
  
<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/s084/s084140/s08414057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { 0 } ^  \infty  e ^ {- \lambda t } T ( t)
 +
x  d t  = - R ( \lambda ) x ,
 +
$$
  
giving a bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414058.png" /> which has many properties of a resolvent operator.
+
giving a bounded linear operator $  R ( \lambda ) $
 +
which has many properties of a resolvent operator.
  
A closed operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414059.png" /> with domain of definition dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414060.png" /> is the generating operator of a summable semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414061.png" /> if and only if, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414062.png" />, the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414063.png" /> exists for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414064.png" /> and the following conditions hold: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414066.png" />; b) there exist a non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414069.png" />, jointly continuous in all its variables, and a non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414070.png" />, bounded on any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414071.png" />, such that, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414072.png" />,
+
A closed operator $  A $
 +
with domain of definition dense in $  X $
 +
is the generating operator of a summable semi-group $  T ( t) $
 +
if and only if, for some $  \omega $,  
 +
the resolvent $  R ( \lambda , A ) $
 +
exists for $  \mathop{\rm Re}  \lambda > \omega $
 +
and the following conditions hold: a) $  \| R ( \lambda , A ) \| \leq  M $,  
 +
$  \mathop{\rm Re}  \lambda > \omega $;  
 +
b) there exist a non-negative function $  \phi ( t , x ) $,
 +
$  t > 0 $,  
 +
$  x \in X $,  
 +
jointly continuous in all its variables, and a non-negative function $  \phi ( t) $,  
 +
bounded on any interval $  [ a , b ] \subset  ( 0 , \infty ) $,
 +
such that, for $  \omega _ {1} > \omega $,
  
<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/s084/s084140/s08414073.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty  e ^ {- \omega _ {1} t } \phi ( t , x )  dt  < \infty ,
 +
$$
  
<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/s084/s084140/s08414074.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\; _ {t \rightarrow \infty }  t  ^ {-1}  \mathop{\rm ln}  \phi ( t)  < \infty
 +
,\  \phi ( t , x )  \leq  \phi ( t)  \| x \| ,
 +
$$
  
<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/s084/s084140/s08414075.png" /></td> </tr></table>
+
$$
 +
\| R  ^ {n} ( \lambda , A ) x \|  \leq 
 +
\frac{1}{
 +
( n - 1 ) ! }
 +
\int\limits _ { 0 } ^  \infty  t  ^ {n-1} e ^ {- \lambda t } \phi ( t , x )  dt .
 +
$$
  
 
Under these conditions
 
Under these conditions
  
<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/s084/s084140/s08414076.png" /></td> </tr></table>
+
$$
 +
\| T ( t) x \|  \leq  \phi ( t , x ) ,\ \
 +
\| T ( t) \|  \leq  \phi ( t) .
 +
$$
  
If one requires in addition that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414077.png" /> be summable on finite intervals, a necessary and sufficient condition is the existence of a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414078.png" /> such that, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414079.png" />,
+
If one requires in addition that the function $  \| T ( t) \| $
 +
be summable on finite intervals, a necessary and sufficient condition is the existence of a continuous function $  \phi ( t) $
 +
such that, for $  \omega _ {1} > \omega $,
  
<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/s084/s084140/s08414080.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\int\limits _ { 0 } ^  \infty  \phi ( t) e ^ {- \omega _ {1} t }  dt  < \infty ,
 +
$$
  
<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/s084/s084140/s08414081.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\| R  ^ {n} ( \lambda , A ) \|  \leq 
 +
\frac{1}{( n - 1
 +
) ! }
 +
\int\limits _ { 0 } ^  \infty  t  ^ {n-1} e ^ {- \lambda t } \phi ( t) dt ,
 +
$$
  
<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/s084/s084140/s08414082.png" /></td> </tr></table>
+
$$
 +
\lambda  > \omega ,\  n= 1 , 2 , .  . . .
 +
$$
  
Under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414083.png" />. By choosing different functions satisfying (3), one can define different subclasses of summable semi-groups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414084.png" />, the result is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414085.png" /> and (1) follows from (4). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414087.png" />, condition (4) implies the condition
+
Under these conditions, $  \| T ( t) \| \leq  \phi ( t) $.  
 +
By choosing different functions satisfying (3), one can define different subclasses of summable semi-groups. If $  \phi ( t) = Me ^ {\omega t } $,  
 +
the result is the class $  C _ {0} $
 +
and (1) follows from (4). If $  \phi ( t) = Mt ^ {- \alpha } e ^ {\omega t } $,
 +
0 \leq  \alpha < 1 $,  
 +
condition (4) implies the condition
  
<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/s084/s084140/s08414088.png" /></td> </tr></table>
+
$$
 +
\| R  ^ {n} ( \lambda , A ) \|  \leq 
 +
\frac{M \Gamma ( n - \alpha ) }{( n - 1 ) ! ( \lambda - \omega ) ^ {n - \alpha } }
 +
,\ \
 +
\lambda > \omega ,\  n = 1 , 2 ,\dots.
 +
$$
  
 
==Semi-groups with power singularities.==
 
==Semi-groups with power singularities.==
If in the previous example <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414089.png" />, then the integrals in (4) are divergent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414090.png" />. Hence the generating operator for the corresponding semi-group may not have a resolvent for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414091.png" />, i.e. it may have a spectrum equal to the entire complex plane. However, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414092.png" /> large enough one can define for such operators functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414093.png" /> which coincide with the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414094.png" /> in the previous cases. The operator function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414095.png" /> is called a resolvent of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414097.png" /> if it is analytic in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414098.png" /> and if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s08414099.png" />,
+
If in the previous example $  \alpha \geq  1 $,  
 +
then the integrals in (4) are divergent for $  n \leq  \alpha - 1 $.  
 +
Hence the generating operator for the corresponding semi-group may not have a resolvent for any $  \lambda $,  
 +
i.e. it may have a spectrum equal to the entire complex plane. However, for $  n $
 +
large enough one can define for such operators functions $  S _ {n} ( \lambda , A ) $
 +
which coincide with the functions $  R  ^ {n+1} ( \lambda , A ) $
 +
in the previous cases. The operator function $  S _ {n} ( \lambda , A ) $
 +
is called a resolvent of order $  n $
 +
if it is analytic in some domain $  G \subset  \mathbf C $
 +
and if for $  \lambda \in G $,
 +
 
 +
$$
 +
S _ {n} ( \lambda , A ) Ax  = A S _ {n} ( \lambda , A ) x ,\ \
 +
x \in D ( A) ,
 +
$$
  
<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/s084/s084140/s084140100.png" /></td> </tr></table>
+
$$
 +
S _ {n} ( \lambda , A ) ( A - \lambda )  ^ {n+1} x  = x ,\  x \in D ( A  ^ {n+1} ) ,
 +
$$
  
<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/s084/s084140/s084140101.png" /></td> </tr></table>
+
and if  $  S _ {n} ( \lambda , A ) x = 0 $
 +
for all  $  \lambda \in G $
 +
implies  $  x = 0 $.
 +
If  $  \overline{D}\; ( A  ^ {n+1} ) = X $,
 +
the operator may have a unique resolvent of order  $  n $,
 +
for which there is a maximal domain of analyticity, known as the resolvent set of order  $  n $.
  
and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140102.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140103.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140104.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140105.png" />, the operator may have a unique resolvent of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140106.png" />, for which there is a maximal domain of analyticity, known as the resolvent set of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140108.png" />.
+
Let  $  T ( t) $
 +
be a strongly-continuous semi-group such that the inequality
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140109.png" /> be a strongly-continuous semi-group such that the inequality
+
$$
 +
\| T ( t) \|  \leq  M t ^ {- \alpha } e ^ {\omega t }
 +
$$
  
<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/s084/s084140/s084140110.png" /></td> </tr></table>
+
holds for  $  \alpha \geq  1 $.
 +
Then its generating operator  $  B $
 +
has a resolvent of order  $  n $
 +
for  $  n > \alpha - 1 $,
 +
and, moreover,
  
holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140111.png" />. Then its generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140112.png" /> has a resolvent of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140113.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140114.png" />, and, moreover,
+
$$
 +
S _ {n} ( \lambda , B ) 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/s084/s084140/s084140115.png" /></td> </tr></table>
+
\frac{1}{n!}
 +
\int\limits _ { 0 } ^  \infty  t  ^ {n} e ^ {- \lambda t } T
 +
( t) x  dt ,\  \mathop{\rm Re}  \lambda > \omega ,
 +
$$
  
<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/s084/s084140/s084140116.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
\left \|
 +
\frac{d  ^ {k} S _ {n} ( \lambda , B ) }{d \lambda  ^ {k} }
 +
x \right \|  \leq 
 +
\frac{M \Gamma ( k + n + 1 - \alpha ) }{n
 +
! ( \mathop{\rm Re}  \lambda - \omega ) ^ {k + n + 1 - \alpha } }
 +
,
 +
$$
  
<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/s084/s084140/s084140117.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Re}  \lambda  > \omega ,\  k  = 0 , 1 ,\dots .
 +
$$
  
Conversely, suppose that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140118.png" /> the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140119.png" /> has a resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140120.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140121.png" /> satisfying (5) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140122.png" />. Then there exists a unique semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140123.png" /> such that
+
Conversely, suppose that for $  \mathop{\rm Re}  \lambda > 0 $
 +
the operator $  B $
 +
has a resolvent $  S _ {n} ( \lambda , B ) $
 +
of order $  n $
 +
satisfying (5) with $  n > \alpha - 1 $.  
 +
Then there exists a unique semi-group $  T ( t) $
 +
such that
  
<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/s084/s084140/s084140124.png" /></td> </tr></table>
+
$$
 +
\| T ( t) \|  \leq  M t ^ {- \alpha } e ^ {\omega t } ,
 +
$$
  
and the generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140125.png" /> of this semi-group is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140126.png" />.
+
and the generating operator $  A $
 +
of this semi-group is such that $  S _ {n} ( \lambda , A ) = S _ {n} ( \lambda , B ) $.
  
 
==Smooth semi-groups.==
 
==Smooth semi-groups.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140127.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140128.png" /> is continuously differentiable and
+
If $  x \in D ( A _ {0} ) $,  
 +
the function $  T ( t) x $
 +
is continuously differentiable and
 +
 
 +
$$
  
<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/s084/s084140/s084140129.png" /></td> </tr></table>
+
\frac{d T ( t) }{dt}
 +
= A _ {0} T
 +
( t) x  = T ( t) A _ {0} x .
 +
$$
  
There exist semi-groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140130.png" /> such that, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140131.png" />, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140132.png" /> are non-differentiable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140133.png" />. However, there are important classes of semi-groups for which the degree of smoothness increases with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140134.png" />. If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140136.png" />, are differentiable for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140137.png" />, then it follows from the semi-group property that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140138.png" /> are twice differentiable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140139.png" />, three times differentiable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140140.png" />, etc. Therefore, if these functions are differentiable at any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140141.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140142.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140143.png" /> is infinitely differentiable.
+
There exist semi-groups of class $  C _ {0} $
 +
such that, if $  x \notin D ( A _ {0} ) = D ( A) $,  
 +
the functions $  T ( t) x $
 +
are non-differentiable for all $  t $.  
 +
However, there are important classes of semi-groups for which the degree of smoothness increases with increasing $  t $.  
 +
If the functions $  T ( t) x $,
 +
$  t > t _ {0} $,  
 +
are differentiable for any $  x \in X $,  
 +
then it follows from the semi-group property that the $  T ( t) x $
 +
are twice differentiable if $  t > 2 t _ {0} $,  
 +
three times differentiable if $  t > 3 t _ {0} $,  
 +
etc. Therefore, if these functions are differentiable at any $  t > 0 $
 +
for $  x \in X $,  
 +
then $  T ( t) x $
 +
is infinitely differentiable.
  
Given a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140144.png" />, a necessary and sufficient condition for the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140145.png" /> to be differentiable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140146.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140147.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140148.png" />, is that there exist numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140149.png" /> such that the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140150.png" /> is defined in the domain
+
Given a semi-group of class $  C _ {0} $,  
 +
a necessary and sufficient condition for the functions $  T ( t) x $
 +
to be differentiable for all $  x \in X $
 +
and  $  t > t _ {0} $,  
 +
where $  t _ {0} \geq  0 $,  
 +
is that there exist numbers $  a , b , c > 0 $
 +
such that the resolvent $  R ( \lambda , A ) $
 +
is defined in the domain
  
<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/s084/s084140/s084140151.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Re}  \lambda  > a - b  \mathop{\rm ln}  |  \mathop{\rm Im}  \lambda | ,
 +
$$
  
 
while in this domain
 
while in this domain
  
<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/s084/s084140/s084140152.png" /></td> </tr></table>
+
$$
 +
\| R( \lambda , A ) \|  \leq  c  |  \mathop{\rm Im}  \lambda | .
 +
$$
  
A necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140153.png" /> to be infinitely differentiable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140155.png" /> is that, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140156.png" />, there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140157.png" /> such that the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140158.png" /> is defined in the domain
+
A necessary and sufficient condition for $  T ( t) x $
 +
to be infinitely differentiable for all $  x \in X $
 +
and  $  t > 0 $
 +
is that, for every $  b > 0 $,  
 +
there exist $  a _ {b} , c _ {b} $
 +
such that the resolvent $  R ( \lambda , A ) $
 +
is defined in the domain
  
<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/s084/s084140/s084140159.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Re}  \lambda  > a _ {b} - b  \mathop{\rm ln}  |  \mathop{\rm Im}  \lambda | ,
 +
$$
  
 
and such that
 
and such that
  
<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/s084/s084140/s084140160.png" /></td> </tr></table>
+
$$
 +
\| R ( \lambda , A ) \|  \leq  c _ {b}  |  \mathop{\rm Im}  \lambda | .
 +
$$
  
Sufficient conditions are: If there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140161.png" /> for which
+
Sufficient conditions are: If there exists a $  \mu > \omega $
 +
for which
  
<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/s084/s084140/s084140162.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\; _ {\tau \rightarrow \infty }  \mathop{\rm ln}  | \tau |  \| R ( \mu + i \tau , A ) \|  = t _ {0< \infty ,
 +
$$
  
then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140163.png" /> are differentiable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140164.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140165.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140166.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140167.png" /> are infinitely differentiable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140168.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140169.png" />.
+
then the $  T ( t) x $
 +
are differentiable for $  t > t _ {0} $
 +
and $  x \in X $;  
 +
if $  t _ {0} = 0 $,  
 +
then the $  T ( t) x $
 +
are infinitely differentiable for all $  t > 0 $
 +
and $  x \in X $.
  
The degree of smoothness of a semi-group may sometimes be inferred from its behaviour at zero; for example, suppose that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140170.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140171.png" /> such that, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140172.png" />,
+
The degree of smoothness of a semi-group may sometimes be inferred from its behaviour at zero; for example, suppose that for every $  c > 0 $
 +
there exists a $  \delta _ {c} $
 +
such that, for $  0 < t < \delta _ {c} $,
  
<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/s084/s084140/s084140173.png" /></td> </tr></table>
+
$$
 +
\| I - T ( t) \|  \leq  2 - ct  \mathop{\rm ln}  t  ^ {-1} ,
 +
$$
  
then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140174.png" /> are infinitely differentiable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140175.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140176.png" />.
+
then the $  T ( t) x $
 +
are infinitely differentiable for all $  t > 0 $,  
 +
$  x \in X $.
  
There are smoothness conditions for summable semi-groups and semi-groups of polynomial growth. If a semi-group has polynomial growth of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140177.png" /> and is infinitely differentiable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140178.png" />, then the function
+
There are smoothness conditions for summable semi-groups and semi-groups of polynomial growth. If a semi-group has polynomial growth of degree $  \alpha $
 +
and is infinitely differentiable for $  t > 0 $,  
 +
then the function
  
<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/s084/s084140/s084140179.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{d T ( t) }{dt}
 +
= A T ( t) x
 +
$$
  
 
also has polynomial growth:
 
also has polynomial growth:
  
<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/s084/s084140/s084140180.png" /></td> </tr></table>
+
$$
 +
\| A T ( t) \|  \leq  M _ {1} t ^ {- \beta } e ^ {\omega t } .
 +
$$
  
In the general case there is no rigorous relationship between the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140181.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140182.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140183.png" /> can be utilized for a more detailed classification of infinitely-differentiable semi-groups of polynomial growth.
+
In the general case there is no rigorous relationship between the numbers $  \alpha $
 +
and $  \beta $,  
 +
and $  \beta $
 +
can be utilized for a more detailed classification of infinitely-differentiable semi-groups of polynomial growth.
  
 
==Analytic semi-groups.==
 
==Analytic semi-groups.==
An important class of semi-groups, related to partial differential equations of parabolic type, comprises those semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140184.png" /> which admit an analytic continuation to some sector of the complex plane containing the positive real axis. A semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140185.png" /> has this property if and only if its resolvent satisfies the following inequality in some right half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140186.png" />:
+
An important class of semi-groups, related to partial differential equations of parabolic type, comprises those semi-groups $  T ( t) $
 +
which admit an analytic continuation to some sector of the complex plane containing the positive real axis. A semi-group of class $  C _ {0} $
 +
has this property if and only if its resolvent satisfies the following inequality in some right half-plane $  \mathop{\rm Re}  \lambda > \omega $:
  
<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/s084/s084140/s084140187.png" /></td> </tr></table>
+
$$
 +
\| R ( \lambda , A ) \|  \leq  M  | \lambda - \omega |  ^ {-1} .
 +
$$
  
 
Another necessary and sufficient conditions is: The semi-group is strongly differentiable and its derivative satisfies the estimate
 
Another necessary and sufficient conditions is: The semi-group is strongly differentiable and its derivative satisfies the estimate
  
<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/s084/s084140/s084140188.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\frac{d T }{dt}
 +
( t) \right \|  \leq  M t  ^ {-1} e ^ {\omega t } .
 +
$$
  
 
Finally, the inequality
 
Finally, the inequality
  
<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/s084/s084140/s084140189.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\; _ {t \rightarrow 0 }  \| I - T ( t) \|  < 2
 +
$$
  
is also a sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140190.png" /> to be analytic.
+
is also a sufficient condition for $  T ( t) $
 +
to be analytic.
  
If a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140191.png" /> has an analytic continuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140192.png" /> to a sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140193.png" /> and has polynomial growth at zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140194.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140195.png" />, then the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140196.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140197.png" /> of its generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140198.png" /> has an analytic continuation to the sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140199.png" />, and satisfies the following estimate in any sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140200.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140201.png" />:
+
If a semi-group $  T ( t) $
 +
has an analytic continuation $  T ( z) $
 +
to a sector $  |  \mathop{\rm arg}  z | < \phi \leq  \pi / 2 $
 +
and has polynomial growth at zero, $  \| T ( z) \| \leq  c  | z |  ^  \alpha  $,
 +
$  \alpha > 0 $,  
 +
then the resolvent $  S _ {n} ( \lambda , A ) $
 +
of order $  n > \alpha - 1 $
 +
of its generating operator $  A $
 +
has an analytic continuation to the sector $  |  \mathop{\rm arg}  \lambda | \leq  \pi / 2 + \phi $,  
 +
and satisfies the following estimate in any sector $  |  \mathop{\rm arg}  \lambda | \leq  \pi / 2 + \psi $,  
 +
$  \psi < \phi $:
  
<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/s084/s084140/s084140202.png" /></td> </tr></table>
+
$$
 +
\| S _ {n} ( \lambda , A ) \|  \leq  | \lambda | ^
 +
{\alpha - n - 1 } M ( \psi ) .
 +
$$
  
Conversely, suppose that the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140203.png" /> of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140204.png" /> is defined in a sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140205.png" /> and that
+
Conversely, suppose that the resolvent $  S _ {n} ( \lambda , B ) $
 +
of an operator $  B $
 +
is defined in a sector $  |  \mathop{\rm arg}  \lambda | \leq  \pi / 2 + \psi $
 +
and that
  
<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/s084/s084140/s084140206.png" /></td> </tr></table>
+
$$
 +
\| S _ {n} ( \lambda , B ) \|  \leq  \lambda ^ {\alpha - n - 1 } M .
 +
$$
  
Then there exists a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140207.png" /> of growth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140208.png" />, analytic in the sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140209.png" />, whose generating operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140210.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140211.png" />.
+
Then there exists a semi-group $  T ( z) $
 +
of growth $  \alpha $,  
 +
analytic in the sector $  |  \mathop{\rm arg}  z | < \psi $,  
 +
whose generating operator $  A $
 +
is such that $  S _ {n} ( \lambda , A ) = S _ {n} ( \lambda , B ) $.
  
 
==Distribution semi-groups.==
 
==Distribution semi-groups.==
In accordance with the general concept of the theory of distributions (cf. [[Generalized function|Generalized function]]), one can drop the requirement that the operator-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140212.png" /> be defined for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140213.png" />, demanding only that it be possible to evaluate the integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140214.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140215.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140216.png" /> of infinitely-differentiable functions with compact support. Hence the following definition: A distribution semi-group on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140217.png" /> is a continuous linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140218.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140219.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140220.png" /> of all bounded linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140221.png" />, with the following properties: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140222.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140223.png" />; b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140224.png" /> are functions in the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140225.png" /> of all functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140226.png" /> with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140227.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140228.png" />, where the star denotes convolution:
+
In accordance with the general concept of the theory of distributions (cf. [[Generalized function|Generalized function]]), one can drop the requirement that the operator-valued function $  T ( t) $
 +
be defined for every $  t > 0 $,  
 +
demanding only that it be possible to evaluate the integrals $  \int _ {- \infty }  ^ {+ \infty } T ( t) \phi ( t)  dt $
 +
for all $  \phi $
 +
in the space $  D ( \mathbf R ) $
 +
of infinitely-differentiable functions with compact support. Hence the following definition: A distribution semi-group on a Banach space $  X $
 +
is a continuous linear mapping $  T ( \phi ) $
 +
of $  D ( \mathbf R ) $
 +
into the space $  L ( X) $
 +
of all bounded linear operators on $  X $,  
 +
with the following properties: a) $  T ( \phi ) = 0 $
 +
if $  \supp  \phi \subset  ( - \infty , 0 ) $;  
 +
b) if $  \phi , \psi $
 +
are functions in the subspace $  D  ^ + ( \mathbf R ) $
 +
of all functions in $  D ( \mathbf R ) $
 +
with support in $  ( 0 , \infty ) $,
 +
then $  T ( \phi * \psi ) = T ( \phi ) T ( \psi ) $,  
 +
where the star denotes convolution:
  
<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/s084/s084140/s084140229.png" /></td> </tr></table>
+
$$
 +
\phi * \psi  = \int\limits _ {- \infty } ^  \infty  \phi ( t - s ) \psi ( s)  d s
 +
$$
  
(the semi-group property); c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140230.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140231.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140232.png" />; d) the linear hull of the set of all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140233.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140234.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140235.png" />, is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140236.png" />; e) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140237.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140238.png" />, there exists a continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140239.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140240.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140241.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140242.png" /> and
+
(the semi-group property); c) if $  T ( \phi ) x = 0 $
 +
for all $  \phi \in D  ^ + ( \mathbf R ) $,  
 +
then $  x = 0 $;  
 +
d) the linear hull of the set of all values of $  T ( \phi ) x $,  
 +
$  \phi \in D  ^ + ( \mathbf R ) $,  
 +
$  x \in X $,  
 +
is dense in $  X $;  
 +
e) for any $  y = T ( \psi ) x $,  
 +
$  \psi \in D  ^ + ( \mathbf R ) $,  
 +
there exists a continuous $  u ( t) $
 +
on $  ( 0 , \infty ) $
 +
with values in $  X $,  
 +
so that $  u( 0) = y $
 +
and
  
<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/s084/s084140/s084140243.png" /></td> </tr></table>
+
$$
 +
T ( \phi ) y  = \int\limits _ { 0 } ^  \infty  \phi ( t) u ( t)  dt
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140244.png" />.
+
for all $  \phi \in D ( \mathbf R ) $.
  
The infinitesimal operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140245.png" /> of a distribution semi-group is defined as follows. If there exists a delta-sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140246.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140247.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140248.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140249.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140250.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140251.png" />. The infinitesimal operator has a closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140252.png" />, known as the infinitesimal generator of the distribution semi-group. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140253.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140254.png" /> and contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140255.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140256.png" />.
+
The infinitesimal operator $  A _ {0} $
 +
of a distribution semi-group is defined as follows. If there exists a delta-sequence $  \{ \rho _ {n} \} \subset  D  ^ + ( \mathbf R ) $
 +
such that $  T ( \rho _ {n} ) x \rightarrow x $
 +
and $  T ( - \rho _ {n}  ^  \prime  ) x \rightarrow y $
 +
as $  n \rightarrow \infty $,  
 +
then $  x \in D ( A _ {0} ) $
 +
and $  y = A _ {0} x $.  
 +
The infinitesimal operator has a closure $  A = \overline{ {A _ {0} }}\; $,  
 +
known as the infinitesimal generator of the distribution semi-group. The set $  \cap_{n=1}^  \infty  D ( A _ {0}  ^ {n} ) $
 +
is dense in $  X $
 +
and contains $  T ( \phi ) X $
 +
for any $  \phi \in D  ^ + ( \mathbf R ) $.
  
A closed linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140257.png" /> with a dense domain of definition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140258.png" /> is the infinitesimal generator of a distribution semi-group if and only if there exist numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140259.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140260.png" /> and a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140261.png" /> such that the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140262.png" /> exists for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140263.png" /> and satisfies the inequality
+
A closed linear operator $  A $
 +
with a dense domain of definition in $  X $
 +
is the infinitesimal generator of a distribution semi-group if and only if there exist numbers $  a , b \geq  0 $,
 +
$  c > 0 $
 +
and a natural number $  m $
 +
such that the resolvent $  R ( \lambda , A ) $
 +
exists for $  \mathop{\rm Re}  \lambda \geq  a  \mathop{\rm ln}  ( 1 + | \lambda | ) + b $
 +
and satisfies the inequality
  
<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/s084/s084140/s084140264.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
\| R ( \lambda , A ) \|  \leq  c ( 1 + | \lambda | ) ^ {m} .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140265.png" /> is a closed linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140266.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140267.png" /> can be made into a [[Fréchet space|Fréchet space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140268.png" /> by introducing the system of norms
+
If $  A $
 +
is a closed linear operator on $  X $,  
 +
then the set $  \cap_{n=1}^  \infty  D ( A  ^ {n} ) $
 +
can be made into a [[Fréchet space|Fréchet space]] $  X _  \infty  $
 +
by introducing the system of norms
  
<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/s084/s084140/s084140269.png" /></td> </tr></table>
+
$$
 +
\| x \| _ {n} = \sum_{k=0}^n  \| A  ^ {k} x \| .
 +
$$
  
The restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140270.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140271.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140272.png" /> leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140273.png" /> invariant. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140274.png" /> is the infinitesimal generator of a semi-group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140275.png" /> is the infinitesimal generator of a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140276.png" /> (continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140277.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140278.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140279.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140280.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140281.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140282.png" /> has a non-empty resolvent set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140283.png" /> is the infinitesimal generator of a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140284.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140285.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140286.png" /> is the infinitesimal generator of a distribution semi-group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140287.png" />.
+
The restriction $  A _  \infty  $
 +
of $  A $
 +
to $  X _  \infty  $
 +
leaves $  X _  \infty  $
 +
invariant. If $  A $
 +
is the infinitesimal generator of a semi-group, then $  A _  \infty  $
 +
is the infinitesimal generator of a semi-group of class $  C _ {0} $(
 +
continuous for $  t \geq  0 $,  
 +
$  T ( 0 ) = I $)  
 +
on $  X _  \infty  $.  
 +
Conversely, if $  X _  \infty  $
 +
is dense in $  X $,  
 +
the operator $  A $
 +
has a non-empty resolvent set and $  A $
 +
is the infinitesimal generator of a semi-group of class $  C _ {0} $
 +
on $  X _  \infty  $,  
 +
then $  A $
 +
is the infinitesimal generator of a distribution semi-group on $  X $.
  
A distribution semi-group has exponential growth of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140288.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140289.png" />, if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140290.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140291.png" /> is a continuous mapping in the topology induced on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140292.png" /> by the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140293.png" /> of rapidly-decreasing functions. A closed linear operator is the infinitesimal generator of a distribution semi-group with the above property if and only if it has a resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140294.png" /> which satisfies (6) in the domain
+
A distribution semi-group has exponential growth of order at most $  q $,  
 +
$  1 \leq  q < \infty $,  
 +
if there exists an $  \omega > 0 $
 +
such that $  \mathop{\rm exp} ( - \omega t  ^ {q} ) T ( \phi ) $
 +
is a continuous mapping in the topology induced on $  D  ^ + $
 +
by the space $  S ( \mathbf R ) $
 +
of rapidly-decreasing functions. A closed linear operator is the infinitesimal generator of a distribution semi-group with the above property if and only if it has a resolvent $  R ( \lambda , A ) $
 +
which satisfies (6) in the domain
  
<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/s084/s084140/s084140295.png" /></td> </tr></table>
+
$$
 +
\{  \lambda  : { \mathop{\rm Re}  \lambda  \geq  [ \alpha  \mathop{\rm ln}  ( 1 + |  \mathop{\rm Im} \
 +
\lambda | + \beta ) ] ^ {1- 1/q } ,  \mathop{\rm Re}  \lambda > \omega } \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140296.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140297.png" /> the semi-group is said to be exponential and inequality (6) is valid in some half-plane. There exists a characterization of the semi-groups of the above types in terms of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140298.png" />. Questions of smoothness and analyticity have also been investigated for distribution semi-groups.
+
where $  \alpha , \beta > 0 $.  
 +
In particular, if $  q = 1 $
 +
the semi-group is said to be exponential and inequality (6) is valid in some half-plane. There exists a characterization of the semi-groups of the above types in terms of the operator $  A _  \infty  $.  
 +
Questions of smoothness and analyticity have also been investigated for distribution semi-groups.
  
==Semi-groups of operators in a (separable) locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140299.png" />.==
+
==Semi-groups of operators in a (separable) locally convex space $  X $.==
The definition of a strongly-continuous semi-group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140300.png" /> continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140301.png" /> remains the same as for a Banach space. Similarly, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140302.png" /> is defined by the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140303.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140304.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140305.png" />. A semi-group is said to be locally equicontinuous (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140308.png" />) if the family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140309.png" /> is equicontinuous when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140310.png" /> ranges over any finite interval in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140311.png" />. In a [[Barrelled space|barrelled space]], a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140312.png" /> is always equicontinuous (cf. [[Equicontinuity|Equicontinuity]]).
+
The definition of a strongly-continuous semi-group of operators $  T ( t ) $
 +
continuous on $  X $
 +
remains the same as for a Banach space. Similarly, the class $  C _ {0} $
 +
is defined by the property $  T ( t ) x \rightarrow x $
 +
as $  t \rightarrow 0 $
 +
for any $  x \in X $.  
 +
A semi-group is said to be locally equicontinuous (of class $  lC _ {0} $)  
 +
if the family of operators $  T ( t ) $
 +
is equicontinuous when $  t $
 +
ranges over any finite interval in $  ( 0 , \infty ) $.  
 +
In a [[Barrelled space|barrelled space]], a semi-group of class $  C _ {0} $
 +
is always equicontinuous (cf. [[Equicontinuity|Equicontinuity]]).
  
A semi-group is said to be equicontinuous (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140315.png" />) if the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140316.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140317.png" />, is equicontinuous.
+
A semi-group is said to be equicontinuous (of class $  uC _ {0} $)  
 +
if the family $  T ( t ) $,
 +
0 \leq  t < \infty $,  
 +
is equicontinuous.
  
 
Infinitesimal operators and infinitesimal generators are defined as in the Banach space case.
 
Infinitesimal operators and infinitesimal generators are defined as in the Banach space case.
  
Assume from now on that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140318.png" /> is sequentially complete. The infinitesimal generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140319.png" /> of a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140320.png" /> is identical to the infinitesimal operator; its domain of definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140321.png" />, is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140322.png" /> and, moreover, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140323.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140324.png" />. The semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140325.png" /> leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140326.png" /> invariant and
+
Assume from now on that the space $  X $
 +
is sequentially complete. The infinitesimal generator $  A $
 +
of a semi-group of class $  l C _ {0} $
 +
is identical to the infinitesimal operator; its domain of definition, $  D ( A) $,  
 +
is dense in $  X $
 +
and, moreover, the set $  \cap_{n=1}^  \infty  D ( A  ^ {n} ) $
 +
is dense in $  X $.  
 +
The semi-group $  T ( t ) $
 +
leaves $  D ( A) $
 +
invariant and
 +
 
 +
$$
  
<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/s084/s084140/s084140327.png" /></td> </tr></table>
+
\frac{dT}{dt}
 +
( t) x  = \
 +
AT ( t ) x  = T ( t ) Ax ,\  0 \leq  t < \infty ,\  x \in D ( A ) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140328.png" /> is the infinitesimal generator of a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140329.png" />, the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140330.png" /> is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140331.png" /> and is the Laplace transform of the semi-group.
+
If $  A $
 +
is the infinitesimal generator of a semi-group of class $  u C _ {0} $,  
 +
the resolvent $  R ( \lambda , A ) $
 +
is defined for $  \mathop{\rm Re}  \lambda > 0 $
 +
and is the Laplace transform of the semi-group.
  
A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140332.png" /> is the infinitesimal generator of a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140333.png" /> if and only if it is closed, has dense domain of definition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140334.png" />, and if there exists a sequence of positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140335.png" /> such that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140336.png" />, the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140337.png" /> is defined and the family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140338.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140339.png" /> is equicontinuous. In this situation the semi-group can be constructed by the formula
+
A linear operator $  A $
 +
is the infinitesimal generator of a semi-group of class $  u C _ {0} $
 +
if and only if it is closed, has dense domain of definition in $  X $,  
 +
and if there exists a sequence of positive numbers $  \lambda _ {k} \rightarrow \infty $
 +
such that, for any $  \lambda _ {k} $,  
 +
the resolvent $  R ( \lambda _ {k} , A ) $
 +
is defined and the family of operators $  [ \lambda _ {k} R ( \lambda _ {k} , A ) ]  ^ {n} $,
 +
$  k , n = 1 , 2 \dots $
 +
is equicontinuous. In this situation the semi-group can be constructed by the formula
  
<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/s084/s084140/s084140340.png" /></td> </tr></table>
+
$$
 +
( t ) x  = \lim\limits _ {k \rightarrow \infty } \
 +
\left (  \mathop{\rm exp} \left [ - \lambda _ {k} - \lambda _ {k}  ^ {2} R ( \lambda _ {k} , A ) \right ] t \right ) x ,
 +
$$
  
<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/s084/s084140/s084140341.png" /></td> </tr></table>
+
$$
 +
t  \geq  0 ,\  x  \in  X .
 +
$$
  
In a non-normed locally convex space, the infinitesimal generator of a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140342.png" /> may have no resolvent at any point. An example is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140343.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140344.png" /> of infinitely-differentiable functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140345.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140346.png" />. As a substitute for the resolvent one can take a continuous operator whose product with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140347.png" />, from the right and the left, differs by a "small amount" from the identity operator.
+
In a non-normed locally convex space, the infinitesimal generator of a semi-group of class $  lC _ {0} $
 +
may have no resolvent at any point. An example is: $  A = d / ds $
 +
in the space $  C  ^  \infty  $
 +
of infinitely-differentiable functions of s $
 +
on $  \mathbf R $.  
 +
As a substitute for the resolvent one can take a continuous operator whose product with $  A - \lambda I $,  
 +
from the right and the left, differs by a "small amount" from the identity operator.
  
A continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140348.png" /> defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140349.png" /> in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140350.png" /> is called an asymptotic resolvent for a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140351.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140352.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140353.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140354.png" /> can be extended from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140355.png" /> to a continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140356.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140357.png" />, and if there exists a limit point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140358.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140359.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140360.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140361.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140362.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140363.png" />, where
+
A continuous operator $  R ( \lambda ) $
 +
defined for $  \lambda $
 +
in a set $  \Lambda \subset  \mathbf C $
 +
is called an asymptotic resolvent for a linear operator $  A $
 +
if $  AR ( \lambda ) $
 +
is continuous on $  X $,  
 +
the operator $  R ( \lambda ) A $
 +
can be extended from $  D ( A) $
 +
to a continuous operator $  B ( \lambda ) $
 +
on $  X $,  
 +
and if there exists a limit point $  \lambda _ {0} $
 +
of the set $  \Lambda $
 +
such that $  H  ^ + ( \lambda ) x \rightarrow 0 $,  
 +
$  H  ^ {-} ( \lambda ) x \rightarrow 0 $
 +
as $  \lambda \rightarrow \lambda _ {0} $
 +
for any $  x \in X $,  
 +
where
  
<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/s084/s084140/s084140364.png" /></td> </tr></table>
+
$$
 +
H  ^ + ( \lambda )  = ( A - \lambda I ) R ( \lambda ) - I ,\ \
 +
H  ^ {-} ( \lambda )  = B ( \lambda ) - \lambda R ( \lambda ) - I .
 +
$$
  
 
An asymptotic resolvent possesses various properties resembling those of the ordinary resolvent.
 
An asymptotic resolvent possesses various properties resembling those of the ordinary resolvent.
  
A closed linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140365.png" /> with a dense domain of definition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140366.png" /> is the infinitesimal generator of a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140367.png" /> if and only if there exist numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140368.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140369.png" /> such that, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140370.png" />, there exists an asymptotic resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140371.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140372.png" /> with the properties: the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140373.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140374.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140375.png" /> are strongly infinitely differentiable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140376.png" />, and the families of operators
+
A closed linear operator $  A $
 +
with a dense domain of definition in $  X $
 +
is the infinitesimal generator of a semi-group of class $  l C _ {0} $
 +
if and only if there exist numbers $  \omega $
 +
and  $  \alpha > 0 $
 +
such that, for $  \lambda > \omega $,  
 +
there exists an asymptotic resolvent $  R ( \lambda ) $
 +
of $  A $
 +
with the properties: the functions $  R ( \lambda ) $,  
 +
$  H  ^ + ( \lambda ) $,  
 +
$  H  ^ {-} ( \lambda ) $
 +
are strongly infinitely differentiable for $  \lambda > \omega $,  
 +
and the families 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/s084/s084140/s084140377.png" /></td> </tr></table>
+
$$
 +
e ^ {\alpha \lambda }
 +
\frac{d  ^ {n} H  ^  \pm  ( \lambda ) }{d \lambda
 +
^ {n} }
 +
,\ 
 +
\frac{\lambda  ^ {n+1} }{n ! }
 +
 +
\frac{d  ^ {n} R ( \lambda ) }{d \lambda  ^ {n} }
 +
,\  \lambda > \omega ,\  n = 0 , 1 \dots
 +
$$
  
 
are equicontinuous.
 
are equicontinuous.
Line 223: Line 680:
  
 
==Adjoint semi-groups.==
 
==Adjoint semi-groups.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140378.png" /> is a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140379.png" /> on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140380.png" />, then the adjoint operators form a semi-group of bounded operators on the adjoint space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140381.png" />. However, the assertion that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140382.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140383.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140384.png" /> is valid only in the sense of the weak- topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140385.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140386.png" /> is the generating operator, its adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140387.png" /> is a weak infinitesimal generator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140388.png" />, in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140389.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140390.png" /> for which the limit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140391.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140392.png" /> exists in the sense of weak- convergence and is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140393.png" />. The domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140394.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140395.png" /> — again in the sense of the weak- topology — and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140396.png" /> is closed in the weak- topology.
+
If $  T ( t ) $
 +
is a semi-group of class $  C _ {0} $
 +
on a Banach space $  X $,  
 +
then the adjoint operators form a semi-group of bounded operators on the adjoint space $  X  ^  \prime  $.  
 +
However, the assertion that $  T  ^  \prime  ( t ) f \rightarrow f $
 +
as $  t \rightarrow 0 $
 +
for any $  f \in X  ^  \prime  $
 +
is valid only in the sense of the weak- topology $  \sigma ( X  ^  \prime  , X ) $.  
 +
If $  A $
 +
is the generating operator, its adjoint $  A  ^  \prime  $
 +
is a weak infinitesimal generator for $  T  ^  \prime  ( t ) $,  
 +
in the sense that $  D ( A  ^  \prime  ) $
 +
is the set of all $  f $
 +
for which the limit of $  t  ^ {-1} [ T  ^  \prime  ( t ) - I ] f $
 +
as $  t\rightarrow 0 $
 +
exists in the sense of weak- convergence and is equal to $  A  ^  \prime  f $.  
 +
The domain of definition $  D ( A  ^  \prime  ) $
 +
is dense in $  X  ^  \prime  $—  
 +
again in the sense of the weak- topology — and the operator $  A  ^  \prime  $
 +
is closed in the weak- topology.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140397.png" /> be the set of all elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140398.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140399.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140400.png" /> in the strong sense; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140401.png" /> is a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140402.png" /> that is invariant under all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140403.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140404.png" /> the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140405.png" /> form a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140406.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140407.png" /> is also the strong closure of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140408.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140409.png" />. If the original space is reflexive, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140410.png" />. Analogous propositions hold for semi-groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140411.png" /> in locally convex spaces. Semi-groups of classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140412.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140413.png" /> generate semi-groups of the same classes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140414.png" />.
+
Let $  X  ^ + $
 +
be the set of all elements in $  X  ^  \prime  $
 +
such that $  T  ^  \prime  ( f  ) \rightarrow f $
 +
as $  t \rightarrow 0 $
 +
in the strong sense; then $  X  ^ + $
 +
is a closed subspace of $  X  ^  \prime  $
 +
that is invariant under all $  T  ^  \prime  ( t ) $.  
 +
On $  X  ^ + $
 +
the operators $  T  ^  \prime  ( t ) $
 +
form a semi-group of class $  C _ {0} $.  
 +
The space $  X  ^ + $
 +
is also the strong closure of the set $  D ( A  ^  \prime  ) $
 +
in $  X  ^  \prime  $.  
 +
If the original space is reflexive, then $  X  ^ + = X  ^  \prime  $.  
 +
Analogous propositions hold for semi-groups of class $  C _ {0} $
 +
in locally convex spaces. Semi-groups of classes $  l C _ {0} $
 +
and $  u C _ {0} $
 +
generate semi-groups of the same classes in $  X  ^ + $.
  
 
==Distribution semi-groups in a (separable) locally convex space.==
 
==Distribution semi-groups in a (separable) locally convex space.==
A distribution semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140415.png" /> in a sequentially complete locally convex space is defined just as in a Banach space. A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140416.png" /> is said to be locally equicontinuous (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140419.png" />) if, for any compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140420.png" />, the family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140421.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140422.png" />, is equicontinuous. In a barrelled space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140423.png" />, any distribution semi-group is defined by analogy to the Banach case. For semi-groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140424.png" />, the infinitesimal operator is closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140425.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140426.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140427.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140428.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140429.png" />,
+
A distribution semi-group $  T $
 +
in a sequentially complete locally convex space is defined just as in a Banach space. A semi-group $  T $
 +
is said to be locally equicontinuous (of class $  lD  ^  \prime  $)  
 +
if, for any compact subset $  K \subset  D ( \mathbf R ) $,  
 +
the family of operators $  \{ T ( \phi ) \} $,  
 +
$  \phi \in K $,  
 +
is equicontinuous. In a barrelled space $  X $,  
 +
any distribution semi-group is defined by analogy to the Banach case. For semi-groups of class $  l D  ^  \prime  $,  
 +
the infinitesimal operator is closed $  ( A _ {0} = A ) $,  
 +
$  \cap_{n=1}^  \infty  D ( A  ^ {n} ) $
 +
is dense in $  X $,  
 +
and for any $  x \in X $
 +
and $  \phi \in D ( \mathbf R ) $,
  
<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/s084/s084140/s084140430.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
\left .
 +
\begin{array}{c}
  
A generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140431.png" /> with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140432.png" />, possessing the properties (7), is naturally called the fundamental function of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140434.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140435.png" /> is the infinitesimal operator of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140436.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140437.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140438.png" /> is the fundamental function of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140439.png" />. The converse statement is true under certain additional assumptions about the order of singularity of the fundamental function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140440.png" /> (or, more precisely, of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140441.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140442.png" />).
+
{T ( \phi ) x \in D ( A ) ,\
 +
T  ^  \prime  ( \phi ) = AT ( \phi ) x + \phi ( 0 ) x , }
 +
\\
  
A useful notion for the characterization of semi-groups in a locally convex space is that of the generalized resolvent. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140443.png" /> denote the Laplace transform of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140444.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140445.png" /> be the space of all such transforms. A topology is induced in this space, via the Laplace transform, from the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140446.png" />. The Laplace transform of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140447.png" />-valued generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140448.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140449.png" />. Under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140450.png" /> is a continuous mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140451.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140452.png" /> of continuous linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140453.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140454.png" /> be the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140455.png" /> obtained from functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140456.png" /> with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140457.png" />, with the natural topology. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140458.png" /> is a linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140459.png" />, it can be "lifted" to an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140460.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140461.png" /> via the equality
+
{T  ^  \prime  ( \phi ) x  = T ( \phi ) Ax + \phi ( 0 ) x ,\  x \in D ( A ). }
 +
\end{array}
  
<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/s084/s084140/s084140462.png" /></td> </tr></table>
+
\right \}
 +
$$
  
Thus, it is defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140463.png" /> such that the right-hand side of the equality is defined for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140464.png" /> and it extends to a generalized function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140465.png" />. The continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140466.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140467.png" /> is defined by
+
A generalized function  $  T $
 +
with support in  $  [ 0 , \infty ) $,
 +
possessing the properties (7), is naturally called the fundamental function of the operator  $  ( d / dt ) - A $.  
 +
Thus, if  $  A $
 +
is the infinitesimal operator of a semi-group  $  T $
 +
of class  $  l D  ^  \prime  $,
 +
then  $  T $
 +
is the fundamental function of the operator  $  ( d / dt ) - A $.  
 +
The converse statement is true under certain additional assumptions about the order of singularity of the fundamental function  $  T $(
 +
or, more precisely, of the function $  f ( T ( \phi ) x ) $,
 +
where  $  f \in X  ^  \prime  $).
  
<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/s084/s084140/s084140468.png" /></td> </tr></table>
+
A useful notion for the characterization of semi-groups in a locally convex space is that of the generalized resolvent. Let  $  \widehat \phi  $
 +
denote the Laplace transform of a function  $  \phi \in D ( \mathbf R ) $,
 +
and let  $  \widehat{D}  ( \mathbf R ) $
 +
be the space of all such transforms. A topology is induced in this space, via the Laplace transform, from the topology of  $  D ( \mathbf R ) $.
 +
The Laplace transform of an  $  X $-
 +
valued generalized function  $  F $
 +
is defined by  $  \widehat{F}  ( \widehat \phi  ) = F ( \phi ) $.  
 +
Under these conditions,  $  \widehat{F}  $
 +
is a continuous mapping of  $  \widehat{D}  ( \mathbf R ) $
 +
into the space  $  L ( X ) $
 +
of continuous linear operators on  $  X $.  
 +
Let  $  \widehat{D}  {} _ +  ^  \prime  $
 +
be the space of all  $  \widehat{F}  $
 +
obtained from functions  $  F $
 +
with support in  $  ( 0 , \infty ) $,
 +
with the natural topology. If  $  A $
 +
is a linear operator on  $  X $,
 +
it can be "lifted" to an operator  $  \widetilde{A}  $
 +
on  $  \widehat{D}  {} _ +  ^  \prime  $
 +
via the equality
  
If the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140469.png" /> has a continuous inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140470.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140471.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140472.png" /> is called the generalized resolvent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140473.png" />.
+
$$
 +
( \widetilde{A}  \widehat{F}  )  = A ( \widehat{F}  ( \widehat \phi  ) )  = A F ( \phi ) .
 +
$$
  
An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140474.png" /> has a generalized resolvent if and only if the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140475.png" /> has a locally equicontinuous fundamental function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140476.png" />, constructed by the formula
+
Thus, it is defined for all  $  \widehat{F}  \in \widehat{D}  {} _ +  ^  \prime  $
 +
such that the right-hand side of the equality is defined for any  $  \phi \in D ( \mathbf R ) $
 +
and it extends to a generalized function in  $  \widehat{D}  {} _ +  ^  \prime  $.  
 +
The continuous operator  $  \widetilde \lambda  $
 +
on  $  \widehat{D}  {} _ +  ^  \prime  $
 +
is defined by
  
<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/s084/s084140/s084140477.png" /></td> </tr></table>
+
$$
 +
( \widetilde \lambda  \widehat{F}  ) ( \widehat \phi  )  = \lambda \widehat{F}  ( \widehat \phi  )  = F  ^  \prime
 +
( \phi )  = - F ( \phi  ^  \prime  ) .
 +
$$
  
where
+
If the operator  $  \widetilde{A}  - \widetilde \lambda  $
 +
has a continuous inverse  $  \widetilde{R}  $
 +
on  $  \widehat{D}  {} _ +  ^  \prime  $,
 +
then  $  \widetilde{R}  $
 +
is called the generalized resolvent of  $  A $.
  
<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/s084/s084140/s084140478.png" /></td> </tr></table>
+
An operator  $  A $
 +
has a generalized resolvent if and only if the operator  $  ( d / dt ) - A $
 +
has a locally equicontinuous fundamental function  $  T $,
 +
constructed by the formula
  
Subject to certain additional assumptions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140479.png" /> is a distribution semi-group. An extension theorem for semi-groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140480.png" /> has also been proved in terms of generalized resolvents.
+
$$
 
+
T ( \phi ) = ( \widetilde{R} ( 1 \otimes x ) ) ( \widehat \phi  ) ,\  \phi \in D ( \mathbf R ) ,\ \
See also [[Semi-group of non-linear operators|Semi-group of non-linear operators]].
+
x \in X ,
 
+
$$
====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) {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.M. Vuvunikyan, "Evolutionary representations of algebras of generalized functions" , ''Theory of operators in function spaces'' , Novosibirsk (1977) pp. 99–120 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.P. Zabreiko, A.V. Zafievskii, "On a certain class of semigroups" ''Soviet Math. Dokl.'' , '''10''' : 6 (1969) pp. 1523–1526 ''Dokl. Akad. Nauk SSSR'' , '''189''' : 5 (1969) pp. 934–937 {{MR|264459}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.V. Zafievskii, ''Trudy Mat. Inst. Voronezh. Univ.'' , '''1''' (1970) pp. 206–210</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S.G. Krein, "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc. (1971) (Translated from Russian) {{MR|0342804}} {{ZBL|0179.20701}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> Yu.T. Sil'chenko, "An evolutionary equation with an operator generating a nonlinear semigroup" ''Differential Equations'' , '''15''' : 2 (1979) pp. 255–258 ''Differentsial'nye Uravneniya'' , '''15''' : 2 (1979) pp. 363–366 {{MR|}} {{ZBL|0505.34046}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> J. Chazarain, "Problèmes de Cauchy abstracts et applications à quelques problèmes mixtes" ''J. Funct. Anal.'' , '''7''' : 3 (1971) pp. 386–446</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> I. Ciorânescu, "La caracterisation spectrale d'opérateur, générateurs des semi-groupes distributions d'ordre fini de croissance" ''J. Math. Anal. Appl.'' , '''34''' (1971) pp. 34–41</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> I. Ciorânescu, "A characterization of distribution semigroups of finite growth order" ''Rev. Roum. Math. Pures Appl.'' , '''22''' : 8 (1977) pp. 1053–1068 {{MR|500280}} {{ZBL|0374.46032}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> T. Kato, "A characterization of holomorphic semigroups" ''Proc. Amer. Math. Soc.'' , '''25''' : 3 (1970) pp. 495–498 {{MR|0264456}} {{ZBL|0199.45604}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J. Lions, "Les semigroupes distributions" ''Portugal. Math.'' , '''19''' (1960) pp. 141–164</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> A. Pazy, "On the differentiability and compactness of semi-groups of linear operators" ''J. Math. Mech.'' , '''17''' : 12 (1968) pp. 1131–1141 {{MR|231242}} {{ZBL|0162.45903}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A. Pazy, "Approximations of the identity operator by semigroups of linear operators" ''Proc. Amer. Math. Soc.'' , '''30''' : 1 (1971) pp. 147–150 {{MR|0287362}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> T. Ushijima, "On the abstract Cauchy problems and semi-groups of linear operators in locally convex spaces" ''Sci. Papers College Gen. Educ. Univ. Tokyo'' , '''21''' (1971) pp. 93–122 {{MR|0312324}} {{ZBL|0239.47031}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> T. Ushijima, "On the generation and smoothness of semi-groups of linear operators" ''J. Fac. Sci. Univ. Tokyo, Sec. 1A'' , '''19''' : 1 (1972) pp. 65–127 {{MR|0308854}} {{ZBL|0239.47032}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> C. Wild, "Semi-groupes de croissance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140481.png" /> holomorphes" ''C.R. Acad. Sci. Paris Sér. A'' , '''285''' (1977) pp. 437–440 (English abstract) {{MR|448159}} {{ZBL|0359.47024}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> J.A. Gol'dstein, "Semigroups of linear operators and application" , Oxford Univ. Press (1985) (Translated from Russian)</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> A. Pazy, "Semigroups of linear operators and application to partial differential equations" , Springer (1983) {{MR|0710486}} {{ZBL|}} </TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, "One-parameter semigroups" , ''CWI Monographs'' , '''5''' , North-Holland (1987) {{MR|0915552}} {{ZBL|0636.47051}} </TD></TR></table>
 
  
 +
where
  
 +
$$
 +
( 1 \otimes x ) ( \widehat \phi  )  =  ( \delta \otimes x ) \phi  =  \phi ( 0 ) x .
 +
$$
  
====Comments====
+
Subject to certain additional assumptions,  $  T $
 +
is a distribution semi-group. An extension theorem for semi-groups of class  $  l C _ {0} $
 +
has also been proved in terms of generalized resolvents.
  
 +
See also [[Semi-group of non-linear operators]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Butzer, H. Berens, "Semigroups of operators and approximation" , Springer (1967) {{MR|230022}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Kellermann, M. Hieber, "Integrated semigroups" ''J. Funct. Anal.'' , '''84''' (1989) pp. 160–180 {{MR|0999494}} {{ZBL|0604.47025}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> I. Miyadera, N. Tanaka, "Exponentially bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140482.png" />-semigroups and integrated semigroups" ''Tokyo J. Math.'' , '''12''' (1989) pp. 99–115 {{MR|1001735}} {{ZBL|}} </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) {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.M. Vuvunikyan, "Evolutionary representations of algebras of generalized functions" , ''Theory of operators in function spaces'' , Novosibirsk (1977) pp. 99–120 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.P. Zabreiko, A.V. Zafievskii, "On a certain class of semigroups" ''Soviet Math. Dokl.'' , '''10''' : 6 (1969) pp. 1523–1526 ''Dokl. Akad. Nauk SSSR'' , '''189''' : 5 (1969) pp. 934–937 {{MR|264459}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.V. Zafievskii, ''Trudy Mat. Inst. Voronezh. Univ.'' , '''1''' (1970) pp. 206–210</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S.G. Krein, "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc. (1971) (Translated from Russian) {{MR|0342804}} {{ZBL|0179.20701}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> Yu.T. Sil'chenko, "An evolutionary equation with an operator generating a nonlinear semigroup" ''Differential Equations'' , '''15''' : 2 (1979) pp. 255–258 ''Differentsial'nye Uravneniya'' , '''15''' : 2 (1979) pp. 363–366 {{MR|}} {{ZBL|0505.34046}} </TD></TR>
 +
<TR><TD valign="top">[8]</TD> <TD valign="top"> J. Chazarain, "Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes" ''J. Funct. Anal.'' , '''7''' : 3 (1971) pp. 386–446 {{ZBL|0211.12902}} </TD></TR>
 +
<TR><TD valign="top">[9]</TD> <TD valign="top"> I. Ciorânescu, "La caracterisation spectrale d'opérateur, générateurs des semi-groupes distributions d'ordre fini de croissance" ''J. Math. Anal. Appl.'' , '''34''' (1971) pp. 34–41</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> I. Ciorânescu, "A characterization of distribution semigroups of finite growth order" ''Rev. Roum. Math. Pures Appl.'' , '''22''' : 8 (1977) pp. 1053–1068 {{MR|500280}} {{ZBL|0374.46032}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> T. Kato, "A characterization of holomorphic semigroups" ''Proc. Amer. Math. Soc.'' , '''25''' : 3 (1970) pp. 495–498 {{MR|0264456}} {{ZBL|0199.45604}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J. Lions, "Les semigroupes distributions" ''Portugal. Math.'' , '''19''' (1960) pp. 141–164</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> A. Pazy, "On the differentiability and compactness of semi-groups of linear operators" ''J. Math. Mech.'' , '''17''' : 12 (1968) pp. 1131–1141 {{MR|231242}} {{ZBL|0162.45903}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A. Pazy, "Approximations of the identity operator by semigroups of linear operators" ''Proc. Amer. Math. Soc.'' , '''30''' : 1 (1971) pp. 147–150 {{MR|0287362}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> T. Ushijima, "On the abstract Cauchy problems and semi-groups of linear operators in locally convex spaces" ''Sci. Papers College Gen. Educ. Univ. Tokyo'' , '''21''' (1971) pp. 93–122 {{MR|0312324}} {{ZBL|0239.47031}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> T. Ushijima, "On the generation and smoothness of semi-groups of linear operators" ''J. Fac. Sci. Univ. Tokyo, Sec. 1A'' , '''19''' : 1 (1972) pp. 65–127 {{MR|0308854}} {{ZBL|0239.47032}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> C. Wild, "Semi-groupes de croissance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140481.png" /> holomorphes" ''C.R. Acad. Sci. Paris Sér. A'' , '''285''' (1977) pp. 437–440 (English abstract) {{MR|448159}} {{ZBL|0359.47024}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> J.A. Gol'dstein, "Semigroups of linear operators and application" , Oxford Univ. Press (1985) (Translated from Russian)</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> A. Pazy, "Semigroups of linear operators and application to partial differential equations" , Springer (1983) {{MR|0710486}} {{ZBL|}} </TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, "One-parameter semigroups" , ''CWI Monographs'' , '''5''' , North-Holland (1987) {{MR|0915552}} {{ZBL|0636.47051}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Butzer, H. Berens, "Semigroups of operators and approximation" , Springer (1967) {{MR|230022}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Kellermann, M. Hieber, "Integrated semigroups" ''J. Funct. Anal.'' , '''84''' (1989) pp. 160–180 {{MR|0999494}} {{ZBL|0604.47025}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> I. Miyadera, N. Tanaka, "Exponentially bounded $c$-semigroups and integrated semigroups" ''Tokyo J. Math.'' , '''12''' (1989) pp. 99–115 {{MR|1001735}} {{ZBL|}} </TD></TR></table>

Latest revision as of 09:12, 21 January 2024


A family $ \{ T \} $ of operators on a Banach space or topological vector space with the property that the composite of any two operators in the family is again a member of the family. If the operators $ T $ are "indexed" by elements of some abstract semi-group $ \mathfrak A $ and the binary operation of the latter is compatible with the composition of operators, $ \{ T \} $ is known as a representation of the semi-group $ \mathfrak A $. The most detailed attention has been given to one-parameter semi-groups (cf. One-parameter semi-group) of bounded linear operators on a Banach space $ X $, which yield a representation of the additive semi-group of all positive real numbers, i.e. families $ T ( t) $ with the property

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

If $ T ( t) $ is strongly measurable, $ t > 0 $, then $ T ( t) $ is a strongly-continuous semi-group; this will be assumed in the sequel.

The limit

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

exists; it is known as the type of the semi-group. The functions $ T ( t) x $ increase at most exponentially.

An important characteristic is the infinitesimal operator (infinitesimal generator) of the semi-group:

$$ A _ {0} x = \lim\limits _ {t \rightarrow 0 } t ^ {-1} [ T ( t) x - x ] , $$

defined on the linear set $ D ( A _ {0} ) $ of all elements $ x $ for which the limit exists; the closure, $ A $, of this operator (if it exists) is known as the generating operator, or generator, of the semi-group. Let $ X _ {0} $ be the subspace defined as the closure of the union of all values $ T ( t) x $; then $ D ( A _ {0} ) $ is dense in $ X _ {0} $. If there are no non-zero elements in $ X _ {0} $ such that $ T ( t) x \equiv 0 $, then the generating operator $ A $ exists. In the sequel it will be assumed that $ X _ {0} = X $ and that $ T ( t) x \equiv 0 $ implies $ x = 0 $.

The simplest class of semi-groups, denoted by $ C _ {0} $, is defined by the condition: $ T ( t) x \rightarrow x $ as $ t \rightarrow 0 $ for any $ x \in X $. This is equivalent to the condition: The function $ \| T ( t) \| $ is bounded on any interval $ ( 0 , a ] $. In that case $ T ( t) $ has a generating operator $ A = A _ {0} $ whose resolvent $ R ( \lambda , A ) = ( A - \lambda I ) ^ {-1} $ satisfies the inequalities

$$ \tag{1 } \| R ^ {n} ( \lambda , A ) \| \leq M ( \lambda - \omega ) ^ {-} n ,\ \ n = 1 , 2 , . . . ; \ \lambda > \omega , $$

where $ \omega $ is the type of the semi-group. Conversely, if $ A $ is a closed operator with domain of definition dense in $ X $ and with a resolvent satisfying (1), then it is the generating operator of some semi-group $ T ( t) $ of class $ C _ {0} $ such that $ \| T ( t) \| \leq M e ^ {\omega t } $. Condition (1) is satisfied if

$$ \| R ( \lambda , A ) \| \leq ( \lambda - \omega ) ^ {-1} $$

(the Hill–Yosida condition). If, moreover, $ \omega = 0 $, then $ T ( t) $ is a contraction semi-group: $ \| T ( t) \| \leq 1 $.

A summable semi-group is a semi-group for which the functions $ \| T ( t) x \| $ are summable on any finite interval for all $ x \in X $. A summable semi-group has a generating operator $ A = \overline{ {A _ {0} }}\; $. The operator $ A _ {0} $ is closed if and only if, for every $ x \in X $,

$$ \lim\limits _ {t \rightarrow 0 } \frac{1}{t} \int\limits _ { 0 } ^ { t } T ( s) x d s = x . $$

For $ \mathop{\rm Re} \lambda > \omega $ one can define the Laplace transform of a summable semi-group,

$$ \tag{2 } \int\limits _ { 0 } ^ \infty e ^ {- \lambda t } T ( t) x d t = - R ( \lambda ) x , $$

giving a bounded linear operator $ R ( \lambda ) $ which has many properties of a resolvent operator.

A closed operator $ A $ with domain of definition dense in $ X $ is the generating operator of a summable semi-group $ T ( t) $ if and only if, for some $ \omega $, the resolvent $ R ( \lambda , A ) $ exists for $ \mathop{\rm Re} \lambda > \omega $ and the following conditions hold: a) $ \| R ( \lambda , A ) \| \leq M $, $ \mathop{\rm Re} \lambda > \omega $; b) there exist a non-negative function $ \phi ( t , x ) $, $ t > 0 $, $ x \in X $, jointly continuous in all its variables, and a non-negative function $ \phi ( t) $, bounded on any interval $ [ a , b ] \subset ( 0 , \infty ) $, such that, for $ \omega _ {1} > \omega $,

$$ \int\limits _ { 0 } ^ \infty e ^ {- \omega _ {1} t } \phi ( t , x ) dt < \infty , $$

$$ \overline{\lim\limits}\; _ {t \rightarrow \infty } t ^ {-1} \mathop{\rm ln} \phi ( t) < \infty ,\ \phi ( t , x ) \leq \phi ( t) \| x \| , $$

$$ \| R ^ {n} ( \lambda , A ) x \| \leq \frac{1}{ ( n - 1 ) ! } \int\limits _ { 0 } ^ \infty t ^ {n-1} e ^ {- \lambda t } \phi ( t , x ) dt . $$

Under these conditions

$$ \| T ( t) x \| \leq \phi ( t , x ) ,\ \ \| T ( t) \| \leq \phi ( t) . $$

If one requires in addition that the function $ \| T ( t) \| $ be summable on finite intervals, a necessary and sufficient condition is the existence of a continuous function $ \phi ( t) $ such that, for $ \omega _ {1} > \omega $,

$$ \tag{3 } \int\limits _ { 0 } ^ \infty \phi ( t) e ^ {- \omega _ {1} t } dt < \infty , $$

$$ \tag{4 } \| R ^ {n} ( \lambda , A ) \| \leq \frac{1}{( n - 1 ) ! } \int\limits _ { 0 } ^ \infty t ^ {n-1} e ^ {- \lambda t } \phi ( t) dt , $$

$$ \lambda > \omega ,\ n= 1 , 2 , . . . . $$

Under these conditions, $ \| T ( t) \| \leq \phi ( t) $. By choosing different functions satisfying (3), one can define different subclasses of summable semi-groups. If $ \phi ( t) = Me ^ {\omega t } $, the result is the class $ C _ {0} $ and (1) follows from (4). If $ \phi ( t) = Mt ^ {- \alpha } e ^ {\omega t } $, $ 0 \leq \alpha < 1 $, condition (4) implies the condition

$$ \| R ^ {n} ( \lambda , A ) \| \leq \frac{M \Gamma ( n - \alpha ) }{( n - 1 ) ! ( \lambda - \omega ) ^ {n - \alpha } } ,\ \ \lambda > \omega ,\ n = 1 , 2 ,\dots. $$

Semi-groups with power singularities.

If in the previous example $ \alpha \geq 1 $, then the integrals in (4) are divergent for $ n \leq \alpha - 1 $. Hence the generating operator for the corresponding semi-group may not have a resolvent for any $ \lambda $, i.e. it may have a spectrum equal to the entire complex plane. However, for $ n $ large enough one can define for such operators functions $ S _ {n} ( \lambda , A ) $ which coincide with the functions $ R ^ {n+1} ( \lambda , A ) $ in the previous cases. The operator function $ S _ {n} ( \lambda , A ) $ is called a resolvent of order $ n $ if it is analytic in some domain $ G \subset \mathbf C $ and if for $ \lambda \in G $,

$$ S _ {n} ( \lambda , A ) Ax = A S _ {n} ( \lambda , A ) x ,\ \ x \in D ( A) , $$

$$ S _ {n} ( \lambda , A ) ( A - \lambda ) ^ {n+1} x = x ,\ x \in D ( A ^ {n+1} ) , $$

and if $ S _ {n} ( \lambda , A ) x = 0 $ for all $ \lambda \in G $ implies $ x = 0 $. If $ \overline{D}\; ( A ^ {n+1} ) = X $, the operator may have a unique resolvent of order $ n $, for which there is a maximal domain of analyticity, known as the resolvent set of order $ n $.

Let $ T ( t) $ be a strongly-continuous semi-group such that the inequality

$$ \| T ( t) \| \leq M t ^ {- \alpha } e ^ {\omega t } $$

holds for $ \alpha \geq 1 $. Then its generating operator $ B $ has a resolvent of order $ n $ for $ n > \alpha - 1 $, and, moreover,

$$ S _ {n} ( \lambda , B ) x = \ \frac{1}{n!} \int\limits _ { 0 } ^ \infty t ^ {n} e ^ {- \lambda t } T ( t) x dt ,\ \mathop{\rm Re} \lambda > \omega , $$

$$ \tag{5 } \left \| \frac{d ^ {k} S _ {n} ( \lambda , B ) }{d \lambda ^ {k} } x \right \| \leq \frac{M \Gamma ( k + n + 1 - \alpha ) }{n ! ( \mathop{\rm Re} \lambda - \omega ) ^ {k + n + 1 - \alpha } } , $$

$$ \mathop{\rm Re} \lambda > \omega ,\ k = 0 , 1 ,\dots . $$

Conversely, suppose that for $ \mathop{\rm Re} \lambda > 0 $ the operator $ B $ has a resolvent $ S _ {n} ( \lambda , B ) $ of order $ n $ satisfying (5) with $ n > \alpha - 1 $. Then there exists a unique semi-group $ T ( t) $ such that

$$ \| T ( t) \| \leq M t ^ {- \alpha } e ^ {\omega t } , $$

and the generating operator $ A $ of this semi-group is such that $ S _ {n} ( \lambda , A ) = S _ {n} ( \lambda , B ) $.

Smooth semi-groups.

If $ x \in D ( A _ {0} ) $, the function $ T ( t) x $ is continuously differentiable and

$$ \frac{d T ( t) }{dt} x = A _ {0} T ( t) x = T ( t) A _ {0} x . $$

There exist semi-groups of class $ C _ {0} $ such that, if $ x \notin D ( A _ {0} ) = D ( A) $, the functions $ T ( t) x $ are non-differentiable for all $ t $. However, there are important classes of semi-groups for which the degree of smoothness increases with increasing $ t $. If the functions $ T ( t) x $, $ t > t _ {0} $, are differentiable for any $ x \in X $, then it follows from the semi-group property that the $ T ( t) x $ are twice differentiable if $ t > 2 t _ {0} $, three times differentiable if $ t > 3 t _ {0} $, etc. Therefore, if these functions are differentiable at any $ t > 0 $ for $ x \in X $, then $ T ( t) x $ is infinitely differentiable.

Given a semi-group of class $ C _ {0} $, a necessary and sufficient condition for the functions $ T ( t) x $ to be differentiable for all $ x \in X $ and $ t > t _ {0} $, where $ t _ {0} \geq 0 $, is that there exist numbers $ a , b , c > 0 $ such that the resolvent $ R ( \lambda , A ) $ is defined in the domain

$$ \mathop{\rm Re} \lambda > a - b \mathop{\rm ln} | \mathop{\rm Im} \lambda | , $$

while in this domain

$$ \| R( \lambda , A ) \| \leq c | \mathop{\rm Im} \lambda | . $$

A necessary and sufficient condition for $ T ( t) x $ to be infinitely differentiable for all $ x \in X $ and $ t > 0 $ is that, for every $ b > 0 $, there exist $ a _ {b} , c _ {b} $ such that the resolvent $ R ( \lambda , A ) $ is defined in the domain

$$ \mathop{\rm Re} \lambda > a _ {b} - b \mathop{\rm ln} | \mathop{\rm Im} \lambda | , $$

and such that

$$ \| R ( \lambda , A ) \| \leq c _ {b} | \mathop{\rm Im} \lambda | . $$

Sufficient conditions are: If there exists a $ \mu > \omega $ for which

$$ \overline{\lim\limits}\; _ {\tau \rightarrow \infty } \mathop{\rm ln} | \tau | \| R ( \mu + i \tau , A ) \| = t _ {0} < \infty , $$

then the $ T ( t) x $ are differentiable for $ t > t _ {0} $ and $ x \in X $; if $ t _ {0} = 0 $, then the $ T ( t) x $ are infinitely differentiable for all $ t > 0 $ and $ x \in X $.

The degree of smoothness of a semi-group may sometimes be inferred from its behaviour at zero; for example, suppose that for every $ c > 0 $ there exists a $ \delta _ {c} $ such that, for $ 0 < t < \delta _ {c} $,

$$ \| I - T ( t) \| \leq 2 - ct \mathop{\rm ln} t ^ {-1} , $$

then the $ T ( t) x $ are infinitely differentiable for all $ t > 0 $, $ x \in X $.

There are smoothness conditions for summable semi-groups and semi-groups of polynomial growth. If a semi-group has polynomial growth of degree $ \alpha $ and is infinitely differentiable for $ t > 0 $, then the function

$$ \frac{d T ( t) }{dt} x = A T ( t) x $$

also has polynomial growth:

$$ \| A T ( t) \| \leq M _ {1} t ^ {- \beta } e ^ {\omega t } . $$

In the general case there is no rigorous relationship between the numbers $ \alpha $ and $ \beta $, and $ \beta $ can be utilized for a more detailed classification of infinitely-differentiable semi-groups of polynomial growth.

Analytic semi-groups.

An important class of semi-groups, related to partial differential equations of parabolic type, comprises those semi-groups $ T ( t) $ which admit an analytic continuation to some sector of the complex plane containing the positive real axis. A semi-group of class $ C _ {0} $ has this property if and only if its resolvent satisfies the following inequality in some right half-plane $ \mathop{\rm Re} \lambda > \omega $:

$$ \| R ( \lambda , A ) \| \leq M | \lambda - \omega | ^ {-1} . $$

Another necessary and sufficient conditions is: The semi-group is strongly differentiable and its derivative satisfies the estimate

$$ \left \| \frac{d T }{dt} ( t) \right \| \leq M t ^ {-1} e ^ {\omega t } . $$

Finally, the inequality

$$ \overline{\lim\limits}\; _ {t \rightarrow 0 } \| I - T ( t) \| < 2 $$

is also a sufficient condition for $ T ( t) $ to be analytic.

If a semi-group $ T ( t) $ has an analytic continuation $ T ( z) $ to a sector $ | \mathop{\rm arg} z | < \phi \leq \pi / 2 $ and has polynomial growth at zero, $ \| T ( z) \| \leq c | z | ^ \alpha $, $ \alpha > 0 $, then the resolvent $ S _ {n} ( \lambda , A ) $ of order $ n > \alpha - 1 $ of its generating operator $ A $ has an analytic continuation to the sector $ | \mathop{\rm arg} \lambda | \leq \pi / 2 + \phi $, and satisfies the following estimate in any sector $ | \mathop{\rm arg} \lambda | \leq \pi / 2 + \psi $, $ \psi < \phi $:

$$ \| S _ {n} ( \lambda , A ) \| \leq | \lambda | ^ {\alpha - n - 1 } M ( \psi ) . $$

Conversely, suppose that the resolvent $ S _ {n} ( \lambda , B ) $ of an operator $ B $ is defined in a sector $ | \mathop{\rm arg} \lambda | \leq \pi / 2 + \psi $ and that

$$ \| S _ {n} ( \lambda , B ) \| \leq \lambda ^ {\alpha - n - 1 } M . $$

Then there exists a semi-group $ T ( z) $ of growth $ \alpha $, analytic in the sector $ | \mathop{\rm arg} z | < \psi $, whose generating operator $ A $ is such that $ S _ {n} ( \lambda , A ) = S _ {n} ( \lambda , B ) $.

Distribution semi-groups.

In accordance with the general concept of the theory of distributions (cf. Generalized function), one can drop the requirement that the operator-valued function $ T ( t) $ be defined for every $ t > 0 $, demanding only that it be possible to evaluate the integrals $ \int _ {- \infty } ^ {+ \infty } T ( t) \phi ( t) dt $ for all $ \phi $ in the space $ D ( \mathbf R ) $ of infinitely-differentiable functions with compact support. Hence the following definition: A distribution semi-group on a Banach space $ X $ is a continuous linear mapping $ T ( \phi ) $ of $ D ( \mathbf R ) $ into the space $ L ( X) $ of all bounded linear operators on $ X $, with the following properties: a) $ T ( \phi ) = 0 $ if $ \supp \phi \subset ( - \infty , 0 ) $; b) if $ \phi , \psi $ are functions in the subspace $ D ^ + ( \mathbf R ) $ of all functions in $ D ( \mathbf R ) $ with support in $ ( 0 , \infty ) $, then $ T ( \phi * \psi ) = T ( \phi ) T ( \psi ) $, where the star denotes convolution:

$$ \phi * \psi = \int\limits _ {- \infty } ^ \infty \phi ( t - s ) \psi ( s) d s $$

(the semi-group property); c) if $ T ( \phi ) x = 0 $ for all $ \phi \in D ^ + ( \mathbf R ) $, then $ x = 0 $; d) the linear hull of the set of all values of $ T ( \phi ) x $, $ \phi \in D ^ + ( \mathbf R ) $, $ x \in X $, is dense in $ X $; e) for any $ y = T ( \psi ) x $, $ \psi \in D ^ + ( \mathbf R ) $, there exists a continuous $ u ( t) $ on $ ( 0 , \infty ) $ with values in $ X $, so that $ u( 0) = y $ and

$$ T ( \phi ) y = \int\limits _ { 0 } ^ \infty \phi ( t) u ( t) dt $$

for all $ \phi \in D ( \mathbf R ) $.

The infinitesimal operator $ A _ {0} $ of a distribution semi-group is defined as follows. If there exists a delta-sequence $ \{ \rho _ {n} \} \subset D ^ + ( \mathbf R ) $ such that $ T ( \rho _ {n} ) x \rightarrow x $ and $ T ( - \rho _ {n} ^ \prime ) x \rightarrow y $ as $ n \rightarrow \infty $, then $ x \in D ( A _ {0} ) $ and $ y = A _ {0} x $. The infinitesimal operator has a closure $ A = \overline{ {A _ {0} }}\; $, known as the infinitesimal generator of the distribution semi-group. The set $ \cap_{n=1}^ \infty D ( A _ {0} ^ {n} ) $ is dense in $ X $ and contains $ T ( \phi ) X $ for any $ \phi \in D ^ + ( \mathbf R ) $.

A closed linear operator $ A $ with a dense domain of definition in $ X $ is the infinitesimal generator of a distribution semi-group if and only if there exist numbers $ a , b \geq 0 $, $ c > 0 $ and a natural number $ m $ such that the resolvent $ R ( \lambda , A ) $ exists for $ \mathop{\rm Re} \lambda \geq a \mathop{\rm ln} ( 1 + | \lambda | ) + b $ and satisfies the inequality

$$ \tag{6 } \| R ( \lambda , A ) \| \leq c ( 1 + | \lambda | ) ^ {m} . $$

If $ A $ is a closed linear operator on $ X $, then the set $ \cap_{n=1}^ \infty D ( A ^ {n} ) $ can be made into a Fréchet space $ X _ \infty $ by introducing the system of norms

$$ \| x \| _ {n} = \sum_{k=0}^n \| A ^ {k} x \| . $$

The restriction $ A _ \infty $ of $ A $ to $ X _ \infty $ leaves $ X _ \infty $ invariant. If $ A $ is the infinitesimal generator of a semi-group, then $ A _ \infty $ is the infinitesimal generator of a semi-group of class $ C _ {0} $( continuous for $ t \geq 0 $, $ T ( 0 ) = I $) on $ X _ \infty $. Conversely, if $ X _ \infty $ is dense in $ X $, the operator $ A $ has a non-empty resolvent set and $ A $ is the infinitesimal generator of a semi-group of class $ C _ {0} $ on $ X _ \infty $, then $ A $ is the infinitesimal generator of a distribution semi-group on $ X $.

A distribution semi-group has exponential growth of order at most $ q $, $ 1 \leq q < \infty $, if there exists an $ \omega > 0 $ such that $ \mathop{\rm exp} ( - \omega t ^ {q} ) T ( \phi ) $ is a continuous mapping in the topology induced on $ D ^ + $ by the space $ S ( \mathbf R ) $ of rapidly-decreasing functions. A closed linear operator is the infinitesimal generator of a distribution semi-group with the above property if and only if it has a resolvent $ R ( \lambda , A ) $ which satisfies (6) in the domain

$$ \{ \lambda : { \mathop{\rm Re} \lambda \geq [ \alpha \mathop{\rm ln} ( 1 + | \mathop{\rm Im} \ \lambda | + \beta ) ] ^ {1- 1/q } , \mathop{\rm Re} \lambda > \omega } \} , $$

where $ \alpha , \beta > 0 $. In particular, if $ q = 1 $ the semi-group is said to be exponential and inequality (6) is valid in some half-plane. There exists a characterization of the semi-groups of the above types in terms of the operator $ A _ \infty $. Questions of smoothness and analyticity have also been investigated for distribution semi-groups.

Semi-groups of operators in a (separable) locally convex space $ X $.

The definition of a strongly-continuous semi-group of operators $ T ( t ) $ continuous on $ X $ remains the same as for a Banach space. Similarly, the class $ C _ {0} $ is defined by the property $ T ( t ) x \rightarrow x $ as $ t \rightarrow 0 $ for any $ x \in X $. A semi-group is said to be locally equicontinuous (of class $ lC _ {0} $) if the family of operators $ T ( t ) $ is equicontinuous when $ t $ ranges over any finite interval in $ ( 0 , \infty ) $. In a barrelled space, a semi-group of class $ C _ {0} $ is always equicontinuous (cf. Equicontinuity).

A semi-group is said to be equicontinuous (of class $ uC _ {0} $) if the family $ T ( t ) $, $ 0 \leq t < \infty $, is equicontinuous.

Infinitesimal operators and infinitesimal generators are defined as in the Banach space case.

Assume from now on that the space $ X $ is sequentially complete. The infinitesimal generator $ A $ of a semi-group of class $ l C _ {0} $ is identical to the infinitesimal operator; its domain of definition, $ D ( A) $, is dense in $ X $ and, moreover, the set $ \cap_{n=1}^ \infty D ( A ^ {n} ) $ is dense in $ X $. The semi-group $ T ( t ) $ leaves $ D ( A) $ invariant and

$$ \frac{dT}{dt} ( t) x = \ AT ( t ) x = T ( t ) Ax ,\ 0 \leq t < \infty ,\ x \in D ( A ) . $$

If $ A $ is the infinitesimal generator of a semi-group of class $ u C _ {0} $, the resolvent $ R ( \lambda , A ) $ is defined for $ \mathop{\rm Re} \lambda > 0 $ and is the Laplace transform of the semi-group.

A linear operator $ A $ is the infinitesimal generator of a semi-group of class $ u C _ {0} $ if and only if it is closed, has dense domain of definition in $ X $, and if there exists a sequence of positive numbers $ \lambda _ {k} \rightarrow \infty $ such that, for any $ \lambda _ {k} $, the resolvent $ R ( \lambda _ {k} , A ) $ is defined and the family of operators $ [ \lambda _ {k} R ( \lambda _ {k} , A ) ] ^ {n} $, $ k , n = 1 , 2 \dots $ is equicontinuous. In this situation the semi-group can be constructed by the formula

$$ ( t ) x = \lim\limits _ {k \rightarrow \infty } \ \left ( \mathop{\rm exp} \left [ - \lambda _ {k} - \lambda _ {k} ^ {2} R ( \lambda _ {k} , A ) \right ] t \right ) x , $$

$$ t \geq 0 ,\ x \in X . $$

In a non-normed locally convex space, the infinitesimal generator of a semi-group of class $ lC _ {0} $ may have no resolvent at any point. An example is: $ A = d / ds $ in the space $ C ^ \infty $ of infinitely-differentiable functions of $ s $ on $ \mathbf R $. As a substitute for the resolvent one can take a continuous operator whose product with $ A - \lambda I $, from the right and the left, differs by a "small amount" from the identity operator.

A continuous operator $ R ( \lambda ) $ defined for $ \lambda $ in a set $ \Lambda \subset \mathbf C $ is called an asymptotic resolvent for a linear operator $ A $ if $ AR ( \lambda ) $ is continuous on $ X $, the operator $ R ( \lambda ) A $ can be extended from $ D ( A) $ to a continuous operator $ B ( \lambda ) $ on $ X $, and if there exists a limit point $ \lambda _ {0} $ of the set $ \Lambda $ such that $ H ^ + ( \lambda ) x \rightarrow 0 $, $ H ^ {-} ( \lambda ) x \rightarrow 0 $ as $ \lambda \rightarrow \lambda _ {0} $ for any $ x \in X $, where

$$ H ^ + ( \lambda ) = ( A - \lambda I ) R ( \lambda ) - I ,\ \ H ^ {-} ( \lambda ) = B ( \lambda ) - \lambda R ( \lambda ) - I . $$

An asymptotic resolvent possesses various properties resembling those of the ordinary resolvent.

A closed linear operator $ A $ with a dense domain of definition in $ X $ is the infinitesimal generator of a semi-group of class $ l C _ {0} $ if and only if there exist numbers $ \omega $ and $ \alpha > 0 $ such that, for $ \lambda > \omega $, there exists an asymptotic resolvent $ R ( \lambda ) $ of $ A $ with the properties: the functions $ R ( \lambda ) $, $ H ^ + ( \lambda ) $, $ H ^ {-} ( \lambda ) $ are strongly infinitely differentiable for $ \lambda > \omega $, and the families of operators

$$ e ^ {\alpha \lambda } \frac{d ^ {n} H ^ \pm ( \lambda ) }{d \lambda ^ {n} } ,\ \frac{\lambda ^ {n+1} }{n ! } \frac{d ^ {n} R ( \lambda ) }{d \lambda ^ {n} } ,\ \lambda > \omega ,\ n = 0 , 1 \dots $$

are equicontinuous.

Generation theorems have also been proved for other classes of semi-groups of operators on a locally convex space.

Adjoint semi-groups.

If $ T ( t ) $ is a semi-group of class $ C _ {0} $ on a Banach space $ X $, then the adjoint operators form a semi-group of bounded operators on the adjoint space $ X ^ \prime $. However, the assertion that $ T ^ \prime ( t ) f \rightarrow f $ as $ t \rightarrow 0 $ for any $ f \in X ^ \prime $ is valid only in the sense of the weak- topology $ \sigma ( X ^ \prime , X ) $. If $ A $ is the generating operator, its adjoint $ A ^ \prime $ is a weak infinitesimal generator for $ T ^ \prime ( t ) $, in the sense that $ D ( A ^ \prime ) $ is the set of all $ f $ for which the limit of $ t ^ {-1} [ T ^ \prime ( t ) - I ] f $ as $ t\rightarrow 0 $ exists in the sense of weak- convergence and is equal to $ A ^ \prime f $. The domain of definition $ D ( A ^ \prime ) $ is dense in $ X ^ \prime $— again in the sense of the weak- topology — and the operator $ A ^ \prime $ is closed in the weak- topology.

Let $ X ^ + $ be the set of all elements in $ X ^ \prime $ such that $ T ^ \prime ( f ) \rightarrow f $ as $ t \rightarrow 0 $ in the strong sense; then $ X ^ + $ is a closed subspace of $ X ^ \prime $ that is invariant under all $ T ^ \prime ( t ) $. On $ X ^ + $ the operators $ T ^ \prime ( t ) $ form a semi-group of class $ C _ {0} $. The space $ X ^ + $ is also the strong closure of the set $ D ( A ^ \prime ) $ in $ X ^ \prime $. If the original space is reflexive, then $ X ^ + = X ^ \prime $. Analogous propositions hold for semi-groups of class $ C _ {0} $ in locally convex spaces. Semi-groups of classes $ l C _ {0} $ and $ u C _ {0} $ generate semi-groups of the same classes in $ X ^ + $.

Distribution semi-groups in a (separable) locally convex space.

A distribution semi-group $ T $ in a sequentially complete locally convex space is defined just as in a Banach space. A semi-group $ T $ is said to be locally equicontinuous (of class $ lD ^ \prime $) if, for any compact subset $ K \subset D ( \mathbf R ) $, the family of operators $ \{ T ( \phi ) \} $, $ \phi \in K $, is equicontinuous. In a barrelled space $ X $, any distribution semi-group is defined by analogy to the Banach case. For semi-groups of class $ l D ^ \prime $, the infinitesimal operator is closed $ ( A _ {0} = A ) $, $ \cap_{n=1}^ \infty D ( A ^ {n} ) $ is dense in $ X $, and for any $ x \in X $ and $ \phi \in D ( \mathbf R ) $,

$$ \tag{7 } \left . \begin{array}{c} {T ( \phi ) x \in D ( A ) ,\ T ^ \prime ( \phi ) x = AT ( \phi ) x + \phi ( 0 ) x , } \\ {T ^ \prime ( \phi ) x = T ( \phi ) Ax + \phi ( 0 ) x ,\ x \in D ( A ). } \end{array} \right \} $$

A generalized function $ T $ with support in $ [ 0 , \infty ) $, possessing the properties (7), is naturally called the fundamental function of the operator $ ( d / dt ) - A $. Thus, if $ A $ is the infinitesimal operator of a semi-group $ T $ of class $ l D ^ \prime $, then $ T $ is the fundamental function of the operator $ ( d / dt ) - A $. The converse statement is true under certain additional assumptions about the order of singularity of the fundamental function $ T $( or, more precisely, of the function $ f ( T ( \phi ) x ) $, where $ f \in X ^ \prime $).

A useful notion for the characterization of semi-groups in a locally convex space is that of the generalized resolvent. Let $ \widehat \phi $ denote the Laplace transform of a function $ \phi \in D ( \mathbf R ) $, and let $ \widehat{D} ( \mathbf R ) $ be the space of all such transforms. A topology is induced in this space, via the Laplace transform, from the topology of $ D ( \mathbf R ) $. The Laplace transform of an $ X $- valued generalized function $ F $ is defined by $ \widehat{F} ( \widehat \phi ) = F ( \phi ) $. Under these conditions, $ \widehat{F} $ is a continuous mapping of $ \widehat{D} ( \mathbf R ) $ into the space $ L ( X ) $ of continuous linear operators on $ X $. Let $ \widehat{D} {} _ + ^ \prime $ be the space of all $ \widehat{F} $ obtained from functions $ F $ with support in $ ( 0 , \infty ) $, with the natural topology. If $ A $ is a linear operator on $ X $, it can be "lifted" to an operator $ \widetilde{A} $ on $ \widehat{D} {} _ + ^ \prime $ via the equality

$$ ( \widetilde{A} \widehat{F} ) = A ( \widehat{F} ( \widehat \phi ) ) = A F ( \phi ) . $$

Thus, it is defined for all $ \widehat{F} \in \widehat{D} {} _ + ^ \prime $ such that the right-hand side of the equality is defined for any $ \phi \in D ( \mathbf R ) $ and it extends to a generalized function in $ \widehat{D} {} _ + ^ \prime $. The continuous operator $ \widetilde \lambda $ on $ \widehat{D} {} _ + ^ \prime $ is defined by

$$ ( \widetilde \lambda \widehat{F} ) ( \widehat \phi ) = \lambda \widehat{F} ( \widehat \phi ) = F ^ \prime ( \phi ) = - F ( \phi ^ \prime ) . $$

If the operator $ \widetilde{A} - \widetilde \lambda $ has a continuous inverse $ \widetilde{R} $ on $ \widehat{D} {} _ + ^ \prime $, then $ \widetilde{R} $ is called the generalized resolvent of $ A $.

An operator $ A $ has a generalized resolvent if and only if the operator $ ( d / dt ) - A $ has a locally equicontinuous fundamental function $ T $, constructed by the formula

$$ T ( \phi ) x = ( \widetilde{R} ( 1 \otimes x ) ) ( \widehat \phi ) ,\ \phi \in D ( \mathbf R ) ,\ \ x \in X , $$

where

$$ ( 1 \otimes x ) ( \widehat \phi ) = ( \delta \otimes x ) \phi = \phi ( 0 ) x . $$

Subject to certain additional assumptions, $ T $ is a distribution semi-group. An extension theorem for semi-groups of class $ l C _ {0} $ has also been proved in terms of generalized resolvents.

See also Semi-group of non-linear operators.

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How to Cite This Entry:
Semi-group of operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-group_of_operators&oldid=24563
This article was adapted from an original article by Yu.M. VuvunikyanS.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article