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The semi-group on a dual [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104001.png" /> composed of the adjoint operators of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104002.png" />-semi-group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104003.png" /> (cf. also [[Semi-group of operators|Semi-group of operators]]).
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104004.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104006.png" />-semi-group on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104007.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104008.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104009.png" /> and
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i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040010.png" />, the identity operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040011.png" />;
+
The semi-group on a dual [[Banach space|Banach space]]  $  X  ^ {*} $
 +
composed of the adjoint operators of a  $  C _ {0} $-
 +
semi-group on $  X $(
 +
cf. also [[Semi-group of operators|Semi-group of operators]]).
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040013.png" />;
+
Let  $  \mathbf T = \{ T ( t ) \} _ {t \geq  0 }  $
 +
be a  $  C _ {0} $-
 +
semi-group on a Banach space  $  X $,
 +
i.e. $  T ( t ) \in {\mathcal L} ( X ) $
 +
for all $  t $
 +
and
  
iii) the orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040014.png" /> are strongly continuous (cf. [[Strongly-continuous semi-group|Strongly-continuous semi-group]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040016.png" />. On the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040017.png" />, the adjoint semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040018.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040019.png" />, satisfies i) and ii), but not necessarily iii). Therefore one defines
+
i) $  T ( 0 ) = I $,  
 +
the identity operator on  $  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/a/a110/a110400/a11040020.png" /></td> </tr></table>
+
ii)  $  T ( t + s ) = T ( t ) T ( s ) $
 +
for all  $  t,s \geq  0 $;
  
<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/a/a110/a110400/a11040021.png" /></td> </tr></table>
+
iii) the orbits  $  t \mapsto T ( t ) x $
 +
are strongly continuous (cf. [[Strongly-continuous semi-group|Strongly-continuous semi-group]]) on  $  [ 0, \infty ) $
 +
for all  $  x \in X $.
 +
On the dual space  $  X  ^ {*} $,
 +
the adjoint semi-group  $  \mathbf T  ^ {*} = \{ T  ^ {*} ( t ) \} _ {t \geq 0 }  $,
 +
with  $  T  ^ {*} ( t ) = ( T ( t ) )  ^ {*} $,
 +
satisfies i) and ii), but not necessarily iii). Therefore one defines
  
This is a norm-closed, weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040022.png" />-dense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040023.png" />-invariant subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040024.png" />, and the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040025.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040026.png" />-semi-group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040027.png" />, called the strongly continuous adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040028.png" />. Its infinitesimal generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040029.png" /> is the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040032.png" /> is the adjoint of the infinitesimal generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040034.png" />. Its spectrum satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040036.png" /> is reflexive (cf. [[Reflexive space|Reflexive space]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040037.png" /> [[#References|[a9]]].
+
$$
 +
X  ^  \odot  = \{ { x  ^ {*} \in X  ^ {*}  } : { \textrm{ the orbit  } }
 +
$$
  
Starting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040038.png" />, one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040040.png" />. The natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040042.png" />, is an isomorphic imbedding with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040043.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040044.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040046.png" />-reflexive with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040047.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040048.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040049.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040050.png" />. This is the case if and only if the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040051.png" /> is weakly compact for some (hence for all) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040052.png" /> [[#References|[a7]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040053.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040054.png" />-reflexive with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040056.png" />, then the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040057.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040058.png" /> generates a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040059.png" />-semi-group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040060.png" /> [[#References|[a1]]].
+
$$
 +
\
 +
{} {t \mapsto T  ^ {*} ( t ) x  ^ {*}  \textrm{ is  strongly  continuous  on  }  [ 0, \infty )  } \} .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040061.png" /> be the quotient mapping. If, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040062.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040063.png" /> is separably-valued, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040064.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040065.png" />. Hence, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040066.png" /> extends to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040067.png" />-group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040068.png" /> is either trivial or non-separable [[#References|[a4]]].
+
This is a norm-closed, weak $  * $-
 +
dense, $  \mathbf T  ^ {*} $-
 +
invariant subspace of  $  X  ^ {*} $,  
 +
and the restriction  $  \mathbf T  ^  \odot  = \{ T  ^ {*} ( t ) \mid  _ {X  ^  \odot  } \} _ {t \geq  0 }  $
 +
is a $  C _ {0} $-
 +
semi-group on  $  X  ^  \odot  $,  
 +
called the strongly continuous adjoint of  $  \mathbf T $.  
 +
Its infinitesimal generator  $  A  ^  \odot  $
 +
is the part of  $  A  ^ {*} $
 +
in  $  X  ^  \odot  $,  
 +
where  $  A  ^ {*} $
 +
is the adjoint of the infinitesimal generator  $  A $
 +
of  $  \mathbf T $.  
 +
Its spectrum satisfies  $  \sigma ( A  ^  \odot  ) = \sigma ( A  ^ {*} ) = \sigma ( A ) $.  
 +
If  $  X $
 +
is reflexive (cf. [[Reflexive space|Reflexive space]]), then $  X  ^  \odot  = X  ^ {*} $[[#References|[a9]]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040069.png" /> is a positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040070.png" />-semi-group on a [[Banach lattice|Banach lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040071.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040072.png" /> need not be a sublattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040073.png" /> [[#References|[a2]]]. If, however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040074.png" /> has order-continuous norm, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040075.png" /> is even a projection band in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040076.png" /> [[#References|[a8]]]. For a positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040077.png" />-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040078.png" /> on an arbitrary Banach lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040079.png" /> one has
+
Starting from  $  \mathbf T  ^  \odot  $,
 +
one defines  $  X ^ {\odot * } = ( X  ^  \odot  )  ^ {*} $
 +
and  $  X ^ {\odot \odot } = ( X  ^  \odot  )  ^  \odot  $.  
 +
The natural mapping  $  j : X \rightarrow {X ^ {\odot * } } $,
 +
$  \langle  {jx,x  ^  \odot  } \rangle = \langle  {x  ^  \odot  , x } \rangle $,
 +
is an isomorphic imbedding with values in  $  X ^ {\odot \odot } $,  
 +
and  $  X $
 +
is said to be $  \odot $-
 +
reflexive with respect to  $  \mathbf T $
 +
if  $  j $
 +
maps  $  X $
 +
onto  $  X ^ {\odot \odot } $.  
 +
This is the case if and only if the resolvent  $  ( \lambda - A ) ^ {- 1 } $
 +
is weakly compact for some (hence for all)  $  \lambda \in \varrho ( A ) $[[#References|[a7]]]. If $  X $
 +
is  $  \odot $-
 +
reflexive with respect to  $  \mathbf T $
 +
and  $  B \in {\mathcal L} ( X,X ^ {\odot * } ) $,  
 +
then the part of  $  A ^ {\odot * } + B $
 +
in  $  X $
 +
generates a $  C _ {0} $-
 +
semi-group on  $  X $[[#References|[a1]]].
  
<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/a/a110/a110400/a11040080.png" /></td> </tr></table>
+
Let  $  \pi : {X  ^ {*} } \rightarrow {X  ^ {*} /X  ^  \odot  } $
 +
be the quotient mapping. If, for some  $  x  ^ {*} \in X  ^ {*} $,
 +
the mapping  $  t \mapsto \pi T  ^ {*} ( t ) x  ^ {*} $
 +
is separably-valued, then  $  T  ^ {*} ( t ) x  ^ {*} \in X  ^  \odot  $
 +
for all  $  t > 0 $.  
 +
Hence, if  $  \mathbf T $
 +
extends to a $  C _ {0} $-
 +
group, then  $  X  ^ {*} /X  ^  \odot  $
 +
is either trivial or non-separable [[#References|[a4]]].
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040081.png" />, the disjoint complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040082.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040083.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040084.png" /> has a weak order unit, then for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040086.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040087.png" />, the band generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040088.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040089.png" /> [[#References|[a5]]]. If, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040090.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040091.png" /> is weakly measurable, then, assuming the Martin axiom (cf. [[Suslin hypothesis|Suslin hypothesis]]), for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040092.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040093.png" /> [[#References|[a6]]].
+
If  $  \mathbf T $
 +
is a positive  $  C _ {0} $-
 +
semi-group on a [[Banach lattice|Banach lattice]]  $  X $,
 +
then  $  X  ^  \odot  $
 +
need not be a sublattice of  $  X  ^ {*} $[[#References|[a2]]]. If, however,  $  X  ^ {*} $
 +
has order-continuous norm, then  $  X  ^  \odot  $
 +
is even a projection band in  $  X  ^ {*} $[[#References|[a8]]]. For a positive  $  C _ {0} $-
 +
semi-group  $  \mathbf T $
 +
on an arbitrary Banach lattice  $  X $
 +
one has
 +
 
 +
$$
 +
{\lim\limits  \sup } _ {t \downarrow 0 } \left \| {T  ^ {*} ( t ) x  ^ {*} - x  ^ {*} } \right \| \geq  2 \left \| {x  ^ {*} } \right \|
 +
$$
 +
 
 +
for all  $  x  ^ {*} \in ( X  ^  \odot  )  ^ {d} $,  
 +
the disjoint complement of $  X  ^  \odot  $
 +
in $  X  ^ {*} $.  
 +
If $  ( X  ^  \odot  )  ^ {d} $
 +
has a weak order unit, then for all $  x  ^ {*} \in X  ^ {*} $
 +
and  $  t > 0 $
 +
one has $  T  ^ {*} ( t ) x  ^ {*} \in ( X  ^  \odot  ) ^ {dd } $,  
 +
the band generated by $  X  ^  \odot  $
 +
in $  X  ^ {*} $[[#References|[a5]]]. If, for some $  x  ^ {*} \in X  ^ {*} $,  
 +
the mapping $  t \mapsto T  ^ {*} ( t ) x  ^ {*} $
 +
is weakly measurable, then, assuming the Martin axiom (cf. [[Suslin hypothesis|Suslin hypothesis]]), for all $  t > 0 $
 +
one has $  T  ^ {*} ( t ) x  ^ {*} \in ( X  ^  \odot  ) ^ {dd } $[[#References|[a6]]].
  
 
A general reference is [[#References|[a3]]].
 
A general reference is [[#References|[a3]]].

Latest revision as of 20:04, 4 April 2020


The semi-group on a dual Banach space $ X ^ {*} $ composed of the adjoint operators of a $ C _ {0} $- semi-group on $ X $( cf. also Semi-group of operators).

Let $ \mathbf T = \{ T ( t ) \} _ {t \geq 0 } $ be a $ C _ {0} $- semi-group on a Banach space $ X $, i.e. $ T ( t ) \in {\mathcal L} ( X ) $ for all $ t $ and

i) $ T ( 0 ) = I $, the identity operator on $ X $;

ii) $ T ( t + s ) = T ( t ) T ( s ) $ for all $ t,s \geq 0 $;

iii) the orbits $ t \mapsto T ( t ) x $ are strongly continuous (cf. Strongly-continuous semi-group) on $ [ 0, \infty ) $ for all $ x \in X $. On the dual space $ X ^ {*} $, the adjoint semi-group $ \mathbf T ^ {*} = \{ T ^ {*} ( t ) \} _ {t \geq 0 } $, with $ T ^ {*} ( t ) = ( T ( t ) ) ^ {*} $, satisfies i) and ii), but not necessarily iii). Therefore one defines

$$ X ^ \odot = \{ { x ^ {*} \in X ^ {*} } : { \textrm{ the orbit } } $$

$$ \ {} {t \mapsto T ^ {*} ( t ) x ^ {*} \textrm{ is strongly continuous on } [ 0, \infty ) } \} . $$

This is a norm-closed, weak $ * $- dense, $ \mathbf T ^ {*} $- invariant subspace of $ X ^ {*} $, and the restriction $ \mathbf T ^ \odot = \{ T ^ {*} ( t ) \mid _ {X ^ \odot } \} _ {t \geq 0 } $ is a $ C _ {0} $- semi-group on $ X ^ \odot $, called the strongly continuous adjoint of $ \mathbf T $. Its infinitesimal generator $ A ^ \odot $ is the part of $ A ^ {*} $ in $ X ^ \odot $, where $ A ^ {*} $ is the adjoint of the infinitesimal generator $ A $ of $ \mathbf T $. Its spectrum satisfies $ \sigma ( A ^ \odot ) = \sigma ( A ^ {*} ) = \sigma ( A ) $. If $ X $ is reflexive (cf. Reflexive space), then $ X ^ \odot = X ^ {*} $[a9].

Starting from $ \mathbf T ^ \odot $, one defines $ X ^ {\odot * } = ( X ^ \odot ) ^ {*} $ and $ X ^ {\odot \odot } = ( X ^ \odot ) ^ \odot $. The natural mapping $ j : X \rightarrow {X ^ {\odot * } } $, $ \langle {jx,x ^ \odot } \rangle = \langle {x ^ \odot , x } \rangle $, is an isomorphic imbedding with values in $ X ^ {\odot \odot } $, and $ X $ is said to be $ \odot $- reflexive with respect to $ \mathbf T $ if $ j $ maps $ X $ onto $ X ^ {\odot \odot } $. This is the case if and only if the resolvent $ ( \lambda - A ) ^ {- 1 } $ is weakly compact for some (hence for all) $ \lambda \in \varrho ( A ) $[a7]. If $ X $ is $ \odot $- reflexive with respect to $ \mathbf T $ and $ B \in {\mathcal L} ( X,X ^ {\odot * } ) $, then the part of $ A ^ {\odot * } + B $ in $ X $ generates a $ C _ {0} $- semi-group on $ X $[a1].

Let $ \pi : {X ^ {*} } \rightarrow {X ^ {*} /X ^ \odot } $ be the quotient mapping. If, for some $ x ^ {*} \in X ^ {*} $, the mapping $ t \mapsto \pi T ^ {*} ( t ) x ^ {*} $ is separably-valued, then $ T ^ {*} ( t ) x ^ {*} \in X ^ \odot $ for all $ t > 0 $. Hence, if $ \mathbf T $ extends to a $ C _ {0} $- group, then $ X ^ {*} /X ^ \odot $ is either trivial or non-separable [a4].

If $ \mathbf T $ is a positive $ C _ {0} $- semi-group on a Banach lattice $ X $, then $ X ^ \odot $ need not be a sublattice of $ X ^ {*} $[a2]. If, however, $ X ^ {*} $ has order-continuous norm, then $ X ^ \odot $ is even a projection band in $ X ^ {*} $[a8]. For a positive $ C _ {0} $- semi-group $ \mathbf T $ on an arbitrary Banach lattice $ X $ one has

$$ {\lim\limits \sup } _ {t \downarrow 0 } \left \| {T ^ {*} ( t ) x ^ {*} - x ^ {*} } \right \| \geq 2 \left \| {x ^ {*} } \right \| $$

for all $ x ^ {*} \in ( X ^ \odot ) ^ {d} $, the disjoint complement of $ X ^ \odot $ in $ X ^ {*} $. If $ ( X ^ \odot ) ^ {d} $ has a weak order unit, then for all $ x ^ {*} \in X ^ {*} $ and $ t > 0 $ one has $ T ^ {*} ( t ) x ^ {*} \in ( X ^ \odot ) ^ {dd } $, the band generated by $ X ^ \odot $ in $ X ^ {*} $[a5]. If, for some $ x ^ {*} \in X ^ {*} $, the mapping $ t \mapsto T ^ {*} ( t ) x ^ {*} $ is weakly measurable, then, assuming the Martin axiom (cf. Suslin hypothesis), for all $ t > 0 $ one has $ T ^ {*} ( t ) x ^ {*} \in ( X ^ \odot ) ^ {dd } $[a6].

A general reference is [a3].

References

[a1] Ph. Clément, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, H.R. Thieme, "Perturbation theory for dual semigroups, Part I: The sun-reflexive case" Math. Ann. , 277 (1987) pp. 709–725
[a2] A. Grabosch, R. Nagel, "Order structure of the semigroup dual: A counterexample" Indagationes Mathematicae , 92 (1989) pp. 199–201
[a3] J.M.A.M. van Neerven, "The adjoint of a semigroup of linear operators" , Lecture Notes in Mathematics , 1529 , Springer (1992)
[a4] J.M.A.M. van Neerven, "A dichotomy theorem for the adjoint of a semigroup of operators" Proc. Amer. Math. Soc. , 119 (1993) pp. 765–774
[a5] J.M.A.M. van Neerven, B. de Pagter, "The adjoint of a positive semigroup" Comp. Math. , 90 (1994) pp. 99–118
[a6] J.M.A.M. van Neerven, B. de Pagter, A.R. Schep, "Weak measurability of the orbits of an adjoint semigroup" G. Ferreyra (ed.) G.R. Goldstein (ed.) F. Neubrander (ed.) , Evolution Equations , Lecture Notes in Pure and Appl. Math. , 168 , M. Dekker (1994) pp. 327–336
[a7] B. de Pagter, "A characterization of sun-reflexivity" Math. Ann. , 283 (1989) pp. 511–518
[a8] B. de Pagter, "A Wiener–Young type theorem for dual semigroups" Acta Appl. Math. 27 (1992) pp. 101–109
[a9] R.S. Phillips, "The adjoint semi-group" Pacific J. Math. , 5 (1955) pp. 269–283
How to Cite This Entry:
Adjoint semi-group of operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_semi-group_of_operators&oldid=18347
This article was adapted from an original article by J. van Neerven (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article