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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]]).

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|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

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

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

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

Starting from  $  \mathbf T  ^  \odt  $, 
one defines  $  X ^ {\odt * } = ( X  ^  \odt  )  ^ {*} $
and  $  X ^ {\odt \odt } = ( X  ^  \odt  )  ^  \odt  $. 
The natural mapping  $  j : X \rightarrow {X ^ {\odt * } } $, 
$  \langle  {jx,x  ^  \odt  } \rangle = \langle  {x  ^  \odt  , x } \rangle $, 
is an isomorphic imbedding with values in  $  X ^ {\odt \odt } $, 
and  $  X $
is said to be  $  \odt $-
reflexive with respect to  $  \mathbf T $
if  $  j $
maps  $  X $
onto  $  X ^ {\odt \odt } $. 
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  $  \odt $-
reflexive with respect to  $  \mathbf T $
and  $  B \in {\mathcal L} ( X,X ^ {\odt * } ) $, 
then the part of  $  A ^ {\odt * } + B $
in  $  X $
generates a  $  C _ {0} $-
semi-group on  $  X $[[#References|[a1]]].

Let  $  \pi : {X  ^ {*} } \rightarrow {X  ^ {*} /X  ^  \odt  } $
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  ^  \odt  $
for all  $  t > 0 $. 
Hence, if  $  \mathbf T $
extends to a  $  C _ {0} $-
group, then  $  X  ^ {*} /X  ^  \odt  $
is either trivial or non-separable [[#References|[a4]]].

If  $  \mathbf T $
is a positive  $  C _ {0} $-
semi-group on a [[Banach lattice|Banach lattice]]  $  X $, 
then  $  X  ^  \odt  $
need not be a sublattice of  $  X  ^ {*} $[[#References|[a2]]]. If, however,  $  X  ^ {*} $
has order-continuous norm, then  $  X  ^  \odt  $
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  ^  \odt  )  ^ {d} $, 
the disjoint complement of  $  X  ^  \odt  $
in  $  X  ^ {*} $. 
If  $  ( X  ^  \odt  )  ^ {d} $
has a weak order unit, then for all  $  x  ^ {*} \in X  ^ {*} $
and  $  t > 0 $
one has  $  T  ^ {*} ( t ) x  ^ {*} \in ( X  ^  \odt  ) ^ {dd } $, 
the band generated by  $  X  ^  \odt  $
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  ^  \odt  ) ^ {dd } $[[#References|[a6]]].

A general reference is [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grabosch,   R. Nagel,   "Order structure of the semigroup dual: A counterexample"  ''Indagationes Mathematicae'' , '''92'''  (1989)  pp. 199–201</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.M.A.M. van Neerven,   "The adjoint of a semigroup of linear operators" , ''Lecture Notes in Mathematics'' , '''1529''' , Springer  (1992)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.M.A.M. van Neerven,   B. de Pagter,   "The adjoint of a positive semigroup"  ''Comp. Math.'' , '''90'''  (1994)  pp. 99–118</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  B. de Pagter,   "A characterization of sun-reflexivity"  ''Math. Ann.'' , '''283'''  (1989)  pp. 511–518</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B. de Pagter,   "A Wiener–Young type theorem for dual semigroups"  ''Acta Appl. Math. 27''  (1992)  pp. 101–109</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R.S. Phillips,   "The adjoint semi-group"  ''Pacific J. Math.'' , '''5'''  (1955)  pp. 269–283</TD></TR></table>