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m (tex encoded by computer)
m (\odot fixed)
 
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$$  
 
$$  
X  ^  \odt = \{ { x  ^ {*} \in X  ^ {*}  } :  { \textrm{ the  orbit  } }  
+
X  ^  \odot = \{ { x  ^ {*} \in X  ^ {*}  } :  { \textrm{ the  orbit  } }  
 
$$
 
$$
  
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dense,  $  \mathbf T  ^ {*} $-
 
dense,  $  \mathbf T  ^ {*} $-
 
invariant subspace of  $  X  ^ {*} $,  
 
invariant subspace of  $  X  ^ {*} $,  
and the restriction  $  \mathbf T  ^  \odt = \{ T  ^ {*} ( t ) \mid  _ {X  ^  \odt   } \} _ {t \geq  0 }  $
+
and the restriction  $  \mathbf T  ^  \odot = \{ T  ^ {*} ( t ) \mid  _ {X  ^  \odot   } \} _ {t \geq  0 }  $
 
is a  $  C _ {0} $-
 
is a  $  C _ {0} $-
semi-group on  $  X  ^  \odt $,  
+
semi-group on  $  X  ^  \odot $,  
 
called the strongly continuous adjoint of  $  \mathbf T $.  
 
called the strongly continuous adjoint of  $  \mathbf T $.  
Its infinitesimal generator  $  A  ^  \odt $
+
Its infinitesimal generator  $  A  ^  \odot $
 
is the part of  $  A  ^ {*} $
 
is the part of  $  A  ^ {*} $
in  $  X  ^  \odt $,  
+
in  $  X  ^  \odot $,  
 
where  $  A  ^ {*} $
 
where  $  A  ^ {*} $
 
is the adjoint of the infinitesimal generator  $  A $
 
is the adjoint of the infinitesimal generator  $  A $
 
of  $  \mathbf T $.  
 
of  $  \mathbf T $.  
Its spectrum satisfies  $  \sigma ( A  ^  \odt ) = \sigma ( A  ^ {*} ) = \sigma ( A ) $.  
+
Its spectrum satisfies  $  \sigma ( A  ^  \odot ) = \sigma ( A  ^ {*} ) = \sigma ( A ) $.  
 
If  $  X $
 
If  $  X $
is reflexive (cf. [[Reflexive space|Reflexive space]]), then  $  X  ^  \odt = X  ^ {*} $[[#References|[a9]]].
+
is reflexive (cf. [[Reflexive space|Reflexive space]]), then  $  X  ^  \odot = X  ^ {*} $[[#References|[a9]]].
  
Starting from  $  \mathbf T  ^  \odt $,  
+
Starting from  $  \mathbf T  ^  \odot $,  
one defines  $  X ^ {\odt * } = ( X  ^  \odt )  ^ {*} $
+
one defines  $  X ^ {\odot * } = ( X  ^  \odot )  ^ {*} $
and  $  X ^ {\odt \odt } = ( X  ^  \odt )  ^  \odt $.  
+
and  $  X ^ {\odot \odot } = ( X  ^  \odot )  ^  \odot $.  
The natural mapping  $  j : X \rightarrow {X ^ {\odt * } } $,  
+
The natural mapping  $  j : X \rightarrow {X ^ {\odot * } } $,  
$  \langle  {jx,x  ^  \odt } \rangle = \langle  {x  ^  \odt , x } \rangle $,  
+
$  \langle  {jx,x  ^  \odot } \rangle = \langle  {x  ^  \odot , x } \rangle $,  
is an isomorphic imbedding with values in  $  X ^ {\odt \odt } $,  
+
is an isomorphic imbedding with values in  $  X ^ {\odot \odot } $,  
 
and  $  X $
 
and  $  X $
is said to be  $  \odt $-
+
is said to be  $  \odot $-
 
reflexive with respect to  $  \mathbf T $
 
reflexive with respect to  $  \mathbf T $
 
if  $  j $
 
if  $  j $
 
maps  $  X $
 
maps  $  X $
onto  $  X ^ {\odt \odt } $.  
+
onto  $  X ^ {\odot \odot } $.  
 
This is the case if and only if the resolvent  $  ( \lambda - A ) ^ {- 1 } $
 
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 weakly compact for some (hence for all)  $  \lambda \in \varrho ( A ) $[[#References|[a7]]]. If  $  X $
is  $  \odt $-
+
is  $  \odot $-
 
reflexive with respect to  $  \mathbf T $
 
reflexive with respect to  $  \mathbf T $
and  $  B \in {\mathcal L} ( X,X ^ {\odt * } ) $,  
+
and  $  B \in {\mathcal L} ( X,X ^ {\odot * } ) $,  
then the part of  $  A ^ {\odt * } + B $
+
then the part of  $  A ^ {\odot * } + B $
 
in  $  X $
 
in  $  X $
 
generates a  $  C _ {0} $-
 
generates a  $  C _ {0} $-
 
semi-group on  $  X $[[#References|[a1]]].
 
semi-group on  $  X $[[#References|[a1]]].
  
Let  $  \pi : {X  ^ {*} } \rightarrow {X  ^ {*} /X  ^  \odt } $
+
Let  $  \pi : {X  ^ {*} } \rightarrow {X  ^ {*} /X  ^  \odot } $
 
be the quotient mapping. If, for some  $  x  ^ {*} \in X  ^ {*} $,  
 
be the quotient mapping. If, for some  $  x  ^ {*} \in X  ^ {*} $,  
 
the mapping  $  t \mapsto \pi T  ^ {*} ( t ) x  ^ {*} $
 
the mapping  $  t \mapsto \pi T  ^ {*} ( t ) x  ^ {*} $
is separably-valued, then  $  T  ^ {*} ( t ) x  ^ {*} \in X  ^  \odt $
+
is separably-valued, then  $  T  ^ {*} ( t ) x  ^ {*} \in X  ^  \odot $
 
for all  $  t > 0 $.  
 
for all  $  t > 0 $.  
 
Hence, if  $  \mathbf T $
 
Hence, if  $  \mathbf T $
 
extends to a  $  C _ {0} $-
 
extends to a  $  C _ {0} $-
group, then  $  X  ^ {*} /X  ^  \odt $
+
group, then  $  X  ^ {*} /X  ^  \odot $
 
is either trivial or non-separable [[#References|[a4]]].
 
is either trivial or non-separable [[#References|[a4]]].
  
Line 98: Line 98:
 
is a positive  $  C _ {0} $-
 
is a positive  $  C _ {0} $-
 
semi-group on a [[Banach lattice|Banach lattice]]  $  X $,  
 
semi-group on a [[Banach lattice|Banach lattice]]  $  X $,  
then  $  X  ^  \odt $
+
then  $  X  ^  \odot $
 
need not be a sublattice of  $  X  ^ {*} $[[#References|[a2]]]. If, however,  $  X  ^ {*} $
 
need not be a sublattice of  $  X  ^ {*} $[[#References|[a2]]]. If, however,  $  X  ^ {*} $
has order-continuous norm, then  $  X  ^  \odt $
+
has order-continuous norm, then  $  X  ^  \odot $
 
is even a projection band in  $  X  ^ {*} $[[#References|[a8]]]. For a positive  $  C _ {0} $-
 
is even a projection band in  $  X  ^ {*} $[[#References|[a8]]]. For a positive  $  C _ {0} $-
 
semi-group  $  \mathbf T $
 
semi-group  $  \mathbf T $
Line 110: Line 110:
 
$$
 
$$
  
for all  $  x  ^ {*} \in ( X  ^  \odt )  ^ {d} $,  
+
for all  $  x  ^ {*} \in ( X  ^  \odot )  ^ {d} $,  
the disjoint complement of  $  X  ^  \odt $
+
the disjoint complement of  $  X  ^  \odot $
 
in  $  X  ^ {*} $.  
 
in  $  X  ^ {*} $.  
If  $  ( X  ^  \odt )  ^ {d} $
+
If  $  ( X  ^  \odot )  ^ {d} $
 
has a weak order unit, then for all  $  x  ^ {*} \in X  ^ {*} $
 
has a weak order unit, then for all  $  x  ^ {*} \in X  ^ {*} $
 
and  $  t > 0 $
 
and  $  t > 0 $
one has  $  T  ^ {*} ( t ) x  ^ {*} \in ( X  ^  \odt ) ^ {dd } $,  
+
one has  $  T  ^ {*} ( t ) x  ^ {*} \in ( X  ^  \odot ) ^ {dd } $,  
the band generated by  $  X  ^  \odt $
+
the band generated by  $  X  ^  \odot $
 
in  $  X  ^ {*} $[[#References|[a5]]]. If, for some  $  x  ^ {*} \in X  ^ {*} $,  
 
in  $  X  ^ {*} $[[#References|[a5]]]. If, for some  $  x  ^ {*} \in X  ^ {*} $,  
 
the mapping  $  t \mapsto T  ^ {*} ( t ) 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 $
 
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]]].
+
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=45040
This article was adapted from an original article by J. van Neerven (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article