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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
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