Adjoint semi-group of operators

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