Adjoint semi-group of operators
Let be a -semi-group on a Banach space , i.e. for all and
i) , the identity operator on ;
ii) for all ;
iii) the orbits are strongly continuous (cf. Strongly-continuous semi-group) on for all . On the dual space , the adjoint semi-group , with , satisfies i) and ii), but not necessarily iii). Therefore one defines
This is a norm-closed, weak-dense, -invariant subspace of , and the restriction is a -semi-group on , called the strongly continuous adjoint of . Its infinitesimal generator is the part of in , where is the adjoint of the infinitesimal generator of . Its spectrum satisfies . If is reflexive (cf. Reflexive space), then [a9].
Starting from , one defines and . The natural mapping , , is an isomorphic imbedding with values in , and is said to be -reflexive with respect to if maps onto . This is the case if and only if the resolvent is weakly compact for some (hence for all) [a7]. If is -reflexive with respect to and , then the part of in generates a -semi-group on [a1].
Let be the quotient mapping. If, for some , the mapping is separably-valued, then for all . Hence, if extends to a -group, then is either trivial or non-separable [a4].
If is a positive -semi-group on a Banach lattice , then need not be a sublattice of [a2]. If, however, has order-continuous norm, then is even a projection band in [a8]. For a positive -semi-group on an arbitrary Banach lattice one has
for all , the disjoint complement of in . If has a weak order unit, then for all and one has , the band generated by in [a5]. If, for some , the mapping is weakly measurable, then, assuming the Martin axiom (cf. Suslin hypothesis), for all one has [a6].
A general reference is [a3].
|[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|
Adjoint semi-group of operators. J. van Neerven (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Adjoint_semi-group_of_operators&oldid=18347