# Adjoint semi-group of operators

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The semi-group on a dual Banach space composed of the adjoint operators of a -semi-group on (cf. also 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].