# Adjoint semi-group of operators

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].

#### 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. J. van Neerven (originator),

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