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 |
Adjoint semi-group of operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_semi-group_of_operators&oldid=45040