View source for Adjoint semi-group of operators

The semi-group on a dual [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104001.png" /> composed of the adjoint operators of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104002.png" />-semi-group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104003.png" /> (cf. also [[Semi-group of operators|Semi-group of operators]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104004.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104006.png" />-semi-group on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104007.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104008.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a1104009.png" /> and

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040010.png" />, the identity operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040011.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040013.png" />;

iii) the orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040014.png" /> are strongly continuous (cf. [[Strongly-continuous semi-group|Strongly-continuous semi-group]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040016.png" />. On the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040017.png" />, the adjoint semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040018.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040019.png" />, satisfies i) and ii), but not necessarily iii). Therefore one defines

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This is a norm-closed, weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040022.png" />-dense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040023.png" />-invariant subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040024.png" />, and the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040025.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040026.png" />-semi-group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040027.png" />, called the strongly continuous adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040028.png" />. Its infinitesimal generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040029.png" /> is the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040032.png" /> is the adjoint of the infinitesimal generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040034.png" />. Its spectrum satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040036.png" /> is reflexive (cf. [[Reflexive space|Reflexive space]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040037.png" /> [[#References|[a9]]].

Starting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040038.png" />, one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040040.png" />. The natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040042.png" />, is an isomorphic imbedding with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040043.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040044.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040046.png" />-reflexive with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040047.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040048.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040049.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040050.png" />. This is the case if and only if the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040051.png" /> is weakly compact for some (hence for all) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040052.png" /> [[#References|[a7]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040053.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040054.png" />-reflexive with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040056.png" />, then the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040057.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040058.png" /> generates a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040059.png" />-semi-group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040060.png" /> [[#References|[a1]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040061.png" /> be the quotient mapping. If, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040062.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040063.png" /> is separably-valued, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040064.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040065.png" />. Hence, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040066.png" /> extends to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040067.png" />-group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040068.png" /> is either trivial or non-separable [[#References|[a4]]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040069.png" /> is a positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040070.png" />-semi-group on a [[Banach lattice|Banach lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040071.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040072.png" /> need not be a sublattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040073.png" /> [[#References|[a2]]]. If, however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040074.png" /> has order-continuous norm, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040075.png" /> is even a projection band in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040076.png" /> [[#References|[a8]]]. For a positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040077.png" />-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040078.png" /> on an arbitrary Banach lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040079.png" /> one has

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040081.png" />, the disjoint complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040082.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040083.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040084.png" /> has a weak order unit, then for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040086.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040087.png" />, the band generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040088.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040089.png" /> [[#References|[a5]]]. If, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040090.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040091.png" /> is weakly measurable, then, assuming the Martin axiom (cf. [[Suslin hypothesis|Suslin hypothesis]]), for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040092.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110400/a11040093.png" /> [[#References|[a6]]].

A general reference is [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grabosch,   R. Nagel,   "Order structure of the semigroup dual: A counterexample"  ''Indagationes Mathematicae'' , '''92'''  (1989)  pp. 199–201</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.M.A.M. van Neerven,   "The adjoint of a semigroup of linear operators" , ''Lecture Notes in Mathematics'' , '''1529''' , Springer  (1992)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.M.A.M. van Neerven,   B. de Pagter,   "The adjoint of a positive semigroup"  ''Comp. Math.'' , '''90'''  (1994)  pp. 99–118</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  B. de Pagter,   "A characterization of sun-reflexivity"  ''Math. Ann.'' , '''283'''  (1989)  pp. 511–518</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B. de Pagter,   "A Wiener–Young type theorem for dual semigroups"  ''Acta Appl. Math. 27''  (1992)  pp. 101–109</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R.S. Phillips,   "The adjoint semi-group"  ''Pacific J. Math.'' , '''5'''  (1955)  pp. 269–283</TD></TR></table>