Orlicz Lattices [Phd Thesis]

54 Pages·1977·1.593 MB·English

by  W.J. Claas

ORLICZ LATTICES
getypt door mevrouw A.ten Hoorn
ORLICZ LATTICES
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE
WISKUNDE EN NATUURHETENSCHAPPEN
AAN DE RIJKSUNIVERSITEIT TE LEIDEN, 0P GEZAG VAN DE
RECTOR MAGNIFICUS DR. D.J. KUENEN,
HOOGLERAAR IN DE FACULTEIT DER NISKUNDE EN NATUURNETENSCHAPPEN,
. VOLGENS BESLUIT VAN HET COLLEGE VAN DEKANEN
TE VERDEDIGEN 0P WOENSDAG 1 JUNI 1977
TE KLOKKE 16.15 UUR
door
NILLEM JAN CLAAS
GEBOREN TE HEEMSTEDE IN 1950
druk: Krips Repro Meppel
PROMOTOR:PROF.DR.A.C.ZAANEN
COREFERENT:DR.C.B.HUIJSMANS
Research was supported by the Netherlands Organisation for the Advancement
of Pure Research(Z.H.0.).
Aan Froukje
Aan mijn vader
CONTENTS
INTRODUCTION............................................................... 7
CHAPTER I. PRELIMINARIES.................................................11
1. Riesz spaces...'...........................................11
2. Ideals and bands..........................................13
3. Dedekind completeness.....................................15
4. Freudenthal's spectral theorem............................16
5. Riesz homomorphisms.......................................16
6. Normed Riesz spaces.......................................17
7. The order dual of a Riesz space...........................19
8. Normed Riesz spaces with order continuous norm............21
9. Meyer-Nieberg's lemma and order continuous norms..........24
CHAPTER II. ORLICZ SPACES AND MODULARED SPACES............................28
10. Orlicz spaces.............................................28
11. Modulared Riesz spaces....................................30
12. Orlicz lattices...........................................35
13. Direct sums of Orlicz lattices............................38
CHAPTER III. REPRESENTATION 0F ORLICZ LATTICES.............................41
14. Regular Borel measures....................................41
15. The representation theorem................................43
REFERENCES.................................................................49
SAMENVATTING............................................................... 50
INTRODUCTION
A famous theorem, due to F. Bohnenblust (1940), states that for 1 5 p < a
any L -lattice (i.e., any Banach lattice with the norm satisfying pp(f + g):
= pp(f)+ pp(g) for all f,g in the lattice such that inf(f,g)= 0) is isomorphic
to some real L -space of the form Lp(X,A,u), where (X,A,u) is a measure space.
To explain more explicitly what this means, we recall some definitions. The
real partially ordered vector space L is called a Riesz space (or a vector lat"
tice) if the partial ordering is a lattice ordering (i.e., for any f,g in L the
least upper bound or supremum sup(f,g) exists in L just as well as the greatest
lower bound or infimum inf(f,g)).The Riesz space L is called a normed Riesz
space if there exists a norm p in L with the extra property that |f[ 5 |g| im-
plies p(f)5 9(9). The norm 9 is then called a Riesz norm in L. If L is norm
complete with respect to the Riesz norm p, then L is called a Banach lattice.
It is evident that if 1 5 p < m and (X,A,u) is a measure space, then the real
Banach space Lp(X,A,u) is a Banach lattice with respect to the norm
op(f)= (IX lflpdu)1/p-
It is also evident that the norm in LPLX.A,u) is p-additive, which means by def-
inition that
opp(f + g) = ppp(f)+ ppp(g) whenever f A g = 0.
By Bohnenblust's theorem, as mentioned above, any Banach lattice with p-additive
norm is therefore isomorphic to some Lp(X,A,p). The isomorphism preserves the
norm, so it is an isometric isomorphism.
We shall say a few words about the proof. For this we need some further def-
initions. The Riesz space L is said to be Dedekind complete if for any subset
A of L which is bounded from above the least upper bound sup A exists in L. If
sup A exists for any finite or countable subset A in L, that is bounded from
above,then L is said to be Dedekind o-compZete. Bohnenblust's paper contained
the additional assumptions that the L -lattice is Dedekind o-complete and norm
separable. Later developments have shown that these assumptions may be left out.
We shall prove, actually, as a particular case of a more general theorem, that
any Lp-lattice is Dedekind complete. It should be acknowledged here that the
general theorem referred to (theorem 12.3) is based essentially upon a recent
result due to P. Meyer-Nieberg (1973). Before discussing briefly the generalisa-
-3-
tion of Bohnenblust's theorem to Orlicz lattices, we still observe that the
special case p = 1 (abstract L-spaces) is one of the subjects in a paper by
S. Kakutani (1941).
Let o be a real function, defined for all a 3 0 and such that ¢(0)= 0,
0 5 ¢(a)< m for 0 < a < a and lim o(a)= m as a+w. If, in addition, o is convex,
then a is called an Orlicz function (or a Ibung function). If the Orlicz func-
tion o satisfies ¢(2a) 5 C¢(a) for some constant C > 0 and all a, then o is said
to satisfy the (62,A2)—condition.
Let o be an Orlicz function with the (62,A2)-property and let (X,A,u) be a
measure space. The set
L¢ = (f: JX¢(|f|)du < m)
is now a vector space; the set
B = (f: [X¢(Ifl)du 5 1)
is a convex subset of L¢. The Minkowski functional
p¢(f)= inf(a:a > 0,a-1ch)
of B is a norm in L¢. The normed space L¢ = L@(X,A,u) is called an Orlicz space.
(w. Orlicz, 1932). For ¢(a)= up we get L¢ = Lp. This shows that the class of
Orlicz spaces has the Lp-spaces as a subclass.
For feL¢, write
M(f) = JX¢(IfI>du (1)
and for f # 0, let the function wf be defined by wf(a)= M(af)/M(f) for all
a 3 0. Note that for L¢ = Lp we have that ¢f(a)= up, so w does not depend on f
in this case. Conversely, if we have an Orlicz space L¢ in which wf does not de-
pend on f, then L¢ = Lp for some p 3 1. For simplicity, assume now that u(X) is
finite, so the unit function e, defined by e(x)= 1 for all xeX (i.e., e is the
characteristic function of X), is an element of any L¢. Note that although pf
does depend on f to a certain extent, it is at least true that if we take for
f the characteristic function of any measurable subset E of X, then pf does not
depend on the choice of E, i.e., wf = we in this case. This property is expres-
sed by saying that L¢ is component invariant with respect to the unit e.
Finally, note that in analogy with the p—additivity of the norm in an Lp-space.
we have
M(f + g): M(f) + M(g) whenever inf(f,g)= 0.
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The real function M on the Riesz space L is called a modular if M is non-
negative, M(f)= 0 if and only if f = 0, |f| 5 lg] implies M(f) 5 M(g) and M is
convex, i.e.,
M(af + (1 - a)g)5aM(f) + (1 - a)M(g) for 0 5 a 5 1 and f,geL.
The Riesz space L, equipped with a modular is called a modulared Riesz space,
a notion due to H. Nakano (1950). The set
B = (f:M(f)5 1)
is a convex subset of L. The Minkowski functional
pM(f)= inf(aza > 0,a_1feB)
of B is a norm in L. In analogy with the (52,A2)-condition for Orlicz spaces,
we assume now that M(2f)5 CM(f) for some constant C > 0 and all feL. Finally,
we assume also that M(f + g): M(f) + M(g) whenever inf(f,g)= 0. The modular M
is now called an Orlicz modular and in case L is norm complete with respect
to the corresponding norm pm, the space L is called an Orlicz lattice. Any
real Orlicz space L¢(X,A,u) is an Orlicz lattice with respect to the modular
in formula (1).
S.J. Bernau has given a proof (1973, without using any isomorphism theorem)
that in any Lp-lattice the nonn is p-superadditive, i.e.,
pp(f + 9): op(f)+ pp(g) whenever f,g : 0.
He shall prove that, similarly, in any Orlicz lattice we have
M(f + g): M(f)+ M(g) whenever f,g 3 0.
The positive element e in the Riesz space L is called a weak unit if the
only positive f satisfying inf(f,e)= 0 is the zero element. The positive ele-
ment p in L is called a component of the weak unit e, if inf(p,e - p)= 0. Given
the element f # 0 in the Orlicz lattice L, we define the function of by ¢f(a)=
= M(uf)/M(f) for all a 3 0. If L has a weak unit e, then L is called component
invariant with respect to e if up = we for every component p f 0 of e. We shall
prove that every component invariant Orlicz lattice is isomorphic to some Orlicz
space L¢(X,A,u). This extends Bohnenblust's theorem for Lp-spaces. The concluding
section contains some bibliographical remarks.