© Hindawi Publishing Corp.
ON THE EXTENSION OF POSITIVE LINEAR FUNCTIONALS
ERGASHBOY MUHAMADIEV and ADEL T. DIAB
(Received12 September 1997 andin revisedform 20 April 1998)
Abstract.A necessary andsufficient condition to extenda continuous linear real func-tionals which is positive with respect to a semi-group defined on a subspace of a linear space is discussedin this paper. The case of a closedsubspace of a Banach space is also discussed.
Keywords andphrases. Solidsemi-group, extension functional, positive functional, mono-tonic Banach functional.
2000 Mathematics Subject Classification. 46A22, 46A40.
1. Introduction. LetEbe a real linear space with a linear semi-groupK(we will call the setK⊂Ea linear semi-group if it possesses the following properties:K+K⊂K, andα·K⊂Kfor allα≥0) withK⊂E(the word, linear will be omitted) and letL⊂E be a subspace ofE. Also, letf0be a real linear functional onL. A real linear functional
fonEis calledan extension off0iff (x)=f0(x)for everyx∈L. A linear functional
fwill be calledpositive (with respect toK) iff (x)≥0 for everyx∈K.
In the theory of spaces with an associatedsemi-group, the existence of the extension of positive linear functional plays an important role. Later on, several mathematicians studied a necessary condition to extend a linear continuous positive functional, for example, M. G. Krein theorem [4] states that “ifK≠E is a semi-group in a Banach spaceEwith interior point andthe subspaceL⊂Econtains at least one interior point ofK, then each linear continuous positive functional onL(i.e.,f0:L→Rsuch that
f0(x)≥0 for everyx∈K∩L=KL) can be extended to a linear continuous positive functionalf defined onE(i.e.,f:E→Rsuch thatf0(x)=f (x)for everyx∈L, and
f (x)≥0 for everyx∈K).” Other important theorems on the existence of positive extension of linear functionals are given in [1, 2, 3].
2. Main results. In the present work, we introduce a new condition for the existence of the extension of positive linear functionals which depend upon the definition of Banach functional [5]. Two cases will be considered separately.
Case (A). Eis a real linear space with a semi-groupK. The functionalω:E→Ris calleda Banach functional if (1) ω(x+y)≤ω(x)+ω(y)for everyx,y∈E, (2) ω(λ·x)=λ·ω(x)for everyx∈E, for everyλ≥0.
We say that the Banach functionalωis monotonic (with respect toK), ifω(x+y)≥
Theorem2.1. LetL⊂E be a subspace of a linear space E, andf0: L→R be a
linear functional such thatf0(x)≥0for everyx∈KL. Then a necessary and sufficient
condition for the existence of a linear functionalf:E→R, such thatf0(x)=f (x)for
everyx∈Land f (x)≥0for every x∈K, is the existence of a monotonic Banach functionalωsatisfying the inequality
f0(x)≤ω(x) for everyx∈L. (2.1)
Proof. Let the functionalf be an extension off0. We define the functionalωby
ω(x)=inf{|f (x+y)|:y∈K}, where x∈E. Obviously, ωis a monotonic Banach functional which satisfies inequality (2.1). (Another proof,f is already a monotonic Banach functional andhence we takeω=f.)
Conversely, consider the existence of a monotonic Banach functionalωsatisfying inequality (2.1). Then by the Hahn-Banach theorem there exists a linear functionalf onE such thatf0(x)=f (x)for every x∈Land f (x)≤ω(x)for everyx∈E. It remains to prove thatfis positive. Letx∈K, then from the fact thatωis monotonic, we getω(−x)≤0 andthereforef (−x)≤ω(−x)≤0, which implies thatf (x)≥0. Hence f is a positive extended functional of f0. This completes the proof of the theorem.
In the following theorem, the condition that the Banach functional is monotonic is dropped.
Theorem2.2. LetL⊂Ebe a subspace of a linear spaceE,be a Banach functional (not necessarily monotonic), andf0:L→Rbe a linear functional such that
f0(x)≤(x+y) for everyx∈L, for everyy∈K. (2.2)
Thenf0has an extension functionalfonEsuch thatf (x)≥0for everyx∈K.
Proof. Letω(x)=inf{(x+y):y∈K}for everyx∈E. We start by showing that
(i) ω(x) >−∞for everyx∈E, (ii) f0(x)≤ω(x)for everyx∈L, (iii) ωis a monotonic Banach functional.
Condition (i) is a consequence of inequality (2.2) for, if x = 0, then (y) ≥ 0 for everyy∈K, butis a Banach functional, then 0≤(y)=(x+y+(−x))≤
(x+y)+(−x), which implies that(x+y)≥ −(−x)for everyy∈Kandfor everyx∈E. Therefore,ω(x)≥ −(−x) >−∞ for everyx∈E, this proves condi-tion (i). Sincef0(x)≤(x+y)for everyx∈L, everyy∈K, thenf0(x)≤ω(x)for everyx∈E. This proves condition (ii).
Clearlyω is a Banach functional andit remains to show that ωis a monotonic functional (i.e.,ω(x)≤ω(x+y)for everyy∈Kandeveryx∈E). Sinceω(x+y)= inf{(x+y+y1):y1∈K} =inf{(x+z):z=y+y1∈K,y1∈K} ≥inf{(x+
Case (B). Eis a Banach space with closedsemi-groupK. We introduce the following definition.
Definition. We define the following functional|x|K=inf{x+y:y∈K}for eachx∈E. Obviously,|·|Kis a monotonic functional with respect to the semi-group Kand| · |K is a Banach functional, moreover,|x|K≤ xfor everyx∈E(hence the functional| · |K is a continuous functional onE). Also, from the closeness of a semi-groupK, we have|x|K=0 if andonly if−x∈K.
Example1. LetE=2= {x=(ξi):i|ξi|2<∞},K={(ξi)∈2:ξi≥0,i=1,2,...}. Obviously, for every x = (ξi) ∈ 2, x = x+ −x−, where x+ = 1/2(|ξi| +ξi),
x−=1/2(|ξi|−ξi), thenx+,x−∈Kand|x|K= x+. The following theorem is analogous to Theorem 2.1.
Theorem2.3. LetLbe a closed subspace of a Banach spaceEwith a closed semi-group K, and let f0:L→R be a linear continuous functional such thatf0(x)≥0
for everyx∈KL. Then a necessary and sufficient condition that there exists a linear
continuous functionalf:E→Rsuch thatf0(x)=f (x)for everyx∈Landf (x)≥0
for every x∈K is that there exist numbersα >0, β >0 and a monotonic Banach functionalωthat satisfies the inequalities
f0(x)≤α·ω(x) for everyx∈L, (2.3)
ω(x)≤β·x for everyx∈L. (2.4) Proof. For the proof of the theorem, we needthe following lemma.
Lemma2.1. The linear continuous functional f is positive with respect to a semi-groupKif and only if there exists a numberC >0such that
f (x)≤C·|x|K for everyx∈E. (2.5)
Proof. Iff is positive, thenf (y)≥0 for everyy∈K. Letx∈E, thenx≤x+y for every y ∈K. This implies that f (x+y)=f (x)+f (y)≥f (x) andtherefore f (x)≤f (x+y)≤ |f (x+y)| ≤ fx+y. By taking the infimum of both sides of the previous inequality, we getC= f,f (x)≤C·|x|K. Conversely, letf be a linear continuous functional andthere exists a numberC >0 such that f (x)≤C· |x|K for everyx∈E. In particular ify∈K, then−y∈ −K andwe obtain that−f (y)=
f (−y)≤C|−y|K=0. Hence,f (y)≥0 for everyy∈K. Now we return to the proof of Theorem 2.3.
Let the linear continuous functionalf0on Lhas a linear continuous positive ex-tension functional f on E. Then by Lemma 2.1, there exists a number C >0 such that f (x) ≤C· |x|K for every x ∈E. Put ω(x)= |x|K, α= C and β= 1. Hence f0(x)≤C·ω(x)and ω(x)≤ x for everyx∈L. Conversely, letf0 be a contin-uous linear functional onLwhich satisfies inequalities (2.3) and(2.4). Hence by the Hahn-Banach theorem there exists an extension functionalf of f0on E such that
f (x)≤α·ω(x)for everyx∈E. Hence from the fact thatωis monotonic, we obtain thatf (x)≥0 for everyx∈K, andfrom the following inequality:
we obtain thatf is a linear continuous functional onE. This completes the proof of the theorem.
Corollary2.1. The linear continuous positive functionalf0on a closed subspace
Lof a Banach spaceEhas a linear continuous positive extension functionalf onEif and only if there exists a numberC >0such that
f0(x)≤C·|x|K for everyx∈L. (2.7)
Proof. This follows directly from Theorem 2.3.
In the following section, we discuss the relationship between the two functionals |x|Kand|x|KL, where|x|KL=inf{x+y:y∈KL}for eachx∈L.
It is clear that|x|K≤ |x|KLfor everyx∈L. From Lemma 2.1, iff0is a linear contin-uous functional on a closedsubspaceLin a Banach spaceEsuch thatf0(x)≥0 for everyx∈KL, then there exists a numberC >0 such thatf0(x)≤C· |x|KL for every x∈L.
Theorem2.4. GivenC >0such that|x|KL≤C· |x|Kfor everyx∈L. Then every
continuous linear positive (with respect to a closed semi-groupKL) functionalf0on a
closed subspaceLhas a linear continuous positive (with respect to a closed semi-group
K) extension functionalf onE.
Proof. The proof follows directly from Lemma 2.1 and Corollary 2.1.
We have to note that M. G. Krein theorem can be obtaineddirectly from Theorem 2.4 andthe following lemma.
Lemma2.2. LetK≠Ebe a semi-group in a Banach spaceEwith interior point (i.e.,
Kis a solid semi-group), and let the closed subspaceL⊂Econtains at least one interior point ofK. Then there exists a numberC >0such that|x|KL≤C·|x|Kfor everyx∈L. Proof. Letu0∈L be an interior point ofK, then there existsδ0>0 such that
T (u0,δ0)⊂K, where T (u0,δ0)= {u:u−u0 ≤δ0}. It is well known thatuα=
α·u0+(1−α).y,0< α <1 are the interior points ofKwith radiusα·δ0(i.e.,T (uα,
α·δ0)⊂K)for every y∈K. Now, we suppose that x∈L andx =1 thenZα=
(α·u0−(1−α)·x)∈Land Zα−uα =(1−α)x+y. Therefore, if we choose
α= x+y·(δ0+x+y)−1, thenZα∈Kand |x|KL≤ x+Zα =α·|u0+x
≤u0+1·δ0+x+y−1x+y ≤u0+1δ−1
0 x+y =C·x+y,
(2.8)
whereC=(δ0)−1(uo+1). By taking the infimum of both sides of the last inequality aty∈K, we get
The following example shows that, if the semi-groupKdoes not contain an interior point, then M. G. Krein theorem cannot be appliedbut our work can be applied.
Example2. Back to Example 1, it is known thatKis not solidsemi-group inE=2. Ifx1=(1,1/2,...,1/n,...),x2=(1,1/22,...,1/n2,...), andL= {αx1+βx2:α,β∈R} is a subspace ofE, thenKL=K∩L= {λx1+µx2:λ≥0,λ+µ≥0}. Letf0:L→Rbe defined byf0(x)=f0(αx1+βx2)=αC1+βC2, whereC1and C2are real numbers. Then ifC1−C2>0,C2>0, the functionalf0satisfies inequality (2.7). Therefore, from Corollary 2.1, the linear continuous positive functionalf0onLhas a linear continuous positive extension functionalf onE. We remark here that ifC2=0 andC1>0, then clearly, the positive functionalf0has no positive functional extension onE.
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Muhamadiev: Mathematical and Informational Department, Khujand’s Scientific Center, Academyof Sciences, Khujand, P.O.3Box113, Tajikistan
E-mail address:[email protected], [email protected]
Diab: Mathematics Department, Facultyof Science, Ain Shams University, Cairo, Egypt
