Volume 2008, Article ID 648591,13pages doi:10.1155/2008/648591
Research Article
Fixed Point Theorems for Middle Point Linear
Operators in
L
1
Milena Chermisi1and Anna Martellotti2
1Fachbereich Mathematik, Universit¨at Duisburg-Essen, Lotharstraße 65, 47057 Duisburg, Germany
2Dipartimento di Matematica e Informatica, Universit`a di Perugia, via Vanvitelli 1, 06123 Perugia, Italy
Correspondence should be addressed to Milena Chermisi,[email protected]
Received 3 February 2008; Revised 11 August 2008; Accepted 8 September 2008
Recommended by Klaus Schmitt
We introduce the notion of middle point linear operators. We prove a fixed point result for middle point linear operators inL1. We then present some examples and, as an application, we derive
a Markov-Kakutani type fixed point result for commuting family ofα-nonexpansive and middle point linear operators inL1.
Copyrightq2008 M. Chermisi and A. Martellotti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Furi and Vignoli 1 proved that any α-nonexpansive map T : K→K on a nonempty, bounded, closed, convex subsetKof a Banach spaceXsatisfies
inf
x∈KTx−x0, 1.1
where α is the Kuratowski measure of noncompactness on X. It is of great importance to obtain the existence of fixed points for such mappings in many applications such as eigenvalue problems as well as boundary value problems, including approximation theory, variational inequalities, and complementarity problems. Such results are used in applied mathematics, engineering, and economics.
In this paper, we give optimal sufficient conditions forTto have a fixed point onKin case thatX L1μ, whereμis anσ-finite measure, and as a minor application, in case that Xis a reflexive Banach space.
The study of fixed point theory has been pursued by many authors and many results are known in literature. In order to have an overview of the problem, we present a brief survey of most relevant fixed point theorems. Darbo2showed that anyα-contractionT :
a Banach space. Later, Sadovski˘ı 3extended the Darbo’s result forα-condensing mappings. Belluce and Kirk 4 obtained fixed point results for nonlinear mappings T, defined on a convex and weakly compact subsetKof a Banach space, for whichV :I−Tsatisfies
V
xy
2
≤ 1
2VxVy, for anyx, y∈K. 1.2
Lennard5proved that any nonexpansive map T : K→K has at least one fixed point on every nonempty,·L1-bounded,ρ-compact, convex subset ofL1μ, whereρis the metric of
the convergence locally in measure. For other results, we refer to6–11. The purpose of this paper is two-fold:
ito introduce the notion of middle point linear operator, which extends the notion of convexity in the sense of1.2;
iito show that any continuous operatorT :K, ρ→K, ρhas at least one fixed point in K, wheneverK is a nonempty, ·L1-bounded, ρ-closed, and convex subset of
L1μandT is middle point linear andα-nonexpansive.
The class of middle point linear operators, which are defined in Definition 3.1, comprises not only convex operators in the sense of 1.2 but also affine operators. However, as shown inExample 3.3, middle point linear operators are not necessarily affine. We provide a characterization of middle point linear operators and present some useful properties, such as the stability under pointwise convergence, the convexity of the set of fixed points, and the fact that it suffices to test middle point linearity on a dense subsets of the domain.
The fixed point theorem, which is the main result of the paper, is stated inTheorem 4.5. The idea is to prove, exploiting a result of Bukhvalov12, that the functionalx→ Tx−xL1
attains its minimum value on every nonempty,·L1-bounded,ρ-closed, and convex subset
ofL1μ seeLemma 4.3; the conclusion simply follows as a consequence of Furi-Vignoli’s
Theorem1.
Using a slightly different argument, it is also possibleseeRemark 4.6to prove a fixed point theorem for middle point linear operators defined on a convex and weakly compact subset of an arbitrary Banach space, generalizing some previous results of Belluce and Kirk
4, Theorems 4.1, 4.2. In particular, this implies that anyα-nonexpansive and middle point linear operatorT :K→Khas at least one fixed point on every nonempty, bounded, closed, and convex subsetKof a reflexive Banach space.
We remark that Theorem 4.5 is optimal as Example 4.7 shows that the assumption of middle point linearity onT cannot be avoided, even when K is assumed to be weakly compact. We also present several examples, namely, Examples4.10–4.12, which show that Theorem 4.5applies in situation where neither Sadovski˘ı’s theorem nor Lennard’s theorem does.
A first application ofTheorem 4.5leads to a fixed point result for uniform limits of middle point linear operatorsseeProposition 4.9. As a second application ofTheorem 4.5, we derive a generalization of Markov-Kakutani theorem see 9, 13; more precisely, we show that any commuting familyFofα-nonexpansive and middle point linear operators has a common fixed point onK, wheneverT :K, ρ→K, ρis continuous for anyT ∈ FandK
is a nonempty,·L1-bounded,ρ-closed, convex subset ofL1μ seeTheorem 4.14.
present some useful properties. In Section 4we show all the above-mentioned fixed point results for middle point linear operators inL1μ and in Banach spacesand we present the
examples.
2. Preliminaries
LetΩ,Σ, μbe anσ-finite measure space, and letΩm∞m1be aμ-partition ofΩwithμΩm<
∞. We denote byMμthe collection of all equivalence classes of functionsx:Ω→Rwhich areμ-measurable and finite almost everywhere, modulus theμ-a.e. equivalence.Mμcan be endowed with the metric
ρx, y:
∞
m1
1 2m
1
μΩm
Ωm
|x−y|
1|x−y|dμ, x, y∈Mμ. 2.1
It is well known that the metric ρ is translation-invariant and induces the topology of convergence locally in measure. IfμΩ<∞, then the topology of the convergence locally in measure onMμis equivalent to the topology of the convergence in measure which is induced by the metric
ρx, y:
Ω
|x−y|
1|x−y|dμ. 2.2
Throughout the paper, the symbol·will denote either the norm of a generic normed space or the norm ofL1μ. SinceL1μwill be endowed with the norm topology and the topology
induced byρ, we will say that a subset ofL1μis boundedresp., closed and completeif
it is·L1-boundedresp.,·L1-closed and·L1-complete. We denote byX1the closed unit
ball ofL1μ.
Remark 2.1. Remember thatMμ, ρis a Fr`echet space. Ifμis finite,Mμ, ρis exactly the metric completion of the metric linear spaceL1μ, ρ. We recall also that, any subsetAof L1μis closedand hence completewheneverAisρ-closed.
Given a metric spaceX and a bounded setA ⊂X, we denote byαAtheKuratowski measures of nonompactnessofA, that is,
αA:inf{ε >0 :Acan be covered by finitely many sets of diameter ≤ε}. 2.3
For the properties and examples, we refer to14or11. A mapT:X→Xon a normed spaceX,·is called
nonexpansiveifTx−Ty ≤ x−yfor everyx, y∈X;
α-nonexpansiveif it is continuous andαTA≤αAfor everyA⊂X.
In the sequel, we will simply writenonexpansivefor maps inL1μthat are·
3. Middle point linear operators
We introduce a new class of operators in normed spaces.
Definition 3.1. LetX1be the closed unit ball of a normed spaceXand letKbe a convex subset
ofX alsoK X. A continuous operatorT : K→X is saidmiddle point linearif for every positive numberr >0, the following property holds:
T
xy
2
∈ xy
2 rX1, wheneverx, y∈K, Tx∈xrX1, Ty∈yrX1. 3.1
Remark 3.2. AnyaffineoperatorT :X→Xon a normed spaceX, that is, any map such that
Tax 1−ay aTx 1−aTy, for any x, y∈X, a∈0, 1, 3.2
is middle point linear.
However, as it is shown in the following example, middle point operators need not be affine.
Example 3.3. Let ϕ : 0,∞→0,1 be a nonincreasing continuous function. Define the operatorT :X→Xas
Tx:ϕxx, 3.3
for allx∈X. The operatorTis middle point linear. Indeed, fixr >0 and choosex, y∈Xsuch that
x−Tx 1−ϕxx ≤r, y−Ty 1−ϕyy ≤r. 3.4
Without loss of generality, we can assume thatx ≤ y. This implies thatxy/2 ≤ y
and, by the monotonicity ofϕ, ϕy≤ϕxy/2. Then
x2y−T
xy
2
1−ϕ
xy
2
xy
2
≤1−ϕyy ≤r, 3.5
that is,Txy/2∈xy/2rX1.
Remark 3.4. It is rather natural to compare this new definition with the usual convexity in the real line, namely, in case thatXR.
There exist convex middle point linear mappings, asx→ expxfor anyx ∈0,∞. However, convex functions are not necessarily middle point linear; for instance, the map
h : 0,1→0,1, defined by hx : x2 for allx ∈ 0,1, is not middle point linear since
Remark 3.5. Condition 1.2 implies 3.1, but the converse is not true. For instance, the operatorT : X1→X1, defined as in3.3with ϕr : 1−r2,r ∈ 0,1, is middle point
linear but1.2is not satisfied, since the mapr→1−ϕrris not convex in0,1. Another example is given by the mapSx:x−1e−x2
for anyx∈−M, MwithMlarge enough.
It is easy to prove that ifT is middle point linear then property3.1holds for every convex combination ofxandy, namely, the following statement holds.
Proposition 3.6. LetXbe a normed space andT :X→Xbe a middle point linear operator. For each r >0, ifx, y∈Xare such thatTx∈xrX1andTy∈yrX1, then
Tλx 1−λy∈λx 1−λyrX1, 3.6
for eachλ∈0,1.
Proof. The proof is based on a standard procedure. First, we prove the statement for every dyadic rational in0,1and then, by the continuity ofT, for every numberλ∈0,1.
The following characterization of middle point linear operators is a direct consequence ofDefinition 3.1andProposition 3.6.
Proposition 3.7. LetXbe a normed space andT :X→Xbe a continuous operator. ThenT is middle point linear if and only if the functionalf:X→Rdefined asfx:Tx−xis quasiconvex, that is, the set{x∈X:fx≤r}is convex inXfor anyr >0.
Remark 3.8. The class of affine operators enjoys closedness under convex combination as well as under usual map composition. In general, this is not true for middle point linear operators. To see this, letXRandS, T :R→Rbe defined bySx:x−2x2andTx:x2x4.
FromProposition 3.7,SandT are middle point linear since both the mappingsx → |Sx− x|2x2andx→ |Tx−x|2x4are convex. However,R: 1/2TSis not middle point
linear, sinceRx−x|x4−x2|is not quasiconvex.
Consider now the operatorU:R→R, defined byUx:x−x3. Clearly,Uis middle point linear sinceUx−x|x3|is quasiconvex, butU2is not sinceU2x−x|x3x6−
3x43x2−2|fails to be quasiconvex.
Pointwise convergence respects middle point linearity, namely, the following result can easily be checked.
Proposition 3.9. LetTn:X→Xbe a sequence of middle point linear operators, and letT :X→Xbe
its pointwise limit (i.e.,limn→∞Tnx−Tx0for everyx∈X). ThenT is middle point linear.
It is also interesting to note that it suffices to test property3.1on dense subsets ofX
as the following proposition shows.
Proposition 3.10. LetT : X→X be a continuous operator andDbe a dense subset ofX. If 3.1
holds for any pairx, y∈D, thenTis middle point linear onX.
that
x−xδ ≤δ, y−yδ ≤δ. 3.7
Then
Tx−Txδ ≤ε, Ty−Tyδ ≤ε. 3.8
Sincexδ−Txδ ≤ xδ −xx−TxTx−Txδ ≤ rδε, and analogously yδ−Tyδ ≤rδεthere follows
xδyδ
2 −T
xδyδ 2
≤rδε, 3.9
whence letting firstδand thenεgo to 0, we reach the conclusion.
4. Fixed points for middle point linear operators inL1μ
From now on,Ω,Σ, μwill be an σ-finite measure space. In this section, we present fixed point theorems for middle point operators inL1μ. For convenience of the reader, we first
recall the following result of Bukhvalov, calledoptimization without compactness.
Theorem 4.1see12. LetCnnbe a family of bounded,ρ-closed, and convex sets having the finite
intersection property. Then the intersectionnCnis nonempty.
We now prove, usingTheorem 4.1, the following lemma, which generalizes a result still due to Bukhvalov12 see also15.
Lemma 4.2. LetKbe a nonempty, boundedρ-closed, convex subset ofL1μ. Then any quasiconvex functionalf :K→Rwhich is lower-bounded and lower semicontinuous with respect toρ, attains its minimum value onK.
Proof. Let a : infx∈Kfx and xnn be a minimizing sequence in K such that fxn is decreasing and converges toa. Then
a≤fxn1< fxn, 4.1
for alln. Consider the sublevel sets offdefined as
Fn:{x∈K|fx≤fxn}. 4.2
Using Lemma 4.2, we obtain the following minimum property for middle point operators inL1.
Lemma 4.3. LetKbe a nonempty, bounded,ρ-closed, convex subset ofL1μ.
If T : K, ρ→K, ρ is a continuous, middle point linear operator, then the real-valued functionalfx:Tx−xattains its minimum value onK.
Proof. Clearly, the functional f is lower bounded by 0 and ρ-lower semicontinuous since
T is ρ-continuous and also the norm ·L1 : K→0,∞ is ρ-lower semicontinuous. By
Proposition 3.7, it is quasiconvex. The conclusion then follows fromLemma 4.2.
We need now the following result due to Furi and Vignoli.
Theorem 4.4see1. LetX be a Banach space,αthe Kuratowski measure of noncompactness on
X, andKa nonempty, bounded, closed, convex subset ofX. IfT :K→K isα-nonexpansive, then
infx∈KTx−x0.
FromLemma 4.3andTheorem 4.4, we obtain the following fixed point theorem inL1,
which is the main result of our paper.
Theorem 4.5. LetKbe a nonempty, bounded,ρ-closed, convex subset ofL1μ.
IfT : K, ρ→K, ρis a continuous,α-nonexpansive, middle point linear operator, thenT has at least one fixed point inK.
Remark 4.6. Using a slightly different argument, it is possible to obtain the following fixed point results for middle point operators in Banach spaces.
BSIf K is a nonempty, weakly compact, and convex subset of a Banach space and
T :K→K is a middle point linear operator satisfying infx∈KTx−x 0, thenT has a fixed point inK.
RBSAnyα-nonexpansive and middle point linear operatorT :K→Kon a nonempty, bounded, closed, convex subsetKof a reflexive Banach space has at least one fixed point inK.
Since any bounded, closed, convex subset of a reflexive Banach space is weakly compactsee
16, Corollary III.19, we obtainRBSas a consequence ofBSandTheorem 4.4. Now, to proveBS, let us pickξ∈Ksuch thatc0Tξ−ξ<∞. The set
C:{x∈K:Tx−x ≤c0} 4.3
is weakly compact since it is a closed and convex subset of a weakly compact set. Furthermore, the functionalfx : Tx−x < ∞is quasiconvex and continuous. This implies thatf is lower semicontinuous with respect to the weak topology, since for every
c ∈ Rthe set f−1c,∞is weakly open. Thus,f attains its minimum value onC: there
exists a point x0 ∈ Csuch thatfx0 ≤ fxfor anyx ∈ C. Clearly, fx0 ≤ fxfor any x∈K, that is,x0is a fixed point underT.
mappingsTin Banach spaces for whichV :I−Tsatisfies1.2. Recall that condition1.2is stronger than3.1 seeRemark 3.5.
According to the results in12, one can presume thatρ-closedness of bounded convex sets in L1μ is a sufficient surrogate to compactness. The next example shows that the
assumption that T is middle point linear cannot be avoided, even when K is assumed to be weakly compact.
Example 4.7. In17Alspach has given the following example of fixed-point free map. Let
Ω 0,1with the usual Lebesgue measureμ, and let
K:{x∈L1μ, 0≤x≤2, x1}. 4.4
Then K is convex, closed actually weakly compact, and ρ-closed, since in K we have dominated convergence. Consider the operatorT :K→Kdefined by
Txt:
⎧ ⎪ ⎨ ⎪ ⎩
2x2t∧2 when 0≤t≤ 1
2
2x2t−1−2∨0 when 1
2 < t≤1.
4.5
Then, according to17,Tis an isometrytherefore, it is nonexpansive and norm continuous but it has no fixed point. Again, since in K every convergence is dominated, T is also continuous as a self map ofK, ρ.
According toTheorem 4.5,Tcannot be middle point linear. Indeed for the maps
xt:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 3
2 when 0≤t≤ 1 2 1
2 when 1
2 < t≤1,
y:213/8,1/2∪5/8,1,
4.6
where1Ais the characteristic function of the setA, one findsx−Txy−Ty1/2, while
ξ0:
xy
2
3
410,3/4 7
413/4,1/2 1
411/2,5/4 5
415/4,1 4.7
satisfiesTξ0−ξ047/64>1/2.
Remark 4.8. In 7 the two authors established that an α-Lipschitz map T : Q→Q with constantk≥1 defined on a bounded and convex subsetQof a normed space has the property that
ηT:inf
x∈Qx−Tx ≤
1−1
k
χ0K, 4.8
where χ0K is the infimum of all δ > 0 such that K admits a finite dimensional δ
a minimum: indeed,T as in4.5is an isometry, and henceα-Lipschitz with constantk1. It follows thatηT 0, but there exists no pointx0ofKsatisfying
x0−Tx0 ≤ x−Tx 4.9
for everyx∈K, sinceTis a fixed point free map onK.
The following result concerns approximation of the fixed points ofT.
Proposition 4.9. Assume thatμis a finite measure. LetKbe a nonempty, bounded,ρ-closed, convex subset ofL1μ, and letT
n : K, ρ→K, ρbe a sequence of continuous, middle point linear,
α-nonexpansive operator, uniformly norm-converging to someT(namely,Tnx−Tx →0uniformly
inK). ThenThas a fixed point.
Proof. By Theorem 4.5, we know that eachTn has at least one fixed point xn ∈ K. By12, Theorem 1.4, there exist an increasing sequence of integersn1 < n2 < · · · < nk < · · ·, an
x0∈L1μ, and a sequenceλnn⊂0,1such that
nj
inj−1
λi1, ξj :
nj
inj−1
λixi ρ
−→x0. 4.10
Then, from theρ-continuity of eachTn, Tnξj ρ
→ Tnx0. SinceTnxn xnfor everyn∈N, andTnnconverges uniformly toTinKwe deduce thatTxn−xn ≤εfornsuitably large. ByProposition 3.9,T is middle point linear, and henceTξj−ξj ≤εforjsuitably large, in force ofProposition 3.6.
Now we find
ρTx0, x0≤ρTx0, Tξj ρTξj, ξj ρξj, x0
≤ρTx0, Tnx0 ρTnx0, Tnξj ρTnξj, Tξj Tξj−ξjρξj, x0
≤ Tx0−Tnx0ρTnx0, Tnξj Tnξj−TξjTξj−ξjρξj, x0 < ε
4.11
fornandjsuitably large. ThereforeTx0 x0.
Note that, under the above assumptions,T is middle point linear, byProposition 3.9 and continuous also with respect to the ρ-topology, but we do not know if T is α -nonexpansive; therefore we cannot apply directlyTheorem 4.5.
We give now some applications of Theorem 4.5 in L10,1, μ, where μ denotes
now the Lebesgue measure. The following examples show also thatTheorem 4.5applies in situation where neither Sadovski˘ı’s fixed point theorem3nor Lennard’s fixed point theorem
5does. In fact, in Sadovski˘ı’s theorem,T is required to beα-condensing, while in Lennard’s Theorem,Kneeds to beρ-compact andT nonexpansive.
Example 4.10. LetT :X1→X1be defined as in3.3for allx∈X1, whereϕ :0,1→0,1is
Clearly,T :X1,·→X1,·is continuous since
Tx−Tx0ϕxx−ϕx0x0 ≤ϕxx−x0|ϕx−ϕx0|x0. 4.12
Moreover,T isα-nonexpansive. In fact, ifB⊂X1, then
αTB≤αco{B,0}αB∪ {0} max{αB, α{0}}αB, 4.13
sinceTx ϕxx 1−ϕx0∈co{B,0}for everyx∈B.
To show thatT : X1, ρ→X1, ρis continuous, fixx0 ∈ X1 and a sequencexnn of points ofX1converging locally in measure tox0, that is, limn→∞ρxn, x0 0. Then, sinceρ
is translation invariant andρcx,0≤ρx,0for everyx∈L1μandc∈−1,1, we have
ρTxn, Tx0≤ρϕxnxn−x0,0 ρϕxn−ϕx0x0,0
≤ρxn−x0,0 ρϕxn−ϕx0x0,0.
4.14
The second summand in4.14tends to 0 sinceϕis continuous and limn→∞ρcnx,0 0 for everyx ∈L1μand every sequencec
nnof real numbers with limn→∞cn 0. Hence, the conclusion follows.
Therefore, by Theorem 4.5, T has a fixed point in X1. Notice that T0 0 and Tr∂X1 ϕrr∂X1 for every r ∈ 0,1. Note also that both Sadovski˘ı’s theorem and
Lennard’s theorem cannot be applied, sinceTis neitherα-condensing nor nonexpansive and
X1is notρ-compact.
Example 4.11. LetT:L10,1→L10,1be defined as
Txt: 1 2x
t
2
, t∈0,1. 4.15
The operatorTis linear and hence middle point linear. Since
Tx
1
0
1 2
x
t
2
dμt
1/2
0
|xs|dμs≤ x, ∀x∈L10,1, 4.16
T :K,·→K,·is continuous and nonexpansiveand henceα-nonexpansiveonX1.
Moreover, since for allx0, x∈X1,
ρTx, Tx0 1
0
|xt/2−x0t/2|
2|xt/2−x0t/2|dλt≤ 1
0
|xt/2−x0t/2|
1|xt/2−x0t/2|dλt
2
1/2
0
|xs−x0s|
1|xs−x0s|dλs≤
2ρx, x0,
4.17
Therefore, fromTheorem 4.5,T has a fixed point inX1. In particular,T0 0. Notice
that we cannot apply Lennard’s theorem sinceX1is notρ-compact and that it is not easy to
understand whetherTisα-condensing.
Example 4.12. Consider the multiplication operatorMf : L10,1→L10,1 defined for all
x∈L10,1as
Mfxt:ft·xt, ∀t∈0,1, 4.18
wheref∈L∞0,1×0,1withf∞≤1. The operatorMf is linear, bounded withMf f∞henceMf :K,·→K,·continuous, andMf is nonexpansive.
Moreover, Mf : L10,1, ρ→L10,1, ρ is Lipschitz-continuous; indeed, for all
x, y∈L10,1,
ρMfx, Mfy
1
0
|ft||x−y|
1|ft||x−y|dλ≤ρf∞x−y,0
≤max{1,f∞}ρx−y,0≤ρx, y.
4.19
It is clear thatMf mapsX1intoX1in general,rX1intorX1. Therefore, byTheorem 4.5,Mf has at least one fixed point onX1. Notice that we cannot apply Lennard’s theorem sinceX1is
notρ-compact and thatT is not necessarilyα-condensingit depends on the choice off.
In the literature, one finds several results concerning common fixed point theorems for families of self mapssee, e.g.,9, Chapter 9. All these results however require some form of compactness for the domain of the family.
As an application of Theorem 4.5, we will derive a common fixed point theorem without compactness, that generalizes Markov-Kakutani fixed point theorem see 9, Theorem 9.1.4or13, Section V.10 Theorem 6.
To this aim, we first need the following geometrical property for fixed points of middle point linear operators.
Proposition 4.13. LetXbe a normed space and letT :X→Xbe a middle point linear operator. IfT has two different fixed pointsxandythen any convex combination ofxandyis a fixed point underT.
Proof. SinceTx xandTy y, Tx∈xrX1andTy∈yrX1 for anyr >0. Thus,
byProposition 3.6,
Tαx 1−αy−αx 1−αy ≤r 4.20
for anyr >0 and anyα∈0,1, that gives the conclusion.
Proof. For anyT ∈ F, defineFTto be the set of fixed points ofT. FromTheorem 4.5and Proposition 4.13,FTis nonempty,ρ-closed, and convex.
Moreover, for anyT, S∈ Fandx0∈FT, we have
TSx0 STx0 Sx0, 4.21
that is,Sx0is a fixed point underT. Therefore,SFT⊂FT. Consider now the restriction S:FT→FTofS. FromTheorem 4.5,Shas a fixed point onFT. In other words,SandT
have a common fixed point. This argument can be repeated for every finite subfamily ofF. Thus, the family{FT, T ∈ F}satisfies the finite intersection property, and hence, by Theorem 4.1, there is a pointη∈T∈FFT.
A similar argument had been used in4in the framework of symmetric spaces. Since
L1μis not a symmetric space,Theorem 4.14cannot be derived from there.
On the other side, by means of Remark 4.6, one can derive a slight extension of Markov-Kakutani theorem in the framework of reflexive Banach spaces, since linear maps are middle point linear as well.
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