PII. S0161171202007688 http://ijmms.hindawi.com © Hindawi Publishing Corp.
A NOTE ON UNIFORMLY DOMINATED SETS
OF SUMMING OPERATORS
J. M. DELGADO and C. PIÑEIRO
Received 26 May 2001
LetYbe a Banach space that has no finite cotype andpa real number satisfying 1≤p <∞. We prove that a setᏹ⊂Πp(X, Y )is uniformly dominated if and only if there exists a constantC >0 such that, for every finite set{(xi, Ti):i=1, . . . , n} ⊂X×ᏹ, there is an
operatorT∈Πp(X, Y )satisfyingπp(T )≤CandTixi ≤ T xifori=1, . . . , n.
2000 Mathematics Subject Classification: 47B10.
1. Introduction. LetXandY be Banach spaces andpa real number satisfying 1≤
p <∞. A subsetᏹofΠp(X, Y )is calleduniformly dominatedif there exists a positive Radon measureµdefined on the compact space(BX∗, σ (X∗, X)|BX∗)such that
T xp≤
BX∗
x∗, xpdµx∗ (1.1)
for allx∈Xand allT∈ᏹ. Since the appearance of Grothendieck-Pietsch’s domination theorem forp-summing operators, there is a great interest in finding out the structure of uniformly dominated sets. We will denote byᏰp(µ)the set of all operatorsT ∈ Πp(X, Y )satisfying (1.1) for allx∈X. It is easy to prove thatᏰp(µ)is absolutely convex, closed, and bounded (for thep-summing norm).
In [4], the authors consider the case p=1 and prove thatᏹ⊂Πp(X, Y )is uni-formly dominated if and only ifT∈ᏹT∗(BY∗)lies in the range of a vector measure of bounded variation and valued inX∗.
In [3], the following sufficient condition is proved: “letᏹ⊂Πp(X, Y )and 1≤p < ∞. Suppose that there is a positive constant C >0 such that, for every finite set {x1, . . . , xn}ofX, there existsQ∈ᏹsatisfyingπp(Q)≤Cand
n
i=1 T xi p
≤ n
i=1 Qxi p
(1.2)
for allT∈ᏹ. Thenᏹis uniformly dominated.” They also prove that this condition is necessary in the rather particular case thatᏹ⊂Πp(c0, c0)andᏹ=Ᏸp(µ)for some positive Radon measureµonB1.
In this note, we obtain a necessary and sufficient condition for a setᏹ⊂Πp(X, Y )
to be uniformly dominated, with the only restriction thatY is a Banach space without finite cotype. We refer to [1] for our operator terminology. IfXis a Banach space,BX
will denote its closed unit ball;pa(X) (pw(X))will be the Banach space of the strongly
2. Main result. We need the following characterization of uniformly dominated sets.
Proposition2.1. Let1≤p <∞andᏹ⊂Πp(X, Y ). The following statements are equivalent:
(a)ᏹis uniformly dominated.
(b)For everyε >0and(xn)∈pw(X), there existsn0∈Nsuch that
n≥n0
Tnxn p
< ε (2.1)
for all sequences(Tn)inᏹ.
(c)There exists a constantC >0such that
n
i=1
Tixi p≤Cp sup
x∗∈BX∗ n
i=1
x∗, xip (2.2)
for all{x1, . . . , xn} ⊂Xand{T1, . . . , Tn} ⊂ᏹ.
Proof. (a)⇒(b). In a similar way as in the Pietsch factorization theorem [1], we can obtain, for allT∈ᏹ, operatorsUT:Lp(µ)→∞(BY∗),UT ≤µ(BX∗)1/p, and an
operatorV:X→L∞(µ)such that the following diagram is commutative:
X
V T
Y iY
∞BY∗
L∞(µ)
ip
Lp(µ) UT
(2.3)
Hereipis the canonical injection fromL∞(µ)intoLp(µ)andiY is the isometry from
Y into∞(BY∗)defined byiY(y)=(y∗, y)y∗∈BY∗. Givenε >0 and(xn)∈ p w(X),
we can choosen0∈Nso that
n≥n0
ip◦Vxn p< ε
µBX∗ (2.4)
becauseip◦Visp-summing. Then, if(Tn)is a sequence inᏹ, we have
n≥n0
Tnxn p
=
n≥n0
iY◦Tn xn p
=
n≥n0
UTn◦ip◦Vxn p
≤µBX∗ n≥n0
ip◦Vxn p≤ε.
(2.5)
(b)⇒(c). Using a standard argument, we can prove thatᏹis bounded for the operator norm. Hence, given ˆx=(xn)∈pw(X), there existsMxˆ>0 such that
∞
n=1
for all(Tn)inᏹ. Then, we can consider the linear maps
T:xn∈pw(X)→Tnxn∈pa(Y ) (2.7)
for eachT=(Tn)in ᏹ. They have closed graph; so, by the uniform boundedness principle, there existsM >0 so that
∞
n=1
Tnxn p
1/p
≤Mpxn (2.8)
for all(xn)∈pw(X)and all(Tn)inᏹ(we wrotepfor the norm inpw(X)).
(c)⇒(a). GivenA={T1, . . . , Tn}⊂ᏹandB={x1, . . . , xn}⊂X, we definefA,B:BX∗→Rby
fA,Bx∗=Cp n
i=1
x∗, xip −n
i=1
Tixi p (2.9)
for allx∗∈X∗. We denote byᏼthe set of all functionsfA,B. It is clear thatᏼis convex and disjoint from the coneᏺ= {f∈Ꮿ(BX∗):f (x∗) <0,for allx∗∈BX∗}. In a similar way as in the proof of Pietsch’s domination theorem [1], we can show that there is a probability measureµonBX∗ satisfying
BX∗
T xp−Cpx∗, xp
dµ≤0 (2.10)
for allT∈ᏹand allx∈X.
As an application of this result, we can show a relatively compact set for the p -summing norm which is not uniformly dominated. Put Tn =(1/n)e∗n⊗en, n∈N,
where(en)and (en∗)are the unit basis of c0 and1, respectively. Asπ1(Tn)=1/n,
(Tn)is a null sequence inΠ1(c0, c0), so(Tn)is relatively compact. To see that it is not uniformly dominated, we will use Proposition 2.1: the sequence (en) is weakly summable but, for alln∈N, we have
k≥n
Tkek ∞= k≥n
1
k. (2.11)
We are now ready to introduce our main result.
Theorem2.2. LetY be a Banach space that has no finite cotype,ᏹ⊂Πp(X, Y ), and 1≤p <∞. The following statements are equivalent:
(a)ᏹis uniformly dominated.
(b)There is a constantC >0such that, for every{x1, . . . , xn} ⊂Xand{T1, . . . , Tn}⊂ᏹ, there exists an operatorT∈Πp(X, Y )satisfyingπp(T )≤Cand
Tixi ≤ T xi , i=1, . . . , n. (2.12)
Proof. (a)⇒(b). By hypothesis, there exists a positive Radon measureµ onBX∗
such that
T x ≤
BX∗
x∗, xp dµx∗
1/p
for allT ∈ᏹand allx∈X. SinceY has no finite cotype,Y containsn
∞’s uniformly. By [2], for everyε >0 andn∈N, there is an isomorphismJnfromn
∞onto a subspace ofY satisfyingJn−1 =1 andJn ≤1+εfor alln∈N.
Given{x1, . . . , xn} ⊂Xand{T1, . . . , Tn} ⊂ᏹ, by (2.13) we have
Tixi ≤
BX∗
x∗, xipdµx∗ 1/p
, i=1, . . . , n. (2.14)
For everyi=1, . . . , n, takegi∈Lq(µ)such thatgiq=1 and
BX∗
x∗, xipdµx∗ 1/p
=
BX∗
x∗, xigix∗dµx∗. (2.15)
From (2.14) and (2.15), we obtain
Tixi ≤
BX∗
x∗, xigix∗dµx∗, i=1, . . . , n. (2.16)
Putyi=Jnei, being(ei)n
i=1the unit basis of∞n. We define an operatorT:X→Y by
T x= n
i=1
BX∗
x∗, xgix∗dµx∗
yi. (2.17)
We first prove thatT xp≤(
BX∗|x∗, x|pdµ(x∗))(1+ε)for allx∈X:
T x = sup
y∗∈BY∗
y∗, n
i=1
BX∗
x∗, xgix∗dµx∗
yi
≤ sup
y∗∈BY∗ n
i=1
BX∗
x∗, xgix∗dµx∗
y∗, yi
≤ sup
y∗∈BY∗ n
i=1
BX∗
x∗, xpdµx∗ 1/p
BX∗
gix∗qdµx∗ 1/q
y∗, yi
≤
BX∗
x∗, xp dµx∗
1/p
sup
y∗∈BY∗ n
i=1
y∗, yi
≤
BX∗
x∗, xpdµx∗ 1/p
Jn∗
≤
BX∗
x∗, xpdµx∗ 1/p
(1+ε).
(2.18)
Finally, we need to prove thatTixi ≤ T xi fori=1, . . . , n. Putyi∗=e∗i ◦J−1
also denote byyi∗a Hahn-Banach extension ofe∗i◦J−1
n toY. We have T xi ≥yi∗, T xi
=
yi∗, n j=1 BX∗
x∗, xigjx∗dµx∗ yj = n j=1 BX∗
x∗, xigjx∗dµx∗
yi∗, yj
= n j=1 BX∗
x∗, xigjx∗dµx∗
e∗i ◦Jn−1, Jnej = n j=1 BX∗
x∗, xigjx∗dµx∗
e∗i, ej
=
BX∗
x∗, xigix∗dµx∗
≥ Tixi ,
(2.19)
the last inequality is due to (2.16).
(b)⇒(a). It follows easily usingProposition 2.1(c).
Remarks. (1) It is interesting to give an example of a uniformly dominated setᏹ for which there is no operatorT∈ᏹsatisfyingTixi ≤ T xi,i=1, . . . , n, for some finite set{(xi, Ti):i=1, . . . , n} ⊂X×ᏹ. LetX=1andY =∞and consider the set
ᏹ= {Tβ:β∈B2},Tβ:1→∞being defined byTβ(α)=(αnβn)for allα=(αn)∈ 1. Obviously,ᏹis a uniformly dominated subset ofΠ1(1, ∞).
By contradiction, suppose the following condition holds: “there is a constantC >0 such that, for every finite set{(xi, Ti):i=1, . . . , n} ⊂X×ᏹ, there existsT ∈ᏹ sat-isfyingTixi ≤CT xi,i=1, . . . , n.” Putxi=eiandTi=Tβi fori=1, . . . , n, where
(ei)∞i=1is the unit basis of1andβi=(1/ √
i,(i). . .,1/√i,0, . . .). TakeTγ∈ᏹsuch that Tixi ≤C Tγxi , i=1, . . . , n; (2.20)
this yields
1 √
i≤Cγi, i=1, . . . , n. (2.21)
Then we have
1≥ ∞
i=1 γi2
≥ n
i=1 γi2
≥C12
n
i=1 1
i. (2.22)
So, we have obtained the inequality ni=11/i≤C2 for alln∈Nwhich allows us to state that such an operatorT cannot exist.
(2) Notice that, in the above example,ᏹis absolutely convex and weakly compact in Π1(1, ∞). Then, ᏹ is absolutely convex, closed, and uniformly dominated but
(3) Finally, we give an example of a bounded setᏹof 2-summing operators that does not have property (b) inTheorem 2.2. Consider the setᏹof all 2-summing operators
Tβ:c0→∞defined byTβ(α)=(αnβn)for allα=(αn)∈c0, whereβ=(βn)runs over the unit ball of2. We haveTβ=i◦Sβ,ibeing the identity map from2into∞ andSβ:c0→2defined bySβ(α)=(αnβn). Since2has cotype 2, it follows thatSβ is 2-summing [1]. Nevertheless,ᏹdoes not satisfy property (b) in the above theorem. By contradiction, suppose that there is a constantC >0 such that (b) holds. Again, we take ˜βi=(1/√i,. . .,(i) 1/√i,0, . . .)for alli∈N. By hypothesis, there existsT∈Π2(c
0, ∞) such thatπ2(T )≤CandTβ˜iei ≤ T eifori=1, . . . , n. Then we have
n
i=1 1
i = n
i=1 Tβ˜iei
2 ≤
n
i=1
T ei 2≤C2 (2.23)
for alln∈N. Hence,ᏹdoes not have property (b) inTheorem 2.2.
References
[1] J. Diestel, H. Jarchow, and A. Tonge,Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. [2] R. C. James,Uniformly non-square Banach spaces, Ann. of Math. (2)80(1964), 542–550. [3] R. Khalil and M. Hussain,Uniformly dominated sets ofp-summing operators, Far East J.
Math. Sci. (1998), Special Volume, Part I, 59–68.
[4] B. Marchena and C. Piñeiro,Bounded sets in the range of anX∗∗-valued measure with bounded variation, Int. J. Math. Math. Sci.23(2000), no. 1, 21–30.
J. M. Delgado: Departamento de Matemáticas, Escuela Politécnica Superior,
Univer-sidad de Huelva, La Rábida21819, Huelva, Spain E-mail address:[email protected]
C. Piñeiro: Departamento de Matemáticas, Escuela Politécnica Superior,
