A note on uniformly dominated sets of summing operators

IJMMS 29:5 (2002) 307–312 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 Let Ybe a Banach space that has no finite cotype and pa real number satisfying 1 ≤p<∞. We prove that a set ᏹ⊂Πp(X,Y) is uniformly dominated if and only if there exists a constant C>0 such that, for every finite set {(xi,Ti):i=1,...,n}⊂X×ᏹ, there is an operator T∈Πp(X,Y) satisfying πp(T) ≤Cand Tixi≤Txifor i=1,...,n. 2000 Mathematics Subject Classification: 47B10. 1. Introduction. Let Xand Ybe Banach spaces and pa real number satisfying 1 ≤ p<∞. A subset ᏹof Πp(X,Y) is called uniformly dominated if there exists a positive Radon measure µdefined on the compact space (BX∗,σ(X∗,X)|BX∗)such that Txp≤BX∗ x∗,x pdµx∗(1.1) for all x∈Xand all T∈ᏹ. Since the appearance of Grothendieck-Pietsch’s domination theorem for p-summing operators, there is a great interest in finding out the structure of uniformly dominated sets. We will denote by Ᏸp(µ) the set of all operators T∈ Πp(X,Y ) satisfying (1.1) for all x∈X. It is easy to prove that Ᏸp(µ) is absolutely convex, closed, and bounded (for the p-summing norm). In [4], the authors consider the case p=1 and prove that ᏹ⊂Πp(X,Y ) is uniformly dominated if and only if T∈ᏹT∗(BY∗)lies in the range of a vector measure of bounded variation and valued in X∗. In [3], the following sufficient condition is proved: “let ᏹ⊂Πp(X,Y) and 1 ≤p< ∞. Suppose that there is a positive constant C>0 such that, for every finite set {x1,...,xn}of X, there exists Q∈ᏹsatisfying πp(Q) ≤Cand n  i=1 Txi p≤ n  i=1 Qxi p(1.2) for all T∈ᏹ.Thenᏹis uniformly dominated.” They also prove that this condition is necessary in the rather particular case that ᏹ⊂Πp(c0,c0)and ᏹ=Ᏸp(µ) for some positive Radon measure µon B1. In this note, we obtain a necessary and sufficient condition for a set ᏹ⊂Πp(X,Y ) to be uniformly dominated, with the only restriction that Yis a Banach space without finite cotype. We refer to [1] for our operator terminology. If Xis a Banach space, BX will denote its closed unit ball; p a(X) (p w(X)) will be the Banach space of the strongly (weakly) p-summable sequences. 308 J. M. DELGADO AND C. PIÑEIRO 2. Main result. We need the following characterization of uniformly dominated sets. Proposition 2.1.Let 1≤p<∞and ᏹ⊂Πp(X,Y). The following statements are equivalent: (a) ᏹis uniformly dominated. (b) For every ε>0and (xn)∈p w(X), there exists n0∈Nsuch that  n≥n0 Tnxn p<ε (2.1) for all sequences (Tn)in ᏹ. (c) There exists a constant C>0such that n  i=1 Tixi p≤Cpsup x∗∈BX∗ n  i=1 x∗,xi p(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 all T∈ᏹ, operators UT:Lp(µ) →∞(BY∗),UT≤µ(BX∗)1/p, and an operator V:X→L∞(µ) such that the following diagram is commutative: X V TY iY ∞BY∗ L∞(µ) ipLp(µ) UT (2.3) Here ipis the canonical injection from L∞(µ) into Lp(µ) and iYis the isometry from Yinto ∞(BY∗)defined by iY(y) =(y∗,y)y∗∈BY∗.Givenε>0and(xn)∈p w(X), we can choose n0∈Nso that  n≥n0 ip◦Vxn p<ε µBX∗(2.4) because ip◦Vis p-summing. Then, if (Tn)is a sequence in ᏹ, we have  n≥n0 Tnxn p= n≥n0 iY◦Tnxn p = n≥n0 UTn◦ip◦Vxn p ≤µBX∗ n≥n0 ip◦Vxn p≤ε. (2.5) (b)⇒(c). Using a standard argument, we can prove that ᏹis bounded for the operator norm. Hence, given ˆ x=(xn)∈p w(X), there exists Mˆ x>0 such that ∞  n=1 Tnxn p≤Mˆ x(2.6) A NOTE ON UNIFORMLY DOMINATED SETS OF SUMMING OPERATORS 309 for all (Tn)in ᏹ. Then, we can consider the linear maps  T:xn∈p w(X) →Tnxn∈p a(Y) (2.7) for each  T=(Tn)in ᏹ. They have closed graph; so, by the uniform boundedness principle, there exists M>0 so that  ∞  n=1 Tnxn p  1/p ≤Mpxn(2.8) for all (xn)∈p w(X) and all (Tn)in ᏹ(we wrote pfor the norm in p w(X)). (c)⇒(a). Given A={T1,...,Tn}⊂ᏹand B={x1,...,xn}⊂X, we define fA,B :BX∗→Rby fA,Bx∗=Cp  n  i=1 x∗,xi p − n  i=1 Tixi p(2.9) for all x∗∈X∗.Wedenotebyᏼthe set of all functions fA,B. It is clear that ᏼis convex and disjoint from the cone ᏺ={f∈Ꮿ(BX∗):f(x∗)<0,for all x∗∈BX∗}.Inasimilar way as in the proof of Pietsch’s domination theorem [1], we can show that there is a probability measure µon BX∗satisfying BX∗Txp−Cp x∗,x pdµ ≤0 (2.10) for all T∈ᏹand all x∈X. As an application of this result, we can show a relatively compact set for the psumming norm which is not uniformly dominated. Put Tn=(1/n)e∗ n⊗en,n∈N, where (en)and (e∗ n)are the unit basis of c0and 1, respectively. As π1(Tn)=1/n, (Tn)is a null sequence in Π1(c0,c0),so(Tn)is relatively compact. To see that it is not uniformly dominated, we will use Proposition 2.1:thesequence(en)is weakly summable but, for all n∈N, we have  k≥n Tkek ∞= k≥n 1 k.(2.11) We are now ready to introduce our main result. Theorem 2.2.Let Ybe 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 constant C>0such that, for every {x1,...,xn}⊂Xand {T1,...,Tn}⊂ᏹ, there exists an operator T∈Πp(X,Y) satisfying πp(T) ≤Cand  Tixi ≤ Txi ,i=1,...,n. (2.12) Proof. (a)⇒(b). By hypothesis, there exists a positive Radon measure µon BX∗ such that Tx≤BX∗ x∗,x pdµx∗1/p (2.13) 310 J. M. DELGADO AND C. PIÑEIRO for all T∈ᏹand all x∈X.SinceYhas no finite cotype, Ycontains n ∞’s uniformly. By [2], for every ε>0andn∈N, there is an isomorphism Jnfrom n ∞onto a subspace of Ysatisfying J−1 n=1andJn≤1+εfor all n∈N. Given {x1,...,xn}⊂Xand {T1,...,Tn}⊂ᏹ,by(2.13) we have  Tixi ≤BX∗ x∗,xi pdµx∗1/p ,i=1,...,n. (2.14) For every i=1,...,n,takegi∈Lq(µ) such that giq=1 and BX∗ x∗,xi pdµx∗1/p =BX∗x∗,xigix∗dµx∗.(2.15) From (2.14) and (2.15), we obtain  Tixi ≤BX∗x∗,xigix∗dµx∗,i=1,...,n. (2.16) Put yi=Jnei,being(ei)n i=1the unit basis of n ∞. We define an operator T:X→Yby Tx= n  i=1BX∗x∗,xgix∗dµx∗yi.(2.17) We first prove that Txp≤(BX∗|x∗,x|pdµ(x∗))(1+ε) for all x∈X: Tx= sup y∗∈BY∗    y∗, n  i=1BX∗x∗,xgix∗dµx∗yi     ≤sup y∗∈BY∗ n  i=1BX∗ x∗,x  gix∗ dµx∗ y∗,yi  ≤sup y∗∈BY∗ n  i=1BX∗ x∗,x pdµx∗1/pBX∗ gix∗ qdµx∗1/q y∗,yi  ≤BX∗ x∗,x pdµx∗1/p sup y∗∈BY∗ n  i=1 y∗,yi  ≤BX∗ x∗,x pdµx∗1/p J∗ n  ≤BX∗ x∗,x pdµx∗1/p (1+ε). (2.18) Finally, we need to prove that Tixi≤Txifor i=1,...,n.Puty∗ i=e∗ i◦J−1 n, (e∗ i)n i=1being the unit basis of (n ∞)∗≃n 1. Notice that y∗ i≤1fori=1,...,n.We A NOTE ON UNIFORMLY DOMINATED SETS OF SUMMING OPERATORS 311 also denote by y∗ ia Hahn-Banach extension of e∗ i◦J−1 nto Y. We have  Txi ≥ y∗ i,Txi  =     y∗ i, n  j=1BX∗x∗,xigjx∗dµx∗yj      =      n  j=1BX∗x∗,xigjx∗dµx∗y∗ i,yj      =      n  j=1BX∗x∗,xigjx∗dµx∗e∗ i◦J−1 n,Jnej      =      n  j=1BX∗x∗,xigjx∗dµx∗e∗ i,ej      =BX∗x∗,xigix∗dµx∗ ≥ Tixi , (2.19) the last inequality is due to (2.16). (b)⇒(a). It follows easily using Proposition 2.1(c). Remarks. (1) It is interesting to give an example of a uniformly dominated set ᏹ for which there is no operator T∈ᏹsatisfying Tixi≤Txi,i=1,...,n, for some finite set {(xi,Ti):i=1,...,n}⊂X×ᏹ.LetX=1and Y=∞and consider the set ᏹ={Tβ:β∈B2},Tβ:1→∞being defined by Tβ(α) =(αnβn)for all α=(αn)∈1. Obviously, ᏹis a uniformly dominated subset of Π1(1,∞). By contradiction, suppose the following condition holds: “there is a constant C>0 such that, for every finite set {(xi,Ti):i=1,...,n}⊂X×ᏹ, there exists T∈ᏹsatisfying Tixi≤CTxi,i=1,...,n.” Put xi=eiand Ti=Tβifor i=1,...,n, where (ei)∞ i=1is the unit basis of 1and βi=(1/√i, (i) ...,1/√i,0,...).TakeTγ∈ᏹsuch that  Tixi ≤C Tγxi ,i=1,...,n; (2.20) this yields 1 √i≤C γi ,i=1,...,n. (2.21) Then we have 1≥∞  i=1 γi 2≥ n  i=1 γi 2≥1 C2 n  i=1 1 i.(2.22) So, we have obtained the inequality n i=11/i ≤C2for all n∈Nwhich allows us to state that such an operator Tcannot exist. (2) Notice that, in the above example, ᏹis absolutely convex and weakly compact in Π1(1,∞). Then, ᏹis absolutely convex, closed, and uniformly dominated but ᏹ≠Ᏸ1(µ) for every admissible positive Radon measure µ. 312 J. M. DELGADO AND C. PIÑEIRO (3) Finally, we give an example of a bounded set ᏹof 2-summing operators that does not have property (b) in Theorem 2.2. Consider the set ᏹof all 2-summing operators Tβ:c0→∞defined by Tβ(α) =(αnβn)for all α=(αn)∈c0, where β=(βn)runs over the unit ball of 2. We have Tβ=i◦Sβ,ibeing the identity map from 2into ∞ and Sβ:c0→2defined by Sβ(α) =(αnβn).Since2has cotype 2, it follows that Sβ is 2-summing [1]. Nevertheless, ᏹdoes not satisfy property (b) in the above theorem. By contradiction, suppose that there is a constant C>0 such that (b) holds. Again, we take ˜ βi=(1/√i, (i) ...,1/√i,0,...) for all i∈N. By hypothesis, there exists T∈Π2(c0,∞) such that π2(T) ≤Cand T˜ βiei≤Teifor i=1,...,n. Then we have n  i=1 1 i= n  i=1 T˜ βiei 2≤ n  i=1 Tei 2≤C2(2.23) for all n∈N.Hence,ᏹdoes not have property (b) in Theorem 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 of p-summing operators,FarEastJ. Math. Sci. (1998), Special Volume, Part I, 59–68. [4] B. Marchena and C. Piñeiro, Bounded sets in the range of an X∗∗-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, Universidad de Huelva, La Rábida 21819, Huelva, Spain E-mail address:[email protected] C. 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