BANACH SPACES IN WHICH EVERY p-weakly SUMMABLE SEQUENCE LIES IN THE RANGE OF A VECTOR MEASURE

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 BANACH SPACES IN WHICH EVERY p-weakly SUMMABLE SEQUENCE LIES IN THE RANGE OF A VECTOR MEASURE C. PIÑEIRO (Communicated by Palle E. T. Jorgensen) Abstract. Let X be a Banach space. For 1 <p<+ we prove that the identity map I X is (1, 1,p)-summing if and only if the operator x X xn,x e n l q is nuclear for every unconditionally summable sequence (x n)inx,whereqis the conjugate number for p. Using this result we find a characterization of Banach spaces X in which every p-weakly summable sequence lies inside the range of an X -valued measure (equivalently, every p-weakly summable sequence (x n)inx, satisfying that the operator (α n) l q α nx n X is compact, lies in the range of an X-valued measure) with bounded variation. They are those Banach spaces such that the identity operator I X is (1, 1,p)-summing. Let X be a Banach space. In [AD] it is proved that every sequence (x n )inx satisfying n x n,x 2 < + for all x X lies inside the range of an X-valued measure. Nevertheless, they show a sequence which does not lie in the range of an X-valued measure with bounded variation. In [PR] the authors proved that X is finite dimensional if and only if every nul sequence (equivalently, everyt compact set) in X lies inside the range of an X-valued measure having bounded variation. The purpose of this paper is to characterize, given a real number p (1, + ), the Banach spaces in which every p-weakly summable sequence lies inside the range of an X -valued measure with bounded variation. We start by explaining some basic notation used in this paper. In general, our operator and vector measure terminology and notation follow [Ps] and [DU]. We only consider real Banach spaces. If X is a such space, B X will denote its closed unit ball. The phrase range of an X-valued measure always means a set of the form rg(f) ={F(A):A Σ, where Σ is a σ-algebra of subsets of a set Ω and F :Σ Xis countably additive. Given p 1, lw p (X) will denote the vector space of all sequences (x n)inxsuch that x n,x p < + for all x X. It is easy to see that if (x n ) lw p (X), then ( 1/p ε p ((x n )) = sup x n,x p : x B X < + and (l p w (X),ε p) is itself a Banach space. Received by the editors September 12, 1994 and, in revised form, December 2, Mathematics Subject Classification. Primary 46G10; Secondary 47B10. This research has been partially supported by the D.G.I.C.Y.T., PB c 1996 American Mathematical Society

2 2014 C. PIÑEIRO If ˆx =(x n ) lw p(x)andp is a finite subset of N, ˆx(P)=(x n(p)) is the sequence defined by { x n if n P, x n (P )= 0 if n/ P for all n N. lu p(x) will denote the subspace of lp w (X) consisting of the sequences ˆx =(x n ) such that the net (ˆx(P )) P F(N) converges to (x n )inlw(x), p where F(N) is the set of all finite subsets of N. Recall that lu 1 (X) is formed by the unconditionally summable sequences in X. We need the following propositions that list some privileges that membership in lw p (X) orinlp u (X) entail. Proposition A. Let p > 1and X be a Banach space. The following statements are equivalent: (i) (x n ) lw p (X). (ii) The series α nx n converges unconditionally for every sequence (α n ) l q. (iii) The map (α n ) l q α nx n X defines a bounded operator. Proposition B. Let p 1. If (x n ) lu(x), p then the operator (α n ) l q α nx n X is compact. 1. Main result Throughout this section X will be a Banach space and p (1, + ). Theorem 1. The following statements are equivalent: (i) For every unconditionally sequence (x n ) in X the operator x X x n,x e n l q is nuclear. (ii) There exists a constant c>0such that { n n x k,x k csup x k,x : x 1 (1) ( n 1/p sup x, x k p : x 1 for all {x 1,...,x n X and {x 1,...,x n X. Proof. (i) (ii) We consider the linear map ˆx =(x n ) lu(x) Tˆx 1 N(X,l q ) defined by Tˆx (x )= x n,x e n for all x X ({e n : n N is the unit basis of l q ). It has closed graph, so there exists a positive constant c so that ( ) { ν x n e n : X l q c sup x n,x : x 1 for all (x n ) lu 1 (X). By a standard argument we obtain ( m ) { m (2) ν x n e n : X lq m c sup x n,x : x 1 for all m N and {x 1,...,x m X.

3 BANACH SPACES WITH p-weakly SUMMABLE SEQUENCE 2015 Now, given {x 1,...,x m Xand {x 1,...,x m X, define two operators v : l m q X and u: X l m q by v(α i )= m α i x i and u(x )= m x i,x e i. Note that tr(u v) = m x i,x i,sowehave m x i,x i ν(u v) ν(u) v ( m 1/p =ν(u)sup x, x i p : x 1 and using (2) we obtain { n n x k,x k csup x k,x : x 1 ( n 1/p sup x, x k p : x 1. (ii) (i) Given (x n ) lp w (X ), we define a linear form φ by (x n ) l 1 u(x) x n,x n R. By (ii) φ l 1 u (X) and φ c (x n ) p. So, the linear map x X ( x, x n ) l is integral (see [DU, p. 232]). Equivalently, x X ( x, x n ) c 0 is integral. Then, so is its adjoint (α n ) l 1 α n x n X. Therefore, the linear map ψ defined by (x n) l p w(x ) e n x n I(l 1,X ) is well defined and ψ c. Now denote the restriction map of ψ to l p u(x )byψ u. Since ˆx = lim P ˆx (P ) for all ˆx l p u (X ), it follows that ψ u takes all its values in N (l 1,X ) (note that N (l 1,X ) is a subspace of I(l 1,X ) because (l 1 ) has the metric approximation property). If we also denote the operator (x n) l p u(x ) e n x n N(l 1,X ) by ψ u,then(ψ u ) maps B(X,l 1 )intol p u (X ). In particular, for all (x n ) l 1 u (X), the operator x X x n,x e n l q is integral. This completes the proof because nuclear and integral operators into a reflexive space are the same.

4 2016 C. PIÑEIRO Recall that an operator T : X Y is called (r, q, p)-summing if there is a constant c 0 such that ( n ) 1/r ( n ) 1/q Tx k,yk r c sup x k,x q x B X ( n ) 1/p sup y, yk p y B Y for all finite families of elements x 1,...,x n X and functionals y1,...,y n Y. So, Theorem 1 gives us a characterization of Banach spaces X for which I X is (1, 1, p)-summing. An operator T : X Y is (p, q)-summing if there is a constant c 0 such that ( n ) 1/p ( n 1/q Tx k p c sup x k,x q : x 1 for all finite subset {x 1,...,x n of X. Following [Ps] we will say that a Banach space X satisfies Grothendieck s Theorem (in short, X is a G.T. space) if B(X, l 2 )=Π 1 (X, l 2 ). The next proposition shows the relationship between the Banach spaces X for which I X is absolutely (1, 1, p)-summing and the above classes. Proposition 2. (i) If X is a G.T. space, then I X is (1, 1,p)-summing for 1 <p 2. (ii) If 1 <p<+ and T B(X, Y ), then Tis (1, 1, p)-summing T is (q, 1)-summing. Proof. If (x n ) l 1 u(x), then the operator T : x X x n,x e n l q admits the following factorization X * T l q J l 1 I where I : l 1 l q is the natural inclusion and J : X l 1 is defined by Jx = ( x n,x ) for all x X. I is obviously 1-summing and J is 2-summing by [Ps, 6.6.2], so T is nuclear. (ii) If T is (1, 1,p)-summing there is a constant c 0 such that { n n Tx i,y i csup (3) x i,x : x 1 ε p ((yi ) n ) for all {x 1,...,x n Xand {y1,...,y n Y. Given {x 1,...,x n X,choose yi B Y so that Tx i,yi = Tx i for each i n. By(3)wehave n n α i Tx i = α i Tx i,yi cε 1 ((x i ) n )ε p ((α i yi ) n ) cε 1 ((x i ) n ) (α i ) p

5 BANACH SPACES WITH p-weakly SUMMABLE SEQUENCE 2017 for all (α i ) lp n.then ( n ) 1/q { n Tx i q c sup x i,x : x 1 for all {x 1,...,x n X. In [P, ], Pietsch formulated the following conjecture: for 1/r > 1/q +1/p 1/2, I X is (r, q, p)-summing if and only if X is finite dimensional. The conjecture is true for q = r = 1. Certainly, let p>2. If I X is (1, 1,p)-summing, it follows from Proposition 2(ii) that I x is (q, 1)-summing. By [P, Theorem ] X has to be finite dimensional since q<2. 2. Sequences in the range of a vector measure with bounded variation In this section we use Theorem 1 to obtain a characterization of Banach spaces X for which every p-weakly summable sequence (x n )inxlies inside the range of an X -valued measure having bounded variation. The following lemma collects some elementary facts we need (see [Pi 2]). Lemma 3. Let X be a Banach space. If ˆx =(x n )is a bounded sequence in X, we consider the linear operator Tˆx : l 1 X defined by Tˆx (α n )= α n x n for all (α n ) l 1. Then the following assertions hold: (i) (x n ) lies inside the range of an X -valued measure with bounded variation iff Tˆx is integral. (ii) (x n ) lies inside the range of an X-valued measure with bounded variation iff Tˆx is Pietsch-integral. Now we are ready to face our problem. Theorem 4. Let X a Banach space and 1 <p<+. The following statements are equivalent: (i) Every sequence (x n ) lw p (X) lies inside the range of an X -valued measure with bounded variation. (ii) Every sequence (x n ) lw(x), p satisfying that the operator (α n ) l q αn x n X is compact, lies inside the range of an X-valued measure with bounded variation. (iii) Every sequence (x n ) lu p (X) lies inside the range of an X-valued measure with bounded variation. (iv) I X is (1, 1,p)-summing. Proof. (i) (ii) By Lemma 3, we can consider the linear map φ: ˆx l p w (X) Tˆx I(l 1,X). It is continuous because its graph is closed. Since (l q ) has the approximation property, for each sequence ˆx =(x n ) lw p(x) satisfying that the operator (α n) l q α n x n X is compact, there exists a sequence (ŷ k )inlw(x) p such that ˆx = lim k + ŷ k in lw p (X) and each sequence ŷ k is finite dimensional. Then φ(ŷ k ) belongs to N(l 1,X) for all k N. Bycontinuity,sodoesφ(ˆx) (recall that N (l 1,X) is a closed subspace of I(l 1,X)). Hence, we have proved that such a sequence

6 2018 C. PIÑEIRO (x n ) l p w (X) actually lies inside a sum of segments { [ zn,z n ]= αn z n :(α n ) l, (α n ) 1 where z n < + (see [Pi 1]). (ii) (iii) It is obvious because the operator (α n ) l q α n x n X is compact for each sequence (x n ) l p n (X). (iii) (iv) Now we consider the linear map ψ :ˆx l p u (X) Tˆx I(l 1,X). Having a closed graph, ψ is continuous. Since ˆx = lim P ˆx(P ) for every sequence ˆx l p u (X), it follows that ψ takes its values into N (l 1,X). As mentioned earlier, using the trace duality it is easy to prove that ψ takes every (x n) l 1 u(x )in x n e n I(X, l q ). Again the reflexivity of l q yields (iv). (iv) (i) In the same way as in the proof of Theorem 1 we can prove that the linear map (x n ) l p w(x ) e n x n I(l 1,X ) is well defined and continuous. In particular, it follows from the above lemma that every (x n ) l p w (X) lies inside the range of an X -valued measure of bounded variation. In view of Theorem 4 and the notes at the end of section 1, for p>2, only finitedimensional Banach spaces X have the property that every sequence (x n ) l p w(x) lies inside the range of an X-valued measure having bounded variation. That is why from now on we only consider p [1, 2]. 3. Final notes and examples It is well known that every sequence (x n ) lw 1 (X) lies inside the range of an X-valued measure with bounded variation. In fact, the vector measure F defined by ( ) F (A) =2 r n (t)dt x n, A for any Lebesgue measurable subset A of [0, 1], has bounded variation whenever (x n ) lw(x). 1 In [AD] it is proved that {x n : n N rg(f). Then, given an infinite-dimensional Banach space X, we can consider the set r(x) formedbyall real numbers r [1, 2] such that every sequence (x n ) lw(x) r lies inside the range of an X-valued measure having bounded variation. Then r(x) isanintervalwhose bounds are 1 and sup(r(x)). In the following we will determine the set r(x) for some classical Banach spaces. (i) r(x) =[1,2] for every Banach space X satisfying: (a) X is a G.T. space, (b) X is a dual space.

7 BANACH SPACES WITH p-weakly SUMMABLE SEQUENCE 2019 By Proposition 2(i), I X is (1, 1,r)-summing for all r (1, 2]. Then Theorem 4 implies that r r(x) for all r [1, 2]. In particular, if µ is a σ-finite positive measure, r(l (µ)) = [1, 2]. (ii) r(l p )={1for 1 p<+. I p will denote the identity map l p l p. First, we consider the case p =1. If (e n ) denotes the unit basis of l =(l 1 ),then(e n ) l1 w (l ). Since e n s = for s 1 it follows that I cannot be (s, 1)-summing for s 1. So, Proposition 2(ii) tells us that I is not (1, 1,r)-summing for r>1. By Theorem 4, r(l 1 )={1. Now suppose 1 <p<+. Claim. r(l p ) (1,q)=. Let r r(l p ) (1,q). Theorems 1 and 4 assure us that there is a constant c 0 such that n (4) x i,x i cε 1 ((x i ) n )ε r ((x i ) n ) for all {x 1,...,x n l p and {x 1,...,x n l q. Given (α n ) l q and (β n ) l u with u = rq(q r) 1, define x n = α ne n and x n = β n e n for all n N. From (4) we get m α i β i cε 1 ((α i e i ) m )ε r ((β i e i ) m ) for all m N. Applying Holder s inequality we obtain ε 1 ((α i e i )m ) (α n) q ε p ((e n )) = (α n) q and ε r ((β i e i ) m ) ε q ((e n )) (β n ) u = (β n ) u. Then, for all m N and (α n ) l q,wehave m α i β i c (α n ) q (β n ) u. This implies that (β n ) l p =(l q ).Choosing(β n ) l u \l p we fall in a contradiction since rq(q r) 1 >p. With our claim established we already have proved that r(l p )={1for p<2. Finally, we are going to show that r(l p ) [q, 2] = for p 2. This is the easy part. Certainly, the identity map l 1 l p is not nuclear, hence Lemma 3(ii) allows us to conclude that the sequence (e n ) does not lie inside the range of an l p -valued measure of bounded variation. Nevertheless, (e n ) lw(l r p ) for all r q. Thus [q, 2] r(l p )=. (iii) r(x) ={1for all infinite-dimensional L p -space X with 1 p<+. By [LP, Proposition 7.3], X has a complemented subspace H isomorphic to l p. Then r(x) r(h) =r(l p )={1. References [AD] R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Anna. Soc. Math. Polon. Ser. I Comment. Math. Prace Mat. 30 (1991), MR 92g:46049 [DU] J. Diestel and J. J. Uhl, Vector measures, Math. Surveys Monographs, vol. 15, Amer. Math. Soc., Providence, RI, MR 56:12216 [LP] J. Lindenstrauss and Pelczynski, Absolutely summing operators in L p-spaces and their applications, Studia Math. 29 (1968), MR 37:6743 [P] A. Pietsch, Operator ideals, North-Holland, Amsterdam, MR 81j:47001

8 2020 C. PIÑEIRO [Pi 1] C. Piñeiro, Operators on Banach spaces taking compact sets inside ranges of vector measures, Proc. Amer. Math. Soc. 116 (1992), MR 93b:47076 [Pi 2], Sequences in the range of a vector measure with bounded variations, Proc. Amer. Math. Soc. 123 (1995), CMP 95:16 [PR] C. Piñeiro and L. Rodriguez-Piazza, Banach spaces in which every compact lies inside the range of a vector measure, Proc. Amer. Math. Soc. 114 (1992), MR 92e:46038 [Ps] G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conf. Ser. in Math., vol. 60, Amer. Math. Soc., Providence, RI, MR 88a:47020 [T] N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, Pitman Monographs Surveys Pure Appl. Math., vol. 38, Longman Sci. Tech., Harlow, MR 90k:46039 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla, 41080, Spain Current address: Departamento de Matemáticas, Escuela Politécnica Superior, Universidad de Huelva, La Rábida, Huelva, Spain