View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 331 (2007) 850–865 www.elsevier.com/locate/jmaa Examples of summing, integral and nuclear operators on the space C([0, 1],X)with values in c0 Dumitru Popa Department of Mathematics, University of Constanta, Bd. Mamaia 124, 8700 Constanta, Romania Received 16 January 2006 Available online 16 October 2006 Submitted by J. Diestel Abstract We give necessary and sufficient conditions that some operators on the space C([0, 1],X) with values in c0 be as in the title. Related questions are investigated. © 2006 Elsevier Inc. All rights reserved. Keywords: Banach spaces of continuous functions; Tensor products; Operator ideals; p-Summing operators 1. Introduction Let Ω be a compact Hausdorff space, let X be a Banach space, and let C(Ω,X) stand for the Banach space of continuous X-valued functions on Ω under the uniform norm and C(Ω) when X is the scalar field. It is well known that if Y is a Banach space, then any bounded linear operator U : C(Ω,X) → Y has a finitely additive vector measure G : Σ → L(X, Y ∗∗), where Σ is the σ -field of Borel subsets of Ω, such that ∗ ∗ ∗ y U(f)= fdGy∗ ,f∈ C(Ω,X), y ∈ Y . Ω The measure G is called the representing measure of U. For details, see [2, p. 217], [3, Theo- rem 2.2], [7, p. 182]. E-mail address: [email protected]. 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.09.028 D. Popa / J. Math. Anal. Appl. 331 (2007) 850–865 851 Also, for a bounded linear operator U : C(Ω,X) → Y we can associate, in a natural way, two bounded linear operators # U : C(Ω) → L(X, Y ) and U# : X → L C(Ω),Y defined by # U ϕ (x) = U(ϕ ⊗ x) and (U#x)(ϕ) = U(ϕ ⊗ x), where for ϕ ∈ C(Ω), x ∈ X we denote (ϕ ⊗ x)(ω) = ϕ(ω)x, ω ∈ Ω. We think that it can be of some interest to point out that the operator U # also appears in [8, Theorem 1, p. 377], where it is denoted by U . We denote by (L, op) the class of all bounded linear operators acting between two Banach spaces, (Πr,p,Πr,p(·)) the normed ideal of (r, p)-summing operators, where 1 p r<∞, (Πr ,Πr (·)) for r = p, (As, as) for r = 1, (I, int) the normed ideal of (Grothendieck) integral operators, (Nr , rnuc) the normed ideal of r-nuclear operators, where 1 r<∞, (N , nuc) for r = 1, the normed ideal of nuclear operators. We refer the reader to [4,6,11,12] for details. ∞ ⊂ ∗ ∈ ∗ If X is a Banach space, 1 p< and (xn)n∈N X is such that for any x X the series ∞ | ∗ |p | ∈ N; = ∞ | ∗ |p 1/p ∞ n=1 x (xn) is convergent, then wp(xn n X) supx∗1( n=1 x (xn) ) < and we denote by wp(X) the set of all such a sequences, see [4], [6, Chapter 2], [11, 16.3.1]. We use that T ∈ Πr,p(X, Y ) if and only if for each (xn)n∈N ∈ wp(X) it follows that (T (xn))n∈N ∈ lr (Y ), see [11, 17.2.2 Proposition]. For lp with 1 p<∞ and c0 we denote by (en)n∈N the canonical basis in these spaces and ∞ ∗ 1 + 1 = for 1 <p< we write, as usual, p for the conjugate of p,i.e. p p∗ 1. For a vector measure G : Σ → X we denote |G| and G the variation and semivariation of G, see [7, Chapter I]. We denote by B the σ -field of Borel subsets of [0, 1], μ : B →[0, 1] the Lebesgue measure, (rn)n∈N the sequence of Rademacher functions and if (S,Σ,ν) is a finite measure space, X is a → − Banach space and f : S X is a ν-Bochner integrable function we denote by B ()fdνthe Bochner integral. For all the definitions, notations and results in operator ideal theory we refer the reader to [4,6,11]. The notations and notions used and not defined in this paper can be found in [6]. We recall the general results known in this topic which we use for finding the necessary and sufficient conditions that examples of operators which we study, to be in some normed operator ideals. For a best understanding of the development of the research we indicate also the year publication of the results. Let U : C(Ω,X) → Y be a bounded linear operator, G the representing measure of U and # the operators U : C(Ω) → L(X, Y ), U# : X → L(C(Ω), Y ). Then the following statements are true: Swartz’s Theorem. [21, 1973] U is absolutely summing ⇔ G : Σ → As(X, Y ) is of bounded variation ⇔ U # ∈ As(C(Ω), As(X, Y )). Saab’s Theorem. [19, 1991] U is integral ⇔ G : Σ → I(X,Y) is of bounded variation ⇔ U # ∈ I(C(Ω),I(X,Y)). 852 D. Popa / J. Math. Anal. Appl. 331 (2007) 850–865 Theorem 1. ([1, 1972]; [13, 1990]; [20, 1991]) (a) If U ∈ N (C(Ω, X), Y ), then G : Σ → N (X, Y ) is of bounded variation. (b) Suppose X∗ has the Radon–Nikodym Property. Then U ∈ N (C(Ω, X), Y ) if and only if G : Σ → N (X, Y ) is of bounded variation and G has a |G|nuc-Bochner integrable deriva- tive. For a proof, see [1, Theorem III.4] as is cited in [20], [13, Theorem 1], [20, Theorem 5]. Theorem 2. [16, 2006] Let (U, A) be a maximal normed operator ideal. If G : Σ → (U(X, Y ), A) is of bounded variation, then U ∈ U(C(Ω, X), Y ). For a proof, see [16, Theorem 3]. If we use [7, Theorem 3, p. 162], the condition G : Σ → (U(X, Y ), A) is of bounded variation is equivalent to U # ∈ As(C(Ω), U(X, Y )). Theorem 3. ([10, 1992]; [15, 2001]) Let 1 p r<∞. # (a) If U ∈ Πr,p(C(Ω, X), Y ), then U ∈ Πr,p(C(Ω), Πr,p (X, Y )). (b) If U ∈ Πr,p(C(Ω, X), Y ), then U# ∈ Πr,p(X, Πr,p(C(Ω), Y )). We remark that in [10, Theorem 2.1] is proved (a), for p-summing operators and in [15, Theorem 1], for (r, p)-summing operators, but as is easily seen from the proof, by symmetry, we get that (b) is also true. In [10] and [14] examples which prove that the converse in Theorem 3(a) is not in general true, are given. If Ω is a compact Hausdorff space, Σ is the σ -algebra of all Borel sets in Ω, X a Banach space, we denote by B(Σ,X) the space of X-valued totally measurable functions endowed with the uniform norm. If we use the fact that the bidual of a p-summing operator is p-summing, see [4], [6, 2.19 Proposition, p. 50], [11], that each p-summing operator is weakly compact, see [4], [6, 2.17 The- orem, p. 50], [11], from [2, pp. 220–221] or [3, Theorem 2.2] we get Extension theorem for p-summing operators. Let 1 p<∞. If U : C(Ω,X) → Y is p-summing, then its representing measure G takes its values in L(X, Y ) and its canonical extension U: B(Σ,X) → Y defined by U(f) = fdG for f ∈ B(Σ,X), Ω is p-summing. The main goal of this paper is to obtain necessary and sufficient conditions for some operators to satisfy the hypotheses of the above mentioned general results. This will reveal, among others, what is hidden in these general theorems—in particular, whether the converses of the above general results are true or not. Further, in addition to [10] and [14], we consider whether the converse of Theorem 3(b) is true. D. Popa / J. Math. Anal. Appl. 331 (2007) 850–865 853 2. The examples We begin with an example which appear in this form in [18, Theorems 2.1 and 3.1] and in a more general context in [14, Proposition 2]. ∗ ⊂ ∗ → Example 1. Let X be a Banach space, (xn)n∈N X a bounded sequence and T : X l∞ the = ∗ [ ] → operator defined by T(x) (xn(x))n∈N.LetU : C( 0, 1 ,X) c0 be the operator defined by 1 = ∗ U(f) xnf(t)rn(t) dt . ∈N 0 n (a) Let (U, A) be a maximal normed operator ideal. If T ∈ U(X, l∞), then U ∈ U(C([0, 1], X),c0) and A(U) A(T ). (b) U is (r, p)-summing (respectively integral) ⇔ T is (r, p)-summing (integral). Further Πr,p(U) = Πr,p(T ) (respectively Uint =T int). ∗ → ∗ → = ∗ (c) Suppose xn 0 weak and let T : X c0 be the operator defined by T(x) (xn(x))n∈N. (i) Let (U, A) be a maximal normed operator ideal. If T ∈ U(X, c0), then U ∈ U(C([0, 1],X),c0) and A(U) A(T ). (ii) U is (r, p)-summing (respectively integral) ⇔ T is (r, p)-summing (respectively inte- gral). Further Πr,p(U) = Πr,p(T ) (respectively Uint =T int). (iii) If T is a nuclear operator, then U is a nuclear operator and Unuc T nuc. (iv) If X∗ has the Radon–Nikodym property and U is a nuclear operator, then T is a nuclear operator and T nuc =Unuc.