0 Integral Operators on Spaces of Continuous Vector - valued functions

Let X be a compact Hausdorff space, let E be a Banach space, and let C(X,E) stand for the Banach space of E-valued continuous functions on X under the uniform norm. In this paper we characterize Integral operators (in the sense of Grothendieck) on C(X,E) spaces in term of their representing vector measures. This is then used to give some applications to Nuclear operators on C(X,E) spaces. AMS(MOS) subject Classification (1980). Primary 46E40, 46G10; Secondary 28B05, 28B20. ∗ Supported in part by an NSF Grant DMS-87500750 Introduction Let X be a compact Hausdorff space, let E and F be Banach spaces. Denote by C(X,E) the space of all continuous E-valued functions defined on X under the uniform norm. In [9] C. Swartz showed that a bounded linear operator T :C(X,E) −→ F with representing measure G is absolutely summing if and only if each of the values of G is an absolutely summing operator from E to F and G is of bounded variation as a measure taking its values in the space of absolutely summing operators from E to F equipped with the absolutely summing norm. In this paper we shall extend Swartz’s result to the class of (Grothendieck) integral operators on C(X,E) spaces. More precisely we shall show that a bounded linear operator T : C(X,E) → F with representing measure G is an integral operator if and only if each of the values of G is an integral operator from E to F and G is of bounded variation as a vector measure taking its values in the space of integral operators from E to F equipped with the integral norm. This result is then used to give some applications to Nuclear operators on C(X,E) spaces. I. Preliminaries If X is a compact Hausdorff space and E is a Banach space , then C(X,E) will denote the Banach space of all continuous E-valued functions equipped with the uniform norm. It is well known [4,page 182] that the dual of C(X,E) is isometrically isomorphic to the space M(X,E) of all regular E-valued measures on X that are of bounded variation. When E is the scalar field, we will simply write C(X) and M(X) for C(X,E) and M(X,E). If μ ∈ M(X,E) and e ∈ E, we will denote by |μ| the variation of μ and by 〈e, μ〉 the element of M(X) defined on each Borel subset B of X by 〈e, μ〉(B) = μ(B)(e). The duality between M(X,E) and C(X,E) is then defined as follows: for each f ∈ C(X) and e ∈ E μ(f ⊗ e) = ∫ X fd〈e, μ〉 where f ⊗ e is the element of C(X,E) defined by f ⊗ e(x) = f(x)e for all x ∈ X. 1 If B is a Borel subset of X , then 1B will denote the characteristic function of B, and if e ∈ E we let 1B ⊗ e denote the element of C(X,E) ∗∗ defined by 1B ⊗ e(μ) = 〈e, μ〉(B) = μ(B)(e) for each μ ∈ M(X,E). If X is a compact Hausdorff space , E and F are Banach spaces , every bounded linear operator T :C(X,E) −→ F has a representing measure G. The measure G is defined on the σ-field Σ of Borel subsets of X and takes its values in L(E, F ), the space of all bounded linear operators from E to F . The measure G is such that for each Borel subset B of X and for each e ∈ E G(B)e = T (1B ⊗ e) For y ∈ F , if we denote by Gy∗ the E valued measure on X such that for each Borel subset B of X and each e ∈ E 〈e, Gy∗〉(B) = 〈y , G(B)e〉 then Gy∗ is the unique element of M(X,E ) that represents T y in the sense that for each f ∈ C(X,E) 〈y, T f〉 = ∫ X f(x)dGy∗(x) If E ad F are Banach space, we denote by E ⊗ǫ F the algebraic tensor product of E and F endowed with the norm ||.||ǫ