The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited

2016/12/21 by Muñoz, Fernando, Oja, Eve, Piñeiro, Cándido
#28B05 #46B25 #46B28 #46G10 #47A67 (Primary) #47B38 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1612.07312

Abstract

Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. Denote by C_p(Ω,X) the space of p-continous X-valued functions, 1≤ p≤ ∞. For operators S\inL(C(Ω),L(X,Y)) and U\inL(C_p(Ω,X),Y), we establish integral representation theorems with respect to a vector measure m:Σ→ L(X,Y^**), where Σ denotes the σ-algebra of Borel subsets of Ω. The first theorem extends the classical Bartle-Dunford-Schwartz representation theorem. It is used to prove the second theorem, which extends the classical Dinculeanu-Singer representation theorem, also providing to it an alternative simpler proof. For the latter (and the main) result, we build the needed integration theory, relying on a new concept of the q-semivariation, 1≤ q≤ ∞, of a vector measure m:Σ→ L(X,Y^**).

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