Abstract Cesaro spaces: Integral representations

The Cesaro function spaces Ces(p) = [C, L-p], 1 <= p <= infinity, have received renewed attention in recent years. Many properties of [C, L-p] are known. Less is known about [C, X] when the Cesaro operator takes its values in a rearrangement invariant (r.i.) space X other than L-p. In this paper we study the spaces [C, X] via the methods of vector measures and vector integration. These techniques allow us to identify the absolutely continuous part of [C, X] and the Fatou completion of [C, X]; to show that [C, X] is never reflexive and never r.i.; to identify when [C, X] is weakly sequentially complete, when it is isomorphic to an AL-space, and when it has the Dunford-Pettis property. The same techniques are used to analyze the Cesaro operator C: [C, X] -> X; it is never compact but it can be completely continuous. (C) 2016 Elsevier Inc. All rights reserved.