Abstract Cesàro spaces: Integral representations

The Cesàro function spaces Cesp=[C,Lp]Cesp=[C,Lp], 1≤p≤∞1≤p≤∞, have received renewed attention in recent years. Many properties of [C,Lp][C,Lp] are known. Less is known about [C,X][C,X] when the Cesàro operator takes its values in a rearrangement invariant (r.i.) space X   other than LpLp. In this paper we study the spaces [C,X][C,X] via the methods of vector measures and vector integration. These techniques allow us to identify the absolutely continuous part of [C,X][C,X] and the Fatou completion of [C,X][C,X]; to show that [C,X][C,X] is never reflexive and never r.i.; to identify when [C,X][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 Cesàro operator C:[C,X]→XC:[C,X]→X; it is never compact but it can be completely continuous.