This is an announcement for the paper "Abstract Ces`aro spaces: Integral representations" by Guillermo P. Curbera and Werner J. Ricker.
Abstract: The Ces`aro function spaces $Ces_p=[C,L^p]$, $1\le p\le\infty$, have received renewed attention in recent years. Many properties of $[C,L^p]$ are known. Less is known about $[C,X]$ when the Ces`aro 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 operator $C:[C,X]\to X$; it is never compact but, it can be completely continuous.
Archive classification: math.FA
Mathematics Subject Classification: 46E30, 46G10
Remarks: 21 pages
Submitted from: curbera@us.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1512.02760
or
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alspach＠math.okstate.edu