Abstract Ces\`aro spaces: Integral representations

TL;DR

This paper investigates Cesàro function spaces with values in rearrangement invariant spaces using vector measure techniques, revealing their structural properties and operator behaviors.

Contribution

It introduces new methods to analyze $[C,X]$ spaces, identifying their absolute continuous parts, completions, and properties like reflexivity and compactness.

Findings

$[C,X]$ is never reflexive or rearrangement invariant.

The space $[C,X]$ is never compact but can be completely continuous.

Identifies conditions when $[C,X]$ is weakly sequentially complete or isomorphic to an AL-space.

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.