On $p$-Dunford integrable functions with values in Banach spaces

TL;DR

This paper explores properties of $p$-Dunford integrable functions in Banach spaces, focusing on compactness, composition with $p$-summing operators, and tests for integrability using $w^*$-thick subsets.

Contribution

It provides new insights into the compactness of Dunford operators, conditions for $p$-Bochner integrability of compositions, and introduces tests for $p$-Dunford integrability based on $w^*$-thick subsets.

Findings

Characterization of compactness of Dunford operators

Conditions for $p$-Bochner integrability of compositions

Tests for $p$-Dunford integrability using $w^*$-thick subsets

Abstract

Let $(\Omega,\Sigma,\mu)$ be a complete probability space, $X$ a Banach space and $1\leq p<\infty$. In this paper we discuss several aspects of $p$-Dunford integrable functions $f:\Omega \to X$. Special attention is paid to the compactness of the Dunford operator of $f$. We also study the $p$-Bochner integrability of the composition $u\circ f:\Omega \to Y$, where $u$ is a $p$-summing operator from $X$ to another Banach space $Y$. Finally, we also provide some tests of $p$-Dunford integrability by using $w^*$-thick subsets of $X^*$.