Compactness of integral operators and uniform integrability on measure spaces

TL;DR

This paper characterizes the equivalence between uniform integrability and compactness of integral operators on measure spaces, providing conditions under which these properties coincide.

Contribution

It establishes that uniform integrability and compactness of integral operators are equivalent under certain positivity conditions, linking two important concepts in measure and operator theory.

Findings

$\, ext{Uniform integrability and compactness are equivalent under positivity conditions}$

$\, ext{Characterizes the set of functions ensuring compactness of integral operators}$

$\, ext{Provides conditions for the equality of function sets related to integral operators$

Abstract

Let $(E,\mathcal E,\mu)$ be a measure space and $G\colon E\times E\to [0,\infty]$ be measurable. Moreover, let $\mathcal F\!_{ui}$ denote the set of all $q\in\mathcal E^+$ (measurable numerical functions $q\ge 0$ on $E$) such that $\{G(x,\cdot)q\colon x\in E\}$ is uniformly integrable, and let $\mathcal F\!_{co}$ denote the set of all $q\in\mathcal E^+$ such that the mapping $f\mapsto G(fq) :=\int G(\cdot,y) f(y) q(y)\,d\mu(y)$ is a compact operator on the space $\mathcal E_b$ of bounded measurable functions on $E$ (equipped with the sup-norm). It is shown that $\mathcal F\!_{ui}=\mathcal F\!_{co} $ provided both $\mathcal F\!_{ui}$ and $\mathcal F\!_{co} $contain strictly positive functions.