Improving integrability via absolute summability: a general version of Diestel's Theorem

TL;DR

This paper generalizes Diestel's classical theorem by demonstrating that certain summability conditions on operators can significantly improve the integrability of vector-valued functions, extending results to various Banach spaces and operator classes.

Contribution

It introduces a broad generalization of Diestel's theorem, focusing on $(p,\sigma)$-absolutely continuous operators and their role in enhancing integrability of vector-valued functions.

Findings

Generalized integrability improvement results for vector-valued functions.

Application to classical Banach spaces like $C(K)$, $L^p$, and Hilbert spaces.

Analysis of operators such as $p$-summing, $(q,p)$-summing, and $p$-approximable operators.

Abstract

A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the $(p,\sigma)$-absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces ---including $C(K)$, $L^p$ and Hilbert spaces--- and operators ---$p$-summing, $(q,p)$-summing and $p$-approximable operators---.