Some inclusion results for interpolated summing operator ideals and integrability improvement of vector valued functions

TL;DR

This paper extends the understanding of how interpolated summing operator ideals influence the integrability of vector-valued functions, generalizing classical results and providing new inclusion theorems.

Contribution

It introduces a new abstract inclusion theorem for classes of summing operators and applies it to interpolated operator ideals, advancing integrability results for vector-valued functions.

Findings

Established a new inclusion theorem for summing operator classes.

Extended integrability results for compositions of vector-valued functions and operators.

Connected interpolation techniques with integrability properties of operator compositions.

Abstract

Consider a Banach space valued measurable function $f$ and an operator $u$ from the space where {$f$} takes values. If $f $ is Pettis integrable, a classical result due to J. Diestel shows that composing it with $u$ gives a Bochner integrable function $u \circ f$ whenever $u$ is absolutely summing. In a previous work we have shown that a well-known interpolation technique for operator ideals allows to prove under some requirements that a composition of a $p$-Pettis integrable function with a $q$-summing operator provides an $r$-Bochner integrable function. In this paper a new abstract inclusion theorem for classes of {abstract} summing operators is shown and applied to the class of interpolated operator ideals. Together with the results of the {aforementioned} paper, it provides more results on the relation about the integrability of the function $u \circ f$ and the summability properties of $u$.