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

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 $u$.