Absolutely summing linear operators into spaces with no finite cotype

TL;DR

This paper investigates conditions under which all continuous linear operators from a Banach space X to a space Y with no finite cotype are absolutely summing, with results depending on the cotype of X and applications to multilinear mappings.

Contribution

It characterizes when operators into spaces with no finite cotype are absolutely summing, extending understanding of operator ideals and multilinear mappings.

Findings

Operators into spaces with no finite cotype are absolutely summing under certain cotype conditions.

The problem is fully solved when X assumes its cotype.

Applications to dominated multilinear mappings are provided.

Abstract

Given an infinite-dimensional Banach space $X$ and a Banach space $Y$ with no finite cotype, we determine whether or not every continuous linear operator from $X$ to $Y$ is absolutely $(q;p)$-summing for almost all choices of $p$ and $q$, including the case $p=q$. If $X$ assumes its cotype, the problem is solved for all choices of $p$ and $q$. Applications to the theory of dominated multilinear mappings are also provided.