Some properties of almost summing operators

TL;DR

This paper extends key results from the linear theory of absolutely summing operators to the broader class of almost summing operators, demonstrating that many properties hold under this weaker condition and providing a counterexample.

Contribution

It generalizes three fundamental theorems about $p$-summing operators to almost summing operators, broadening their applicability.

Findings

Almost summing operators preserve properties of $p$-summing operators.

Many classical results remain valid when replacing $p$-summing with almost summing.

An example shows almost summing operators are not necessarily $p$-summing.

Abstract

In this paper we extend the scope of three important results of the linear theory of absolutely summing operators. The first one was proved by Bu and Kranz in \cite{BK} and it asserts that a continuous linear operator between Banach spaces takes almost unconditionally summable sequences into Cohen strongly $q$-summable sequences for any $q\geq2$, whenever its adjoint is $p$-summing for some $p\geq1$. The second of them states that $p$-summing operators with hilbertian domain are Cohen strongly $q$-summing operators ($1<p,q<\infty$), this result is due to Bu \cite{Bu}. The third one is due to Kwapie\'{n} \cite{Kwapien} and it characterizes spaces isomorphic to a Hilbert space using 2-summing operators. We will show that these results are maintained replacing the hypothesis of the operator to be $p$-summing by almost summing. We will also give an example of an almost summing operator that fails to be $p$-summing for every $1\leq p< \infty$.