Almost Summing Polynomials: A Coherent and Compatible Approach the from the Infinite-Dimensional Holomorphy Viewpoint

TL;DR

This paper introduces a new class of almost summing polynomials that form a Banach multi-ideal, aligning with the holomorphy type and extending the concept of almost summing linear operators within infinite-dimensional holomorphy.

Contribution

It defines a natural norm for almost summing polynomials, establishing a coherent, compatible, and holomorphy type structure that improves upon previous concepts.

Findings

The class of almost summing polynomials forms a Banach multi-ideal.

The new class is coherent and compatible with almost summing linear operators.

Similar results do not hold for the original concept by G. Botelho.

Abstract

In this note we explore the notion of everywhere almost summing polynomials and define a natural norm which makes this class a Banach multi-ideal which is a holomorphy type (in the sense of L.Nachbin) and also coherent and compatible (in the sense of D. Carando, V. Dimant and S. Muro) with the notion of almost summing linear operators. Similar results are not valid for the original concept of almost summing polynomials, due to G. Botelho.