On p-Compact mappings and p-approximation

TL;DR

This paper explores the properties of p-compact sets, p-approximation, and p-compact holomorphic functions, introducing a p-compact radius of convergence and linking these concepts to nuclear maps and holomorphy types.

Contribution

It introduces a p-compact radius of convergence for holomorphic functions and characterizes p-compact holomorphic functions using the epsilon-product, advancing the understanding of p-approximation properties.

Findings

p-compact holomorphic functions behave more like nuclear maps than compact maps

A p-compact radius of convergence characterizes p-compact holomorphic functions

The epsilon-product characterizes the p-approximation property in terms of p-compact polynomials and functions

Abstract

The notion of $p$-compact sets arises naturally from Grothendieck's characterization of compact sets as those contained in the convex hull of a norm null sequence. The definition, due to Sinha and Karn (2002), leads to the concepts of $p$-approximation property and $p$-compact operators, which form a ideal with its ideal norm $\kappa_p$. This paper examines the interaction between the $p$-approximation property and the space of holomorphic functions. Here, the $p$-compact analytic functions play a crucial role. In order to understand this type of functions we define a $p$-compact radius of convergence which allow us to give a characterization of the functions in the class. We show that $p$-compact holomorphic functions behave more like nuclear than compact maps. We use the $\epsilon$-product, defined by Schwartz, to characterize the $p$-approximation property of a Banach space in terms of $p$-compact homogeneous polynomials and also in terms of $p$-compact holomorphic functions with range on the space. Finally, we show that $p$-compact holomorphic functions fit in the framework of holomorphy types which allows us to inspect the $\kappa_p$-approximation property. Along these notes we solve several questions posed by Aron, Maestre and Rueda in [2].