$\mathcal A$-compact mappings

TL;DR

This paper explores the concept of $\\mathcal{A}$-compact mappings in Banach spaces, extending the notion from operators to polynomials and holomorphic functions, and characterizes these mappings using the $\\mathcal{A}$-compact radius of convergence.

Contribution

It introduces and studies $\\mathcal{A}$-compact polynomials and holomorphic mappings, extending properties from operators and providing a characterization via the $\\mathcal{A}$-compact radius of convergence.

Findings

Behavior of $\\mathcal{A}$-compact polynomials is determined locally.

Properties of $\\mathcal{A}$-compact operators transfer to polynomials.

Characterization of $\\mathcal{A}$-compact holomorphic functions using the radius of convergence.

Abstract

For a fixed Banach operator ideal $\mathcal A$, we use the notion of $\mathcal A$-compact sets of Carl and Stephani to study $\mathcal A$-compact polynomials and $\mathcal A$-compact holomorphic mappings. Namely, those mappings $g\colon X\rightarrow Y$ such that every $x \in X$ has a neighborhood $V_x$ such that $g(V_x)$ is relatively $\mathcal A$-compact. We show that the behavior of $\mathcal A$-compact polynomials is determined by its behavior in any neighborhood of any point. We transfer some known properties of $\mathcal A$-compact operators to $\mathcal A$-compact polynomials. In order to study $\mathcal A$-compact holomorphic functions, we appeal to the $\mathcal A$-compact radius of convergence which allows us to characterize the functions in this class. Under certain hypothesis on the ideal $\mathcal A$, we give examples showing that our characterization is sharp.