
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 , we use the notion of -compact sets of Carl and Stephani to study -compact polynomials and -compact holomorphic mappings. Namely, those mappings such that every has a neighborhood such that is relatively -compact. We show that the behavior of -compact polynomials is determined by its behavior in any neighborhood of any point. We transfer some known properties of -compact operators to -compact polynomials. In order to study -compact holomorphic functions, we appeal to the -compact radius of convergence which allows us to characterize the functions in this class. Under certain hypothesis on the ideal , we give examples showing that our characterization is sharp.
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