$\mathcal{A}$-compact mappings

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 $g:X\to Y$ 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.

21 Pages; Accepted in RACSAM