The injective hull of ideals of weighted holomorphic mappings

TL;DR

This paper introduces and characterizes the injective hulls of weighted holomorphic mapping ideals, providing new descriptions and a characterization in terms of an Ehrling-type inequality.

Contribution

It defines the injective hull of weighted holomorphic ideals and describes their properties and characterizations, extending existing concepts to this new setting.

Findings

Injective hulls are characterized via a domination property.

Descriptions of injective hulls generated by composition and dual procedures.

A new Ehrling-type inequality characterization for these hulls.

Abstract

We study the injectivity of normed ideals of weighted holomorphic mappings. To be more precise, the concept of injective hull of normed weighted holomorphic ideals is introduced and characterized in terms of a domination property. The injective hulls of those ideals -- generated by the procedures of composition and dual -- are described and these descriptions are applied to some examples of such ideals. A characterization of the closed injective hull of an operator ideal in terms of an Ehrling-type inequality -- due to Jarchow and Pelczy\'nski -- is established for weighted holomorphic mappings.