A-compact holomorphic Lipschitz mappings on the unit ball of a Banach space

TL;DR

This paper investigates a subclass of holomorphic Lipschitz maps on the unit ball of Banach spaces, characterized by A-compactness, and explores their structure as a composition Banach holomorphic Lipschitz ideal.

Contribution

It introduces and studies A-compact holomorphic Lipschitz mappings, linking their properties to A-compact linear operators and establishing their structure as a composition ideal.

Findings

Characterization of A-compact holomorphic Lipschitz maps

Connection between A-compactness and linear operators

Structural analysis as a composition Banach ideal

Abstract

Let X and Y be complex Banach spaces, B_X be the open unit ball of X and HL(B_X,Y) be the Banach space of all holomorphic Lipschitz maps f:B_X->Y such that f(0)=0, endowed with the Lipschitz norm. Given a Banach operator ideal A, we use the property of A-compactness by Carl and Stephani to introduce and study the subclass of those functions in HL(B_X,Y) for which its Lipschitz image is a relatively A-compact subset of Y. We focus our attention on its structure as a composition Banach holomorphic Lipschitz ideal by using its connection with A-compact linear operators through linearization/transposition techniques.