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

TL;DR

The paper extends the theory of operator ideals to non-linear holomorphic Lipschitz maps on the unit ball of a Banach space by introducing $ ext{A}$-compact and $ ext{A}$-bounded holomorphic Lipschitz mappings, measured via the Lipschitz image $ ext{Im}_L(f)$. Central to the approach is the linearization of holomorphic Lipschitz maps through the universal object $oldsymbol{rak G}_0(B_X)$ and the associated operator $T_f$, which allows complete characterizations of non-linear ideals through linear operator ideals (including duals and composition). It develops holomorphic Lipschitz ideals of composition type, proves their closure and regularity properties, and provides precise equivalences with the corresponding linear $ ext{A}$-compact or $ ext{A}$-bounded operators; further, it analyzes weakly compact, Rosenthal, Banach–Saks, and Asplund variants via transposes and duality. The results yield a robust framework for non-linear approximation properties in the holomorphic Lipschitz setting and connect these non-linear ideals with classical operator-ideal theory, enabling transfer of Schauder-type and factorization results to holomorphic maps. This work broadens the scope of density and approximation properties in spaces of holomorphic Lipschitz mappings and clarifies how linearization and transposition govern non-linear ideal behavior.

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.

Theorems & Definitions (53)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Proposition 1.4
• Definition 1.5
• Definition 1.6
• Theorem 1.7
• proof
• Corollary 1.8
• proof