Composition operators for holomorphic Lipschitz functions

TL;DR

This paper investigates composition operators on holomorphic Lipschitz function spaces in complex Banach spaces, providing characterizations, properties, and asymptotic behaviors, especially in finite dimensions.

Contribution

It introduces a linearization approach via Lipschitz-free spaces, characterizes when composition operators are isomorphisms or compact, and analyzes their iterative behavior.

Findings

Compactness and weak compactness coincide in finite dimensions.

Complete characterization of composition operators in terms of the symbol.

Iterates of composition operators converge to zero when the symbol's supremum norm is less than one.

Abstract

We study composition operators on spaces of holomorphic Lipschitz functions defined on the open unit ball of a complex Banach space. Our approach is based on the linearization of the symbol through the holomorphic Lipschitz-free spaces, which allow composition operators to be realized as adjoints of linear operators. For spaces with the bounded approximation property, we characterize composition operators between spaces of holomorphic Lipschitz functions vanishing at the origin and describe when composition operators are onto isomorphisms. We further investigate compactness and weak compactness properties of composition operators. In the finite-dimensional setting, compactness and weak compactness are shown to coincide, and a complete characterization is obtained in terms of the symbol. Finally, we analyze the asymptotic behavior of the iterates of composition operators, proving convergence to zero whenever the supremum norm of the symbol is less than one, and we extend several results to the case not vanishing at 0.