Compact Bloch mappings on the complex unit disc

TL;DR

This paper introduces a new framework for understanding Bloch functions on the unit disc using Bloch molecules and a Bloch-free Banach space, establishing key properties and classical theorems in this context.

Contribution

It develops a novel approach to the duality of Bloch spaces through the concepts of Bloch molecules and Bloch-free Banach spaces, and introduces the notion of compact Bloch mappings.

Findings

Invariance of compact Bloch mappings under M"obius transformations

Linearization of Bloch mappings via Bloch-free Banach space

Classical theorems adapted to Bloch function setting

Abstract

The known duality of the space of Bloch complex-valued functions on the open complex unit disc $\mathbb{D}$ is addressed under a new approach with the introduction of the concepts of Bloch molecules and Bloch-free Banach space of $\mathbb{D}$. We introduce the notion of compact Bloch mapping from $\mathbb{D}$ to a complex Banach space and establish its main properties: invariance by M\"obius transformations, linearization from the Bloch-free Banach space of $\mathbb{D}$, factorization of their derivatives, inclusion properties, Banach ideal property and transposition on the Bloch function space. We state Bloch versions of the classical theorems of Schauder, Gantmacher and Davis-Figiel-Johnson-Pelczy\'nski.