Compact Bloch mappings on the complex unit disc

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.