p-Summing Bloch mappings on the complex unit disc

TL;DR

This paper introduces and studies the properties of p-summing Bloch mappings on the complex unit disc, establishing their invariance, ideal structure, and related domination and factorization theorems in the context of complex Banach spaces.

Contribution

It defines p-summing Bloch mappings, proves their M"obius invariance, and develops Bloch versions of key theorems, connecting these mappings with duals of Bloch molecules.

Findings

The space of p-summing Bloch mappings is M"obius-invariant.

Subspace of zero-preserving mappings forms an injective Banach ideal.

Bloch versions of Pietsch's domination and Maurey's extrapolation theorems are established.

Abstract

The notion of $p$-summing Bloch mapping from the complex unit open disc $\mathbb{D}$ into a complex Banach space $X$ is introduced for any $1\leq p\leq\infty$. It is shown that the linear space of such mappings, equipped with a natural seminorm $\pi^{\mathbb{B}}_p$, is M\"obius-invariant. Moreover, its subspace consisting of all those mappings which preserve the zero is an injective Banach ideal of normalized Bloch mappings. Bloch versions of the Pietsch's domination/factorization Theorem and the Maurey's extrapolation Theorem are presented. We also introduce the spaces of $X$-valued Bloch molecules on $\mathbb{D}$ and identify the spaces of normalized $p$-summing Bloch mappings from $\mathbb{D}$ into $X^*$ under the norm $\pi^{\mathbb{B}}_p$ with the duals of such spaces of molecules under the Bloch version of the $p$-Chevet--Saphar tensor norms $d_p$.