Bloch functions on the unit ball of a Banach space

TL;DR

This paper extends the concept of Bloch functions to the unit ball of complex Banach spaces, analyzing different definitions and their relationships, and showing all bounded analytic functions are Bloch functions under these definitions.

Contribution

It introduces two new ways to define Bloch functions on Banach space unit balls and compares their properties, highlighting differences and similarities.

Findings

The two definitions of Bloch functions are generally different.

All bounded analytic functions on the unit ball are Bloch functions under both definitions.

The relationship between these Bloch spaces is characterized in the paper.

Abstract

The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions $f$ on the unit ball $B_E$ of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch space considering the boundness of $(1-\|x\|^2) \|f'(x)\|$ on $B_E$ and by preserving the invariance of the correspondiing seminorm when we compose with automorphisms $\phi$ of $B_E$. We study the connection between these spaces proving that they are different in general and prove that all bounded analytic functions on $B_{E}$ are Bloch functions in both ways.