Duality of certain Banach spaces of vector-valued holomorphic functions

TL;DR

This paper investigates the duality and isomorphism properties of vector-valued Hardy spaces of holomorphic functions, linking these properties to geometric features of the Banach space F such as UMDP, ARNP, and RNP.

Contribution

It establishes a characterization of when vector-valued Hardy spaces are topologically isomorphic based on the UMDP property of the underlying Banach space.

Findings

H p (D; F) and H q (D; F) are isomorphic if and only if F has UMDP.

The study extends scalar-valued results to vector-valued Hardy spaces.

Connections between Hardy space duality and Banach space properties are clarified.

Abstract

In this work we study the vector-valued Hardy spaces H p (D; F) (1 \leq p \leq \infty) and their relationship with RNP, ARNP and the UMDP properties. By following the approach of Taylor in the scalar-valued case, we prove that, when F and F have the ARNP property, then H p (D; F) and H q (D; F) are canonically topologically isomorphic (for p, q \in (1, \infty) conjugate indices) if and only if F has the UMDP.