Characterising derivations from the disc algebra to its dual

TL;DR

This paper characterizes all bounded derivations from the disc algebra to its dual as elements of the Hardy space $H^1$, showing they are compact and can be factored through specific $L^2$ spaces with constructed measures.

Contribution

It provides a concrete identification of derivations with $H^1$, constructs explicit measures for factorization, and demonstrates the compactness of all such derivations.

Findings

All bounded derivations are compact.

Derivations correspond to elements in $H^1$.

Existence of explicit measures for factorization.

Abstract

We show that the space of all bounded derivations from the disc algebra into its dual can be identified with the Hardy space $H^1$; using this, we infer that all such derivations are compact. Also, given a fixed derivation $D$, we construct a finite, positive Borel measure $\mu_D$ on the closed disc, such that $D$ factors through $L^2(\mu_D)$. Such a measure is known to exist, for any bounded linear map from the disc algebra to its dual, by results of Bourgain and Pietsch, but these results are highly non-constructive.