Characterising derivations from the disc algebra to its dual

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 $μ_D$ on the closed disc, such that $D$ factors through $L^2(μ_D)$. 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.

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