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## Characterizing derivations from the disk algebra to its dual

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Journal publication date 3/08/2011 Proceedings of the American Mathematical Society 3 139 8 1073-1080 Published English

### Abstract

We show that the space of all bounded derivations from the disk 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 disk, such that $D$ factors through $L^2(\mu_D)$. Such a measure is known to exist, for any bounded linear map from the disk algebra to its dual, by results of Bourgain and Pietsch, but these results are highly non-constructive.