Characterizing derivations from the disk algebra to its dual

Choi, Yemon and Heath, Matthew J. (2011) Characterizing derivations from the disk algebra to its dual. Proceedings of the American Mathematical Society, 139 (3). pp. 1073-1080. ISSN 0002-9939

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.