Characterizing derivations from the disk algebra to its dual

Choi, Y.; Heath, M.

doi:10.1090/S0002-9939-2010-10520-8

Characterizing derivations from the disk algebra to its dual
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by Y. Choi and M. J. Heath

Proc. Amer. Math. Soc. 139 (2011), 1073-1080

DOI: https://doi.org/10.1090/S0002-9939-2010-10520-8

Published electronically: August 3, 2010

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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.

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