Characterising derivations from the disc algebra to its dual

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.

v1: 8 pages, preliminary version. v2: expanded the exposition slightly; added new abstract, extra reference, and various top matter. A revised version of this has been accepted for publication by Proc. Amer. Math. Soc.