Characterising derivations from the disc algebra to its dual

Details Characterising derivations from the disc algebra to its dual Characterising derivations from the disc algebra to its dual

2009-09-10

arxiv.org/abs/0909.1867v2

Proc. Amer. Math. Soc. 139 (2011), no. 3, 1073--1080

Mathematics

Functional Analysis

v1: 8 pages, preliminary version. v2: expanded the exposition slightly; added new abstract, extra reference, and various top m

Scientific paper

10.1090/S0002-9939-2010-10520-8

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.

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