Yemon Choi*, Matthew J. Heath
* Corresponding author
Derivation, disk algebra, Hardy space
Appeared as Proc. Amer. Math. Soc. 139 (2011), no. 3, 1073--1080.
Preprint version available at arXiv math.0909.1867
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We show that the space of all bounded derivations from the disk algebra into its dual can be identified with the Hardy space H1; using this, we infer that all such derivations are compact. Also, given a fixed derivation D, we construct a finite, positive Borel measure μD on the closed disk, such that D factors through L2(μD). 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.
The main result of this article answers the question raised by the second author during the problems session of the Banach Algebras 2007 conference (Laval).
I seem to remember that back in 2006, at the BMC in Newcastle, Matt had asked me (or perhaps his supervisor had) about weak compactness of derivations from A(D) to its dual. Owing to some things I had been reading about in the 1st year of my PhD, none of which made it into the final dissertation, I was able to immediately answer: "they are all 2-summing via Bourgain's result". It was therefore rather pleasant to find that on resolving the main question, the techniques used in proving $H^1$-BMOA duality could be adapted to yield an explicit Pietsch measure, without needing Bourgain's theorem.