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Characterizing derivations from the disk algebra to its dual. (English) Zbl 1251.46026
Let $$A$$ be the disk algebra, let $$a\in A$$ and let $$\gamma$$ be a bounded linear functional on $$A$$. Let $$a\gamma$$ be the functional $$x\mapsto\gamma(ax)$$. With this multiplication, the dual $$A^*$$ of $$A$$ becomes an $$A$$-module. Let $$\mathcal{D}(A)$$ denote the Banach space of all bounded derivations from $$A$$ to $$A^*$$. In the present paper an explicit formula is given for a Banach space isomorphism between $$\mathcal{D}(A)$$ and $$\overline{H^1_0}$$, the space of the functions whose complex conjugates belong to the Hardy space $$H^1$$ on the open unit disk and vanish at the origin. The formula is the following: Let $$h\in H^1_0$$, let $$f$$ and $$g$$ be polynomials and let $$u$$ be a primitive of $$f'g$$. Define $D(f)(g):=\int_{|z|=1}u(z)\overline{h(z)}|\mathrm{d}z|\;.$ The authors employ BMOA estimates to prove that this formula defines a uniformly bounded bilinear functional of $$f$$ and $$g$$, so it extends by continuity to an element of $$\mathcal{D}(A)$$. Furthermore, every $$D\in\mathcal{D}(A)$$ is of this form: the existence and uniqueness of $$h$$ follow from the F. and M. Riesz theorem and some integral estimates. The authors derive two notable consequences of this explicit representation. In the third section, they show that if $$h$$ is a polynomial then $$D$$ has finite rank. Hence, $$\mathcal{D}(A)$$ consists entirely of compact operators, settling a conjecture of S. E. Morris [Bounded derivations from uniform algebras. Ph.D. Thesis. University of Cambridge (1993)]. In the last section, given $$D\in\mathcal{D}(A)$$, they construct explicitly a Borel measure $$\mu$$ such that $$\|D(f)\|\leq\|f\|_{L^2(\mu)}$$, giving thus a constructive proof that every $$D\in\mathcal{D}(A)$$ is 2-summing.

##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces
##### Keywords:
derivation; disk algebra; Hardy space
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##### References:
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