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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
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