Interpolating sequences on uniform algebras
Let \(A\) be a uniform algebra with maximal ideal space \(M_A\). A sequence \(\{x_n\}\), \(x_n\in M_A\), is an interpolating sequence for \(A\) if for any bounded sequence \(\{\alpha_n\},\, \alpha_n\in\mathbb C\), there is an \(f\in A\) so that \(\widehat f(x_n)=\alpha_n\) for all \(n\). Given an \(\{\alpha_n\}\in\ell_\infty\), denote \(M_{\{\alpha_n\}}=\inf\{\|f\|_\infty: \widehat f(x_n)=\alpha_n, n\in{\mathbb N}, f\in A\}\). A constant of interpolation for \(\{x_n\}\) is any number \(\geq\sup\{M_{\{\alpha_n\}}: \{\alpha_n\}\in\ell_\infty,\;\|\{\alpha_n\}\|_\infty\leq 1\}\). If there is a linear map \(T: \ell_\infty\to A\) such that \(\widehat{(T\{\alpha_n\})}(\{x_n\})=\{\alpha_n\}\) for every \(\{\alpha_n\}\in\ell_\infty\), then \(\{x_n\}\) is called linear interpolating sequence for \(A\); \(c_0\)-interpolating and \(c_0\)-linear interpolating sequences are defined in the same way. The authors find conditions under which a given interpolating sequence for a uniform algebra is a linear interpolating sequence. Theorem. Let \(A\) be a dual uniform algebra, i.e., \(A\) coincides with the dual space of a Banach space, and let \(\{x_n\}\) be a sequence in \(M_A\cap X\). Then the following are equivalent: (a) Every finite subset of \(\{x_n\}\) is interpolating for \(A\) and there exists a constant of interpolation independent of the number of interpolated terms; (b) \(\{x_n\}\) is \(c_0\)-interpolating for \(A\); (c) \(\{x_n\}\) is linear interpolating for \(A\). It is also shown that, if the Carleson generalized condition, related with the pseudohyberbolic metric, is sufficient for a sequence to be interpolating for the algebra of bounded analytic functions on the unit ball of \(c_0\), then it is sufficient for any dual uniform algebra.
scientific article; zbMATH DE number 5706252