Factorization of Blaschke products and primary ideals in \(H^{\infty }\)
Let \(H^\infty\) be the uniform algebra of all bounded holomorphic functions in the open unit disk and let \(M(H^\infty)\) be the spectrum of \(H^\infty\). For a compact set \(E\subseteq M(H^\infty)\), let \(I=I(E,H^\infty):=\{f\in H^\infty: \hat f\equiv 0 \text{ on } E\}\). Assuming that the hull of \(I\) is contained in the set \(G\) of nontrivial points in \(M(H^\infty)\), the authors continue the study of this special class of ideals and extend the theory developed by \textit{H.-M.\thinspace Lingenberg} [Mich.\ Math.\ J.\ 39, No.\,1, 53--63 (1992; Zbl 0759.46048)] and \textit{P.\,Gorkin, K.\,Izuchi} and \textit{R.\,Mortini} [J.~Funct.\ Anal.\ 177, No.\,1, 107--129 (2000; Zbl 0982.46035)]. New factorization theorems for Carleson-Newman Blaschke products are given, too. The paper is very technical. Let us point out that the authors use the term ``primary ideal'' to refer to ideals of the form \(I(E,H^\infty)\) above, rather than the more conventional meaning (in the commutative algebra setting, the term primary ideal refers to ideals that are contained in a unique maximal ideal).
scientific article; zbMATH DE number 5757618
scientific article