On an example of J. Bourgain on closures of finitely generated ideals
Let \(H^\infty\) be the algebra of bounded analytic functions in the unit disk \(\mathbb{D}\). For two functions \(f\) and \(g\) in \(H^\infty\) let \(I= I(f,g)\) be the ideal generated by \(f\) and \(g\) and let \[ J= J(f,g)= \{h\in H^\infty: |h|\leq C(|f|+|g|)\text{ for some }C>0\}. \] It is obvious that \(J\) is an ideal containing \(I\). We assume that the functions \(f\) and \(g\) have no common factor. In [Math. Ann. 301, No. 1, 135-154 (1995; Zbl 0564.46046)], \textit{P. Gorkin}, \textit{A. Nicolau} and the author proved that \(I(f,g)= J(f,g)\) if and only if \[ |f|+|g|+(1-|z|^2)(|f'|^2+|g'|^2)\geq\delta>0\quad\text{on }\mathbb{D}. \] On the other hand, as was shown by \textit{J. Bourgain} [Ann. Inst. Fourier 35, No. 4, 163-174 (1985; Zbl 0564.46044)], there exists functions \(f\) and \(g\) in \(H^\infty\) for which \(J(f,g)\) is not even contained in the closure \(\overline{I(f,g)}\) of \(I\). He proved the existence of two Blaschke products \(B\) and \(C\) such that \(BC\not\in \overline{I(B^2,C^2)}\). In the paper under review, it is shown that, in contrast to Bourgain's example, the ideal \(J(f,g)\) is contained in the closure of \(I(f,g)\) whenever \(f\) and \(g\) are powers of interpolating Blaschke products. It is also shown that the ideal \(J(B^N,C^N)\) is finitely generated whenever \(B\) and \(C\) are interpolating Blaschke products.
scientific article; zbMATH DE number 1028170