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Blaschke products of Sundberg-Wolff type. (English) Zbl 0885.46046

Summary: Let \(D\) be a Douglas algebra. An interpolating Blaschke product is said to be of Sundberg-Wolff type (for \(D\)) if its zero sequence is interpolating for the algebra \(QD_A=\overline D\cap H^\infty\). We prove that an interpolating Blaschke product \(b\) is of Sundberg-Wolff type if and only if \(b\) is unimodular on the set of trivial Gleason parts contained in the spectrum \(M(D)\) of \(D\) and if every part in \(M(D)\) contains at most one zero of \(b\).
We also show that an inner function \(u\) is invertible in the Douglas algebra \(B\) generated by \(H^\infty\) and the conjugates of \(QD_A\)-interpolating Blaschke products if and only if \(u\) is a finite product of Blaschke products of Sundberg-Wolff type (modulo invertible inner functions in \(D\)).

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46J30 Subalgebras of commutative topological algebras
30D55 \(H^p\)-classes (MSC2000)
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