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Ideals of bounded holomorphic functions on simple \(n\)-sheeted discs. (English) Zbl 0753.30038

We denote by \(H^ \infty(R)\) the Banach algebra of bounded holomorphic functions on a Riemann surface \(R\). An \(n\)-sheeted disc is an unbounded \(n\)-sheeted covering surface of the open unit disc \(D=\{| z|<1\}\). An \(n\)-sheeted disc \(R\) is said to be simple if \(R\) is the Riemann surface of an \(n\)-valued function \(\zeta=B(z)^{1/n}\) for the Blaschke product \(B(z)\) on \(D\) whose zeros are all simple.
For a Riemann surface \(R\), let \(w(R)\) be the smallest positive integer such that, given \(f_ 1,\ldots,f_ m\) and \(g\) arbitrary functions in \(H^ \infty(R)\) with \(\sum^ m_{j=1}| f_ j|^ 2\geq| g|^ 2\), there exist \(h_ 1,\ldots,h_ m\) in \(H^ \infty(R)\) with \(\sum^ m_{j=1}f_ jh_ j=g^{w(R)}\).
Let \(W(n)=\sup_ Rw(R)\) (\(R\) runs over the family of all simple \(n\)- sheeted discs). It is shown that \(n+1\leq W(n)\leq 4n-2\) \((n\geq 2)\).
Reviewer: M.Hara (Nagoya)

MSC:

30H05 Spaces of bounded analytic functions of one complex variable
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