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Interpolating functions of minimal norm, star-invariant subspaces, and kernels of Toeplitz operators. (English) Zbl 0769.30022
Recall that a bounded analytic function on the unit disk is called inner if its boundary values have modulus 1 almost everywhere on the unit circle. The author proves that, given any inner function \(\theta\) there exists a sequence of points \(\{z_ n\}\) in the unit disk and associated complex numbers \(w_ n=\theta(z_ n)\) with \(\sup_ n w_ n<1\) such that \(\theta\) (of norm 1) is the minimal infinity norm solution of the interpolation problem: find \(g\) in \(H^ \infty\) with \(g(z_ n)=w_ n\). This extends a well known property of finite Blaschke products to general inner functions. The sequence of points \(\{z_ n\}\) can in fact be chosen to be an interpolating sequence; recall that a sequence of points in the unit disk is said to be an interpolating sequence if the points are uniformly separated from each other in the hyperbolic metric. The proof relies on the construction of an interpolating Blaschke product satisfying certain prescribed analytic properties due to D. Marshall. Also some sufficient conditions are given for the equality \(T_{\{z_ n\}} K_ \theta^ 2=\ell^ 2\) to hold, where \(Tf=\{(1-| z_ n|)^{1\over 2} f(z_ n)\}_{n=1}^ \infty\) and \(K_ \theta^ 2\) is the star-invariant subspace \(H^ 2\ominus \theta H_ 2\); it is well known that \(TH^ 2=\ell^ 2\) if \(\{z_ n\}\) is an interpolating sequence. In addition the characterization of the extreme points of the unit ball of \(H^ 1\) as outer functions of unit norm due to de Leeuw and Rudin is extended to the setting where the space \(H^ 1\) is replaced by the kernel of a Toeplitz operator acting on \(H^ 1\).

30E05 Moment problems and interpolation problems in the complex plane
30D55 \(H^p\)-classes (MSC2000)
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