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Interpolating functions of minimal norm, star-invariant subspaces and kernels of Toeplitz operators

Author: Konstantin M. Dyakonov
Journal: Proc. Amer. Math. Soc. 116 (1992), 1007-1013
MSC: Primary 30D55; Secondary 30E05, 47B35
MathSciNet review: 1100649
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Abstract: It is proved that for each inner function $ \theta $ there exists an interpolating sequence $ \left\{ {{z_n}} \right\}$ in the disk such that $ {\sup _n}\vert\theta ({z_n})\vert < 1$, but every function $ g$ in $ {H^\infty }$ with $ g({z_n}) = \theta ({z_n})(n = 1,2, \ldots )$ satisfies $ \vert\vert g\vert{\vert _\infty } \geq 1$. Some results are obtained concerning interpolation in the star-invariant subspace $ {H^2} \ominus \theta {H^2}$. This paper also contains a "geometric" result connected with kernels of Toeplitz operators.

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Keywords: Inner function, interpolating Blaschke product, star-invariant subspace, extreme point, Toeplitz operator
Article copyright: © Copyright 1992 American Mathematical Society

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