Two counterexamples involving inner functions
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- by Kenneth Stephenson PDF
- Proc. Amer. Math. Soc. 73 (1979), 22-24 Request permission
Abstract:
Two questions about the existence of bounded analytic functions with prescribed behavior are answered using a construction technique of McLaughlin and Piranian. If A is a relatively closed, countable subset of the unit disc, the construction gives (I) Function $f \in {H^\infty }$ so that the inner factor of $f - \alpha$ is a finite Blaschke product if and only if $\alpha \in A$. (II) Inner function $\phi$ so that $\phi - \alpha$ has a discrete singular inner factor if and only if $\alpha \in A$.References
- Carl C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978), 1–31. MR 482347, DOI 10.1090/S0002-9947-1978-0482347-9
- Renate McLaughlin and George Piranian, The exceptional set of an inner function, Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 185 (1976), no. 1-3, 51–54. MR 0447585
- Kenneth Stephenson, Omitted values of singular inner functions, Michigan Math. J. 25 (1978), no. 1, 91–100. MR 481015
- James Thomson, The commutant of a class of analytic Toeplitz operators. II, Indiana Univ. Math. J. 25 (1976), no. 8, 793–800. MR 417843, DOI 10.1512/iumj.1976.25.25063
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 73 (1979), 22-24
- MSC: Primary 30D50
- DOI: https://doi.org/10.1090/S0002-9939-1979-0512050-4
- MathSciNet review: 512050