Interpolating Blaschke products and angular derivatives

Gallardo-Gutiérrez, Eva; Gorkin, Pamela

doi:10.1090/S0002-9947-2012-05535-8

Interpolating Blaschke products and angular derivatives

Authors: Eva A. Gallardo-Gutiérrez and Pamela Gorkin
Journal: Trans. Amer. Math. Soc. 364 (2012), 2319-2337
MSC (2010): Primary 46J15, 30J10; Secondary 30H10, 47B38
DOI: https://doi.org/10.1090/S0002-9947-2012-05535-8
Published electronically: January 3, 2012
MathSciNet review: 2888208
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Abstract: We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra

$\displaystyle H^\infty [\overline {b}: b~$ $\displaystyle \mbox {has finite angular derivative everywhere}].$

We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.

1. Patrick Ahern, The mean modulus and the derivative of an inner function, Indiana Univ. Math. J. 28 (1979), no. 2, 311–347. MR 523107, https://doi.org/10.1512/iumj.1979.28.28022
2. P. R. Ahern and D. N. Clark, On inner functions with 𝐻^{𝑝}-derivative, Michigan Math. J. 21 (1974), 115–127. MR 0344479
3. John R. Akeroyd, Pratibha G. Ghatage, and Maria Tjani, Closed-range composition operators on 𝔸² and the Bloch space, Integral Equations Operator Theory 68 (2010), no. 4, 503–517. MR 2745476, https://doi.org/10.1007/s00020-010-1806-7
4. G. T. Cargo, Angular and tangential limits of Blaschke products and their successive derivatives, Canad. J. Math. 14 (1962), 334–348. MR 0136743, https://doi.org/10.4153/CJM-1962-026-2
5. Sun Yung A. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976), no. 2, 82–89. MR 0428044, https://doi.org/10.1007/BF02392413
Donald E. Marshall, Subalgebras of 𝐿^{∞} containing 𝐻^{∞}, Acta Math. 137 (1976), no. 2, 91–98. MR 0428045, https://doi.org/10.1007/BF02392414
6. William S. Cohn, On the 𝐻^{𝑝} classes of derivatives of functions orthogonal to invariant subspaces, Michigan Math. J. 30 (1983), no. 2, 221–229. MR 718268
7. Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
8. Konstantin Dyakonov and Artur Nicolau, Free interpolation by nonvanishing analytic functions, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4449–4465. MR 2309193, https://doi.org/10.1090/S0002-9947-07-04186-4
9. Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
10. John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
11. Pamela Gorkin and Raymond Mortini, Cluster sets of interpolating Blaschke products, J. Anal. Math. 96 (2005), 369–395. MR 2177193, https://doi.org/10.1007/BF02787836
12. Pamela Gorkin and Raymond Mortini, Asymptotic interpolating sequences in uniform algebras, J. London Math. Soc. (2) 67 (2003), no. 2, 481–498. MR 1956148, https://doi.org/10.1112/S0024610702004039
13. Håkan Hedenmalm, Thin interpolating sequences and three algebras of bounded functions, Proc. Amer. Math. Soc. 99 (1987), no. 3, 489–495. MR 875386, https://doi.org/10.1090/S0002-9939-1987-0875386-8
14. Kenneth Hoffman, Banach spaces of analytic functions, Dover Publications, Inc., New York, 1988. Reprint of the 1962 original. MR 1102893
15. Keiji Izuchi, Spreading Blaschke products and homeomorphic parts, Complex Variables Theory Appl. 40 (2000), no. 4, 359–369. MR 1772394
16. Sun Yung A. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976), no. 2, 82–89. MR 0428044, https://doi.org/10.1007/BF02392413
Donald E. Marshall, Subalgebras of 𝐿^{∞} containing 𝐻^{∞}, Acta Math. 137 (1976), no. 2, 91–98. MR 0428045, https://doi.org/10.1007/BF02392414
17. Artur Nicolau, Interpolating Blaschke products solving Pick-Nevanlinna problems, J. Anal. Math. 62 (1994), 199–224. MR 1269205, https://doi.org/10.1007/BF02835954
18. Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406
19. J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on 𝐻², Indiana Univ. Math. J. 23 (1973/74), 471–496. MR 0326472, https://doi.org/10.1512/iumj.1973.23.23041
20. Carl Sundberg and Thomas H. Wolff, Interpolating sequences for 𝑄𝐴_{𝐵}, Trans. Amer. Math. Soc. 276 (1983), no. 2, 551–581. MR 688962, https://doi.org/10.1090/S0002-9947-1983-0688962-3