Majorization in de Branges spaces. III. Division by Blaschke products

Baranov, A.; Woracek, H.

doi:10.1090/S1061-0022-2010-01122-1

Majorization in de Branges spaces. III. Division by Blaschke products
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by A. Baranov and H. Woracek

St. Petersburg Math. J. 21 (2010), 843-875

DOI: https://doi.org/10.1090/S1061-0022-2010-01122-1

Published electronically: September 22, 2010

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Abstract:

This paper is a part of a series dealing with subspaces of de Branges spaces of entire functions generated by majorization on subsets of the closed upper half-plane. In the present, third, part the study of a certain Banach space generated by an admissible majorant is continued. The main theme is “invariance of the unit ball with respect to division by Blaschke products”. In connection with this topic, representability via special types of majorants plays an important role. Some (positive and negative) results on invariance under division by Blaschke factors are obtained, and the unit balls representable by $\log$-superharmonic majorants are characterized.

References

Anton D. Baranov, Polynomials in the de Branges spaces of entire functions, Ark. Mat. 44 (2006), no. 1, 16–38. MR 2237209, DOI 10.1007/s11512-005-0006-1
Anton Baranov and Harald Woracek, Subspaces of de Branges spaces generated by majorants, Canad. J. Math. 61 (2009), no. 3, 503–517. MR 2514481, DOI 10.4153/CJM-2009-026-2
Anton Baranov and Harald Woracek, Finite-dimensional de Branges subspaces generated by majorants, Spectral theory in inner product spaces and applications, Oper. Theory Adv. Appl., vol. 188, Birkhäuser Verlag, Basel, 2009, pp. 37–48. MR 2641246, DOI 10.1007/978-3-7643-8911-6_{3}
Anton Baranov and Harald Woracek, Majorization in de Branges spaces. I. Representability of subspaces, J. Funct. Anal. 258 (2010), no. 8, 2601–2636. MR 2593335, DOI 10.1016/j.jfa.2010.01.016
—, Majorization in de Branges spaces. II. Banach spaces generated by majorants, arXiv:0906.2943v1 [math. CV], to appear in Collectanea Mathematica.
A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291–309. MR 147848, DOI 10.1007/BF02545792
Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
Louis de Branges, Some Hilbert spaces of entire functions, Proc. Amer. Math. Soc. 10 (1959), 840–846. MR 114002, DOI 10.1090/S0002-9939-1959-0114002-2
Louis de Branges, Some Hilbert spaces of entire functions, Trans. Amer. Math. Soc. 96 (1960), 259–295. MR 133455, DOI 10.1090/S0002-9947-1960-0133455-X
Louis de Branges, Some Hilbert spaces of entire functions. II, Trans. Amer. Math. Soc. 99 (1961), 118–152. MR 133456, DOI 10.1090/S0002-9947-1961-0133456-2
Louis de Branges, Some Hilbert spaces of entire functions. III, Trans. Amer. Math. Soc. 100 (1961), 73–115. MR 133457, DOI 10.1090/S0002-9947-1961-0133457-4
Louis de Branges, Some Hilbert spaces of entire functions. IV, Trans. Amer. Math. Soc. 105 (1962), 43–83. MR 143016, DOI 10.1090/S0002-9947-1962-0143016-6
Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
Victor Havin and Javad Mashreghi, Admissible majorants for model subspaces of $H^2$. I. Slow winding of the generating inner function, Canad. J. Math. 55 (2003), no. 6, 1231–1263. MR 2016246, DOI 10.4153/CJM-2003-048-8
Victor Havin and Javad Mashreghi, Admissible majorants for model subspaces of $H^2$. II. Fast winding of the generating inner function, Canad. J. Math. 55 (2003), no. 6, 1264–1301. MR 2016247, DOI 10.4153/CJM-2003-049-5
J. Hoffmann-Jørgensen, Stochastic processes on Polish spaces, Various Publications Series (Aarhus), vol. 39, Aarhus Universitet, Matematisk Institut, Aarhus, 1991. MR 1217966
Michael Kaltenbäck and Harald Woracek, De Branges spaces of exponential type: general theory of growth, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 231–284. MR 2160366
Marvin Rosenblum and James Rovnyak, Topics in Hardy classes and univalent functions, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR 1307384, DOI 10.1007/978-3-0348-8520-1
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157