Revista Matemática Complutense

, Volume 32, Issue 1, pp 115–139 | Cite as

Interpolating sequences for weighted spaces of analytic functions on the unit ball of a Hilbert space

  • Oscar Blasco
  • Pablo Galindo
  • Mikael Lindström
  • Alejandro MirallesEmail author


We show that an interpolating sequence for the weighted Banach space of analytic functions on the unit ball of a Hilbert space is hyperbolically separated. In the case of the so-called standard weights, a sufficient condition for a sequence to be linear interpolating is given in terms of Carleson type measures. Other conditions to be linearly interpolating are provided as well. Our results apply to the space of Bloch functions of such unit ball.


Interpolating sequence Hyperbolically separated Bloch function in the ball Infinite dimensional holomorphy Weighted space of analytic functions 

Mathematics Subject Classification

Primary 30D45 46E50 Secondary 46G20 



This paper was completed during the 2016 fall semester while Mikael Lindström was visiting Universidad de Valencia whose hospitality is gratefully acknowledged with special thanks to Pablo Galindo. We warmly thank the referees for their very careful reading and the suggestions provided.


  1. 1.
    Attele, K.R.M.: Interpolating sequences for the derivatives of the Bloch functions. Glasg. Math. J. 34, 35–41 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berndtsson, B.: Interpolating sequences for \(H^\infty \) in the ball. Nederl. Akad. Wetensch. Indag. Math. 47(1), 1–10 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blasco, O., Galindo, P., Miralles, A.: Bloch functions on the unit ball of an infinite dimensional Hilbert space. J. Func. Anal. 267, 1188–1204 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blasco, O., Galindo, P., Lindström, M., Miralles, A.: Composition operators on the Bloch space of the unit ball of a Hilbert space. Banach J. Math. Anal. 11(2), 311–334 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boe, B., Nicolau, A.: Interpolation by functions in the Bloch space. J. Anal. Math. 94, 171–194 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carne, T.K., Cole, B., Gamelin, T.W.: A uniform algebra of analytic functions on a Banach space. Trans. Am. Math. Soc. 314(2), 639–659 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Choe, B.R., Rim, K.S.: Fractional derivatives of Bloch functions, growth rate and interpolation. Acta Math. Hung. 72(1–2), 67–86 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  9. 9.
    Domański, P., Lindström, M.: Sets of interpolation for weighted Banach spaces of holomorphic functions. Ann. Pol. Math. 79(3), 233–264 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duren, P., Schuster, A., Vukotić, D.: On Uniformly Discrete Sequences in the Disk, Operator Theory: Advances and Applications. Birkhäuser, Basel (2005)zbMATHGoogle Scholar
  11. 11.
    Galindo, P., Gamelin, T.W., Lindström, M.: Spectra of composition operators on algebras of analytic functions on Banach spaces. Proc. R. Soc. Edinb. 139A, 107–121 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Galindo, P., Miralles, A.: Interpolating sequences for bounded analytic functions. Proc. Am. Math. Soc. 135(10), 3225–3231 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Galindo, P., Miralles, A., Lindström, M.: Interpolating sequences on uniform algebras. Topology 48, 111–118 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1981)zbMATHGoogle Scholar
  15. 15.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Marcel Dekker Inc, New York (1984)zbMATHGoogle Scholar
  16. 16.
    Hamada, H.: Bloch-type spaces and extended Cesàro operators in the unit ball of a complex Banach space. Sci. China Math. (2018).
  17. 17.
    Madigan, K., Matheson, A.: Compact composition operators on the Bloch space. Trans. Am. Math. Soc. 347, 2679–2687 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Massaneda, X.: \(A^{-p}\) interpolation in the unit ball. J. Lond. Math. Soc. 52(2), 391–401 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mujica, J.: Complex Analysis in Banach Spaces. Dover Books on Mathematics, New York (2010)Google Scholar
  20. 20.
    Mujica, J.: Linearization of holomorphic mappings on infinite-dimensional spaces. Rev. Un. Mater. Argent. 37, 127–134 (1992)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Miralles, A.: Interpolating sequences for \(H^{\infty }(B_H)\). Quaest. Math. 39(6), 785–795 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ng, K.: On a theorem of Dixmier. Math. Scand. 29, 279–280 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rudin, W.: Function Theory in the Unit Ball of \(\mathbb{C}^n\). Springer, New York (1980)CrossRefzbMATHGoogle Scholar
  24. 24.
    Seip, K.: Beurling type density theorems in the unit disk. Invent. Math. 113, 21–39 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Seip, K.: Interpolation and Sampling in Spaces of Analytic Functions, University Lecture Series 33. American Mathematical Society, Providence (2004)Google Scholar
  26. 26.
    Tjani, M.: Distance of a Bloch function to the little Bloch space. Bull. Aust. Math. Soc. 74(1), 101–119 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Xiao, J.: Carleson measure, atomic decomposition and free interpolation from Bloch space. Ann. Acad. Sci. Fenn. Ser. A I Math. 19(1), 35–46 (1994)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Xu, Z.: Bloch type spaces in the unit ball of a Hilbert space (2018). arXiv:1611.10227v2 [math.CV]
  29. 29.
    Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Springer, Berlin (2005)Google Scholar

Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversidad de ValenciaValenciaSpain
  2. 2.Department of MathematicsAbo Akademi UniversityAboFinland
  3. 3.Departament de Matemàtiques and Instituto Universitario de Matemáticas y Aplicaciones de Castellón (IMAC)Universitat Jaume I de Castelló (UJI)CastellóSpain

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