Banach Journal of Mathematical Analysis

Composition operators on the Bloch space of the unit ball of a Hilbert space

Oscar Blasco, Pablo Galindo, Mikael Lindström, and Alejandro Miralles

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Abstract

Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as some examples that clarify the connections among such conditions.

Article information

Source
Banach J. Math. Anal., Volume 11, Number 2 (2017), 311-334.

Dates
Received: 14 March 2016
Accepted: 30 May 2016
First available in Project Euclid: 28 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1485572420

Digital Object Identifier
doi:10.1215/17358787-0000005X

Mathematical Reviews number (MathSciNet)
MR3603343

Zentralblatt MATH identifier
1361.32024

Subjects
Primary: 30D45: Bloch functions, normal functions, normal families
Secondary: 46E50: Spaces of differentiable or holomorphic functions on infinite- dimensional spaces [See also 46G20, 46G25, 47H60] 46G20: Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]

Keywords
composition operator Bloch function in the ball infinite-dimensional holomorphy

Citation

Blasco, Oscar; Galindo, Pablo; Lindström, Mikael; Miralles, Alejandro. Composition operators on the Bloch space of the unit ball of a Hilbert space. Banach J. Math. Anal. 11 (2017), no. 2, 311--334. doi:10.1215/17358787-0000005X. https://projecteuclid.org/euclid.bjma/1485572420


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References

  • [1] O. Blasco, P. Galindo, and A. Miralles, Bloch functions on the unit ball of an infinite dimensional Hilbert space, J. Funct. Anal. 267 (2014), no. 4, 1188–1204.
  • [2] O. Blasco, M. Lindström, and J. Taskinen, Bloch-to-BMOA compositions in several complex variables, Complex Var. Theory Appl. 50 (2005), no. 14, 1061–1080.
  • [3] S. B. Chae, Holomorphy and Calculus in Normed Spaces, Pure and Appl. Math, 92, Marcel Dekker, New York, 1985.
  • [4] J. Dai, Compact composition operators on the Bloch space of the unit ball, J. Math. Anal. Appl. 386 (2012), no. 1, 294–299.
  • [5] D. García, M. Maestre, and P. Sevilla-Peris, Composition operators between weighted spaces of holomorphic functions on Banach spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 81–98.
  • [6] K. Goebel and S. Reich, Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Pure Appl. Math. 83, Marcel Dekker, New York, 1984.
  • [7] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2679–2687.
  • [8] R. M. Timoney, Bloch functions in several complex variables, I, Bull. Lond. Math. Soc. 12 (1980), no. 4, 241–267.
  • [9] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Grad. Texts in Math. 226, Springer, New York, 2005.
  • [10] K. Zhu, Operator Theory in Function Spaces, 2nd ed, Math. Surveys Monogr. 138, American Mathematical Society, Providence, RI, 2007.