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Three Classification Results in the Theory of Weighted Hardy Spaces on the Ball

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Abstract

We present a natural family of Hilbert function spaces on the d-dimensional complex unit ball and classify which of them satisfy that subsets of the ball yield isometrically isomorphic subspaces if and only if there is an analytic automorphism of the ball taking one to the other. We also characterize pairs of weighted Hardy spaces on the unit disk which are isomorphic via a composition operator by a simple criterion on their respective sequences of weights.

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References

  1. Agler, J., McCarthy, J.E.: Pick Interpolation and Hilbert Function Spaces; Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Rhode Island (2000)

    Google Scholar 

  2. Alpay, D., Putinar, M., Vinnikov, V.: A Hilbert space approach to bounded analytic extension in the ball. Commun. Pure Appl. Anal. 2, 139–145 (2003)

    Article  MathSciNet  Google Scholar 

  3. Arcozzi, N., Rochberg, R., Sawyer, E.T.: Carleson measures for the Drury–Arveson Hardy space and other Besov–Sobolev spaces on complex balls. Adv. Math. 218, 1107–1180 (2008)

    Article  MathSciNet  Google Scholar 

  4. Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  5. Davidson, K.R., Hartz, M., Shalit, O.M.: Multipliers of embedded disks. Complex Anal. Oper. Theory 9, 287–321 (2015)

    Article  MathSciNet  Google Scholar 

  6. Davidson, K.R., Ramsey, C., Shalit, O.M.: The isomorphism problem for some universal operator algebras. Adv. Math. 228, 167–218 (2011)

    Article  MathSciNet  Google Scholar 

  7. Davidison, K.R., Ramsey, C., Shalit, O.M.: Operator algebras for analytic varieties. Trans. Am. Math. Soc. 367, 1121–1150 (2015)

    Article  MathSciNet  Google Scholar 

  8. Garnett, J.B.: Bounded Analytic Functions. Springer, San Diego (2007)

    MATH  Google Scholar 

  9. Hartz, M.: Topological isomorphisms for some universal operator algebras. J. Funct. Anal. 263, 3564–3587 (2012)

    Article  MathSciNet  Google Scholar 

  10. Hartz, M.: On the isomorphism problem for multiplier algebras of Nevanlinna–Pick spaces. Canad. J. Math. 69, 54–106 (2017)

    Article  MathSciNet  Google Scholar 

  11. Kerr, M., McCarthy, J.E., Shalit, O.M.: On the isomorphism question for complete Pick multiplier algebras. Integral Equ. Oper. Theory 76, 39–53 (2013)

    Article  MathSciNet  Google Scholar 

  12. Krantz, S.: Complex Analysis: The Geometric Viewpoint; Carus Mathematical Monographs, vol. 23. Mathematical Association of America, Washington (1990)

    Google Scholar 

  13. Krantz, S.: Geometric Function Theory. Cornerstones, Birkhauser, Boston (2006)

    MATH  Google Scholar 

  14. Paulsen, V.I.: An Introduction to the Theory of Reproducing Kernel Hilbert Spaces; Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016)

    Google Scholar 

  15. Rochberg, R.: Complex hyperbolic geometry and Hilbert spaces with complete Pick kernels. J. Funct. Anal. 276, 1622–1679 (2018)

    Article  MathSciNet  Google Scholar 

  16. Salomon, G., Shalit, O.M.: The Isomorphism Problem for Complete Pick Algebras: A Survey; Operator Theory, Function Spaces, and Applications. Operator Theory: Advances and Applications, vol. 255, pp. 167–198. Birkhäuser, Cham (2016)

    MATH  Google Scholar 

  17. Shalit, O.M.: A First Course in Functional Analysis. CRC Press, Boca Raton (2017)

    Book  Google Scholar 

  18. Shalit, O.M.: The Disc Trick (and Some Other Cute Moves); Posted on Shalit’s Blog: “Noncommutative Analysis” (2019). https://noncommutativeanalysis.com/2019/10/10/the-disc-trick-and-some-other-cute-moves/

  19. Walter, R.: Function Theory in the Unit Ball; Classics in Mathematics. Springer, Berlin (2008)

    Google Scholar 

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Acknowledgements

The results presented in this paper were obtained during the summer projects program at the Technion Institute of Technology under the guidance of Orr Moshe Shalit, Satish K. Pandey and Ran Kiri. We would like to thank them for giving us the opportunity to learn and for their attentive and careful instruction. We would like to thank the referee for numerous helpful suggestions to clarify our ideas and to make them more accurate.

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Authors and Affiliations

  1. Tel-Aviv University, Tel Aviv, Israel

    Danny Ofek

  2. Technion Institute of Technology, Haifa, Israel

    Gilad Sofer

Authors
  1. Danny Ofek
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  2. Gilad Sofer
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Correspondence to Gilad Sofer.

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Communicated by Mihai Putinar.

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This article is part of the topical collection “Reproducing kernel spaces and applications” edited by Daniel Alpay.

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Ofek, D., Sofer, G. Three Classification Results in the Theory of Weighted Hardy Spaces on the Ball. Complex Anal. Oper. Theory 15, 65 (2021). https://doi.org/10.1007/s11785-021-01114-6

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  • DOI: https://doi.org/10.1007/s11785-021-01114-6

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