Research Article
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A new characterization of the Hardy space and of other Hilbert spaces of analytic functions

Year 2023, Volume: 1 Issue: 1, 1 - 11, 21.06.2023
https://doi.org/10.26650/ijmath.2023.00005

Abstract

The Fock space can be characterized (up to a positive multiplicative factor) as the only Hilbert space of entire functions in which the adjoint of derivation is multiplication by the complex variable. Similarly (and still up to a positive multiplicative factor) the Hardy space is the only space of functions analytic in the open unit disk for which the adjoint of the backward shift operator is the multiplication operator. In the present paper we characterize the Hardy space and some related reproducing kernel Hilbert spaces in terms of the adjoint of the differentiation operator. We use reproducing kernel methods, which seem to also give a new characterization of the Fock space.

References

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Year 2023, Volume: 1 Issue: 1, 1 - 11, 21.06.2023
https://doi.org/10.26650/ijmath.2023.00005

Abstract

References

  • Aronszajn, N., 1950, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68, 337-404. google scholar
  • Ash, R.B., 1990, Information Ttheory, Dover Publications Inc., New York, Corrected reprint of the 1965 original. google scholar
  • Bargmann, V., 1961, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math., 14, 187-214. google scholar
  • Bargmann, V., 1962, Remarks on a Hilbert space of analytic functions, Proceedings of the National Academy of Arts, 48, 199-204. google scholar
  • Chabat,B., 1990, Introduction a l’analyse complexe. Tome 2, Traduit du Russe: Mathematiques. [Translations of Russian Works: Mathematics]. “Mir”, Moscow, Fonctions de plusieurs variables. [Functions of several variables], Translated from the Russian by Djilali Embarek. google scholar
  • Donoghue, W.F., 1974, Monotone Matrix Functions and Analytic Continuation, volume 207 of Die Grundlehren der mathematischen Wis-senschaften, Springer-Verlag. google scholar
  • Fayngold, M., Fayngold, V., 2013, Quantum mechanics and quantum information: a guide through the quantum world, John Wiley & Sons. google scholar
  • Petz, D., 2008, Quantum Information Theory and Quantum Statistics, Theoretical and Mathematical Physics. Springer-Verlag, Berlin. google scholar
  • Reed, M., Simon, B., 1980, Methods of modern mathematical physics. I, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, second edition, Functional analysis. google scholar
  • Saitoh, S., 1988, Theory of Reproducing Kernels and Its Applications, volume 189, Longman scientific and technical. google scholar
  • Saitoh, S., 1997, Integral Transforms, Reproducing Kernels and Their Applications, volume 369 of Pitman Research Notes in Mathematics Series, Longman, Harlow. google scholar
  • Tung, Y.C.J., 1976, Fock Spaces, A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan, 2005. google scholar
There are 12 citations in total.

Details

Primary Language English
Subjects Pure Mathematics (Other)
Journal Section Research Articles
Authors

Natanael Alpay This is me 0000-0002-1505-7531

Publication Date June 21, 2023
Published in Issue Year 2023 Volume: 1 Issue: 1

Cite

APA Alpay, N. (2023). A new characterization of the Hardy space and of other Hilbert spaces of analytic functions. Istanbul Journal of Mathematics, 1(1), 1-11. https://doi.org/10.26650/ijmath.2023.00005
AMA Alpay N. A new characterization of the Hardy space and of other Hilbert spaces of analytic functions. Istanbul Journal of Mathematics. June 2023;1(1):1-11. doi:10.26650/ijmath.2023.00005
Chicago Alpay, Natanael. “A New Characterization of the Hardy Space and of Other Hilbert Spaces of Analytic Functions”. Istanbul Journal of Mathematics 1, no. 1 (June 2023): 1-11. https://doi.org/10.26650/ijmath.2023.00005.
EndNote Alpay N (June 1, 2023) A new characterization of the Hardy space and of other Hilbert spaces of analytic functions. Istanbul Journal of Mathematics 1 1 1–11.
IEEE N. Alpay, “A new characterization of the Hardy space and of other Hilbert spaces of analytic functions”, Istanbul Journal of Mathematics, vol. 1, no. 1, pp. 1–11, 2023, doi: 10.26650/ijmath.2023.00005.
ISNAD Alpay, Natanael. “A New Characterization of the Hardy Space and of Other Hilbert Spaces of Analytic Functions”. Istanbul Journal of Mathematics 1/1 (June 2023), 1-11. https://doi.org/10.26650/ijmath.2023.00005.
JAMA Alpay N. A new characterization of the Hardy space and of other Hilbert spaces of analytic functions. Istanbul Journal of Mathematics. 2023;1:1–11.
MLA Alpay, Natanael. “A New Characterization of the Hardy Space and of Other Hilbert Spaces of Analytic Functions”. Istanbul Journal of Mathematics, vol. 1, no. 1, 2023, pp. 1-11, doi:10.26650/ijmath.2023.00005.
Vancouver Alpay N. A new characterization of the Hardy space and of other Hilbert spaces of analytic functions. Istanbul Journal of Mathematics. 2023;1(1):1-11.