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A commutator method for the diagonalization of Hankel operators

Abstract

A method for the explicit diagonalization of some Hankel operators is presented. This method makes it possible to give new proofs of classical results on the diagonalization of Hankel operators with absolutely continuous spectrum and obtain new results. The approach relies on the commutation of a Hankel operator with a certain second-order differential operator.

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References

  1. N. I. Akhieser and I. M. Glasman, Theory of Linear Operators in Hilbert Space, vols. I, II, Ungar, New York, 1961, 1963.

    Google Scholar 

  2. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vol. 1, 2, McGraw-Hill, New York-Toronto-London, 1953.

    Google Scholar 

  3. N. Ya. Vilenkin, Special Functions and the Theory of Group Representations [in Russian], Nauka, Moscow, 1965.

    Google Scholar 

  4. T. Carleman, Sur les équations intégrales singuliéres á noyau réel et symetrique, Almqvist and Wiksell, 1923.

  5. W. Magnus, “On the spectrum of Hilberts’ matrix, ” Amer. J. Math., 72 (1950), 699–704.

    MATH  Article  MathSciNet  Google Scholar 

  6. F. G. Mehler, “ Über eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsvertheilung, ” Math. Ann., 18:2 (1881), 161–194.

    Article  MathSciNet  Google Scholar 

  7. F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York-London, 1974.

    Google Scholar 

  8. V. V. Peller, Hankel Operators and Their Applications, Springer-Verlag, New York, 2003.

    MATH  Google Scholar 

  9. M. Rosenblum, “On the Hilbert matrix, I, II, ” Proc. Amer. Math. Soc., 9 (1958), 137–140, 581-585.

    MATH  MathSciNet  Google Scholar 

  10. E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Vol. 1, Clarendon Press, Oxford, 1946.

    Google Scholar 

  11. J. S. Howland, “Spectral theory of operators of Hankel type. I, II, ” Indiana Univ. Math. J., 41:2 (1992), 409–426, 427-434.

    MATH  Article  MathSciNet  Google Scholar 

  12. H. Shanker, “An integral equation for Whittakers’s confluent hypergeometric function,” Proc. Cambridge Philos. Soc., 45 (1949), 482–483.

    MATH  Article  MathSciNet  Google Scholar 

  13. D. R. Yafaev, Mathematical Scattering Theory. Analytic Theory, Amer. Math. Soc., Providence, RI, 2010.

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Translated from Funktsionals’nyi Analiz i Ego Prilozheniya, Vol. 44, No. 4, pp. 65-79, 2010

Original Russian Text Copyright © by D. R. Yafaev

A precise definition of the operator U can be given in terms of the corresponding sesquilinear form.

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Yafaev, D.R. A commutator method for the diagonalization of Hankel operators. Funct Anal Its Appl 44, 295–306 (2010). https://doi.org/10.1007/s10688-010-0040-z

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  • DOI: https://doi.org/10.1007/s10688-010-0040-z

Keywords

  • Hankel operators
  • spectrum
  • eigenfunctions
  • explicit solutions
  • commutators