Journal d'Analyse Mathématique

, Volume 133, Issue 1, pp 133–182 | Cite as

Quasi-diagonalization of Hankel operators

Article

Abstract

We show that all Hankel operators H realized as integral operators with kernels h(t + s) in L 2(R+) can be quasi-diagonalized as H = L*ΣL. Here L is the Laplace transform, Σ is the operator of multiplication by a function (distribution) σ(λ), λ ∈ R. We find a scale of spaces of test functions on which L acts as an isomorphism. Then L* is an isomorphism of the corresponding spaces of distributions. We show that h = L*σ, which yields a one-to-one correspondence between kernels h(t) and sigma-functions σ(λ) of Hankel operators. The sigma-function of a self-adjoint Hankel operator H contains substantial information about its spectral properties. Thus we show that the operators H and Σ have the same number of positive and negative eigenvalues. In particular, we find necessary and sufficient conditions for sign-definiteness of Hankel operators. These results are illustrated with examples of quasi-Carleman operators generalizing the classical Carleman operator with kernel h(t) = t −1 in various directions. The concept of the sigmafunction leads directly to a criterion (equivalent, of course, to the classical Nehari theorem) for boundedness of Hankel operators. Our construction also shows that every Hankel operator is unitarily equivalent by the Mellin transform to a pseudodifferential operator with amplitude which is a product of functions of one variable (x ∈ R and its dual variable).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Akhiezer, The Classical Moment Problem and some Related Questions in Analysis, Oliver and Boyd, Edinburgh and London, 1965.MATHGoogle Scholar
  2. [2]
    S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math., 52 (1929), 1–66.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    T. Carleman, Sur les équations intégrales singulières à noyau réel et symétrique, Almqvist and Wiksell, 1923.MATHGoogle Scholar
  4. [4]
    A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vols. 1, 2, McGraw-Hill, New York-Toronto-London, 1953.MATHGoogle Scholar
  5. [5]
    C. Fefferman, Characterizations of bounded mean oscillation, Bull. Amer.Math. Soc., 77 (1971), 587–588.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    J. B. Garnett, Bounded Analytic Functions Academic Press, New York and London, 1981.MATHGoogle Scholar
  7. [7]
    I. M. Gel’fand and G. E. Shilov, Generalized Functions Vol. 1, Academic Press, New York and London, 1964.MATHGoogle Scholar
  8. [8]
    I. M. Gel’fand and N. Ya. Vilenkin, Generalized Functions Vol. 4, Academic Press, New York and London, 1964.MATHGoogle Scholar
  9. [9]
    H. Hamburger, Über eine Erweiterung des Stieltjesschen Momentensproblems, Math. Ann. 81 (1920), 235–319;, 82 (1921), 120–164, 168–187.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    P. Koosis, Introduction to Hp spaces, Cambridge University Press, Cambridge, 1980.MATHGoogle Scholar
  11. [11]
    W. Magnus, On the spectrum of Hilbert’s matrix, Amer. J. Math., 72 (1950), 699–704.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    F. G. Mehler, Über eine mit den Kugel-und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsvertheilung, Math. Ann. 18 (1881), 161–194.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Z. Nehari, On bounded bilinear forms, Ann. of Math. (2), 65 (1957), 153–162.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    N. K. Nikolski, Operators, Functions, and Systems: an Easy Reading, Vol. 1: Hardy, Hankel, and Toeplitz, Amer. Math. Soc., Providence, RI, 2002.Google Scholar
  15. [15]
    V. V. Peller, Hankel Operators and their Applications, Springer Verlag, New York, 2003.CrossRefMATHGoogle Scholar
  16. [16]
    S. R. Power, Hankel Operators on Hilbert space, Pitman, Boston, 1982.MATHGoogle Scholar
  17. [17]
    M. Rosenblum, On the Hilbert matrix, I, II, Proc. Amer.Math. Soc., 9 (1958), 137–140, 581–585.MathSciNetMATHGoogle Scholar
  18. [18]
    W. Sierpiński, Sur les fonctions convexes mesurables, Fund. Math., 1 (1920), 125–129.CrossRefMATHGoogle Scholar
  19. [19]
    H. Widom, Hankel matrices, Trans. Amer.Math. Soc., 121 (1966), 1–35.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    D. R. Yafaev, The discrete spectrum in the singular Friedrichs model, in Differential Operators and Spectral Theory, American Mathematical Society, Providence, RI, 1999, pp. 255–274.Google Scholar
  21. [21]
    D. R. Yafaev, A commutator method for the diagonalization of Hankel operators, Funct. Anal. Appl. 44 (2010), 295–306.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    D. R. Yafaev, Diagonalizations of two classes of unbounded Hankel operators, Bull. Math. Sci. 4, (2014), 175–198.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    D. R. Yafaev, Criteria for Hankel operators to be sign-definite, Anal. PDE, 8 (2015), 183–221.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    D. R. Yafaev, On finite rank Hankel operators, J. Funct. Anal., 268 (2015), 1808–1839.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    D. R. Yafaev, Quasi-Carleman operators and their spectral properties, Integral Equations Operator Theory, 81 (2015), 499–534.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    D. R. Yafaev, Unbounded Hankel operators and moment problems, Integral Equations Operator Theory, 85 (2016), 289–300.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    D. R. Yafaev, Hankel and Toeplitz operators: continuous and discrete representations, Opuscula Math., 37 (2017), 189–218.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2017

Authors and Affiliations

  1. 1.Irmar Université de Rennes ICampus De BeaulieuRennes CedexFrance

Personalised recommendations