Integral Equations and Operator Theory

, Volume 1, Issue 3, pp 285–309

Products of Toeplitz operators

  • Sheldon Axler
  • Sun-Yung A. Chang
  • Donald Sarason
Article

Abstract

A sufficient condition is found for the product of two Toeplitz operators to be a compact perturbation of a Toeplitz operator. The condition, which comprehends all previously known sufficient conditions, is shown to be necessary under additional hypotheses. The question whether the condition is necessary in general is left open.

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References

  1. [1]
    S. Axler, Doctoral Dissertation, University of California, Berkeley, 1975.Google Scholar
  2. [2]
    S. Axler, I.D. Berg, N.P. Jewell and A.L. Shields, Approximation by compact operators and the space H + C, to appear.Google Scholar
  3. [3]
    C.A. Berger. L.A. Coburn and A. Lebow, Representation and index theory for C*-algebras generated by commuting isometries, J. Functional Anal. 27 (1978), 51–99.Google Scholar
  4. [4]
    A. Brown and P.R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963), 89–102.Google Scholar
  5. [5]
    S-Y. A. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976), 81–89.Google Scholar
  6. [6]
    --, A generalized area integral estimate and applications, to appear.Google Scholar
  7. [7]
    D.N. Clark, On interpolating sequences and the theory of Hankel and Toeplitz matrices, J. Functional Anal. 5 (1970), 247–258.Google Scholar
  8. [8]
    L.A. Coburn, The C*-algebra generated by an isometry, Bull. Amer. Math. Soc. 73 (1967), 722–726.Google Scholar
  9. [9]
    R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York and London, 1972.Google Scholar
  10. [10]
    --, Banach algebra techniques in the theory of Toeplitz operators, Regional Conference Series in Mathematics, no. 15, Amer. Math. Soc., 1973.Google Scholar
  11. [11]
    ——, Local Toeplitz operators, Proc. London Math. Soc. (3) 36 (1978), 243–272.Google Scholar
  12. [12]
    P.L. Duren, Theory of HP Spaces, Academic Press, New York and London, 1970.Google Scholar
  13. [13]
    C. Fefferman and E.M. Stein, HP spaces of several variables, Acta Math. 129 (1972), 137–193.Google Scholar
  14. [14]
    F.R. Gantmacher, The Theory of Matrices, vol. 2, Chelsea, New York, 1964.Google Scholar
  15. [15]
    I.C. Gohberg and N. Ya. Krupnik, On an algebra generated by Toeplitz matrices, Funkcional. Anal. i Priložen. 3 (1969), 46–56.Google Scholar
  16. [16]
    P. Hartman, On completely continuous Hankel matrices, Proc. Amer. Math. Soc. 9 (1958), 862–866.Google Scholar
  17. [17]
    K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.Google Scholar
  18. [18]
    L. Kronecker, Zur Theorie der Elimination einer Variablen aus zwei algebraischen Gleichungen, Monatsber. Königl. Preussischen Akad. Wiss. (Berlin), 1881, 535–600.Google Scholar
  19. [19]
    J. Marcinkiewicz and A. Zygmund, A theorem of Lusin, Duke Math. J. 4 (1938), 473–485.Google Scholar
  20. [20]
    D. Marshall, Subalgebras of L containing H, Acta Math. 137 (1976), 91–98.Google Scholar
  21. [21]
    D. Sarason, Generalized interpolation in H, Trans. Amer. Math. Soc. 127 (1967), 179–203.Google Scholar
  22. [22]
    ——, Algebras of functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973), 286–299.Google Scholar
  23. [23]
    ——, On products of Toeplitz operators, Acta Sci. Math. (Szeged) 35 (1973), 7–12.Google Scholar
  24. [24]
    ——, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405.Google Scholar

Copyright information

© Birkhäuser-Verlag 1978

Authors and Affiliations

  • Sheldon Axler
    • 1
  • Sun-Yung A. Chang
    • 2
  • Donald Sarason
    • 3
  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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