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ArticleOriginal scientific text

Title

Orthogonal polynomials and middle Hankel operators on Bergman spaces

Authors 1, 2, 3

Affiliations

  1. Department of Mathematics, Peking University, Beijing 100871, P.R. China.
  2. Department of Mathematics, Washington University, St. Louis, Missouri 63130, U.S.A.
  3. Department of Mathematics, University of Alabama Tuscaloosa, Alabama 35487, U.S.A.

Abstract

We introduce a sequence of Hankel style operators Hk, k = 1,2,3,..., which act on the Bergman space of the unit disk. These operators are intermediate between the classical big and small Hankel operators. We study the boundedness and Schatten-von Neumann properties of the Hk and show, among other things, that Hk are cut-off at 1/k. Recall that the big Hankel operator is cut-off at 1 and the small Hankel operator at 0.

Bibliography

  1. [AFP] J. Arazy, S. Fisher and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), 989-1054.
  2. [A] S. Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), 315-332.
  3. [J1] S. Janson, Hankel operators between weighted Bergman spaces, Ark. Mat. 26 (1988), 205-219.
  4. [J2] S. Janson, Hankel operators on Bergman spaces with change of weight, Mittag-Leffler report, 1991.
  5. [JR] S. Janson and R. Rochberg, Intermediate Hankel operators on the Bergman space, J. Operator Theory, to appear.
  6. [JP] Q. Jiang and L. Peng, Wavelet transform and Ha-Plitz operators, preprint, 1991.
  7. [N] K. Nowak, Estimate for singular values of commutators on weighted Bergman spaces, Indiana Univ. Math. J., to appear.
  8. [M] M. M. Peloso, Besov spaces, mean oscillation, and generalized Hankel operators, preprint, 1991.
  9. [P] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser. 1, Durham 1976.
  10. [Pel1] V. V. Peller, Vectorial Hankel operators, commutators and related operators of the Schatten-von Neumann class Sp, Integral Equations Operator Theory 5 (1982), 244-272.
  11. [Pel2] V. V. Peller, A description of Hankel operators of class Sp for p > 0, investigation of the rate of rational approximation, and other applications, Math. USSR-Sb. 50 (1985), 465-494.
  12. [PX] L. Peng and C. Xu, Jacobi polynomials and Toeplitz-Hankel type operators on weighted Bergman spaces, preprint, 1991.
  13. [PZ] L. Peng and G. Zhang, Middle Hankel operators on Bergman space, preprint, 1990.
  14. [R1] R. Rochberg, Trace ideal criteria for Hankel operators and commutators, Indiana Univ. Math. J. 31 (1982), 913-925.
  15. [R2] R. Rochberg, Decomposition theorems for Bergman spaces and their applications, in: Operators and Function Theory, Reidel, Dordrecht 1985, 225-277.
  16. [S] S. Semmes, Trace ideal criteria for Hankel operators, and applications to Besov spaces, Integral Equations Operator Theory 7 (1984), 241-281.
  17. [Sz] G. Szegö, Orthogonal Polynomials, Colloq. Publ. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
  18. [Z] G. Zhang, Hankel operators and Plancherel formula, Ph.D thesis, Stockholm University, Stockholm 1991.
Studia Mathematica
1992/102/1
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Pages:
57-75
Main language of publication
English
Received
1991-09-10
Published
1992
Exact and natural sciences