Orthogonal polynomials and middle Hankel operators on Bergman spaces

Lizhong Peng; Richard Rochberg; Zhijian Wu

Studia Mathematica (1992)

  • Volume: 102, Issue: 1, page 57-75
  • ISSN: 0039-3223

Abstract

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We introduce a sequence of Hankel style operators H k , 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 H k and show, among other things, that H k are cut-off at 1/k. Recall that the big Hankel operator is cut-off at 1 and the small Hankel operator at 0.

How to cite

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Peng, Lizhong, Rochberg, Richard, and Wu, Zhijian. "Orthogonal polynomials and middle Hankel operators on Bergman spaces." Studia Mathematica 102.1 (1992): 57-75. <http://eudml.org/doc/215914>.

@article{Peng1992,
abstract = {We introduce a sequence of Hankel style operators $H^k$, 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 $H^k$ and show, among other things, that $H^k$ are cut-off at 1/k. Recall that the big Hankel operator is cut-off at 1 and the small Hankel operator at 0.},
author = {Peng, Lizhong, Rochberg, Richard, Wu, Zhijian},
journal = {Studia Mathematica},
language = {eng},
number = {1},
pages = {57-75},
title = {Orthogonal polynomials and middle Hankel operators on Bergman spaces},
url = {http://eudml.org/doc/215914},
volume = {102},
year = {1992},
}

TY - JOUR
AU - Peng, Lizhong
AU - Rochberg, Richard
AU - Wu, Zhijian
TI - Orthogonal polynomials and middle Hankel operators on Bergman spaces
JO - Studia Mathematica
PY - 1992
VL - 102
IS - 1
SP - 57
EP - 75
AB - We introduce a sequence of Hankel style operators $H^k$, 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 $H^k$ and show, among other things, that $H^k$ are cut-off at 1/k. Recall that the big Hankel operator is cut-off at 1 and the small Hankel operator at 0.
LA - eng
UR - http://eudml.org/doc/215914
ER -

References

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  1. [AFP] J. Arazy, S. Fisher and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), 989-1054. Zbl0669.47017
  2. [A] S. Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), 315-332. Zbl0633.47014
  3. [J1] S. Janson, Hankel operators between weighted Bergman spaces, Ark. Mat. 26 (1988), 205-219. Zbl0676.47013
  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. Zbl0898.47015
  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. Zbl0777.47023
  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. Zbl0356.46038
  10. [Pel1] V. V. Peller, Vectorial Hankel operators, commutators and related operators of the Schatten-von Neumann class S p , Integral Equations Operator Theory 5 (1982), 244-272. Zbl0478.47014
  11. [Pel2] V. V. Peller, A description of Hankel operators of class S p for p > 0, investigation of the rate of rational approximation, and other applications, Math. USSR-Sb. 50 (1985), 465-494. Zbl0561.47022
  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. Zbl0514.47020
  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. Zbl0541.47023
  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. 

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