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COMPUTATION OF HANKEL MATRICES IN TERMS OF CLASSICAL KERNEL FUNCTIONS IN POTENTIAL THEORY

  • Received : 2019.07.11
  • Accepted : 2019.12.12
  • Published : 2020.07.01

Abstract

In this paper, we compute the Hankel matrix representation of the Hankel operator on the Hardy space of a general bounded domain with respect to special orthonormal bases for the Hardy space and its orthogonal complement. Moreover we obtain the compact form of the Hankel matrix for the unit disc case with respect to these bases. One can see that the Hankel matrix generated by this computation turns out to be a generalization of the case of the unit disc from the single simply connected domain to multiply connected domains with much diversities of bases.

Keywords

References

  1. S. Bell, Solving the Dirichlet problem in the plane by means of the Cauchy integral, Indiana Univ. Math. J. 39 (1990), no. 4, 1355-1371. https://doi.org/10.1512/iumj.1990.39.39060
  2. S. Bell, The Szego projection and the classical objects of potential theory in the plane, Duke Math. J. 64 (1991), no. 1, 1-26. https://doi.org/10.1215/S0012-7094-91-06401-X
  3. S. Bell, The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
  4. A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89-102. https://doi.org/10.1007/978-1-4613-8208-9_19
  5. Y.-B. Chung, Classification of Toeplitz operators on Hardy spaces of bounded domains in the plane, Math. Notes 101 (2017), no. 3-4, 529-541. https://doi.org/10.1134/S0001434617030142
  6. P. R. Garabedian, Schwarz's lemma and the Szego kernel function, Trans. Amer. Math. Soc. 67 (1949), 1-35. https://doi.org/10.2307/1990414
  7. K. Zhu, Operator theory in function spaces, second edition, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/138