• Received : 2014.09.12
  • Accepted : 2014.11.24
  • Published : 2014.12.25


We construct an orthonormal basis for the Bergman space associated to a simply connected domain. We use the or-thonormal basis for the Hardy space consisting of the Szegő kernel and the Riemann mapping function and rewrite their area integrals in terms of arc length integrals using the complex Green's identity. And we make a note about the matrix of a Toeplitz operator with respect to the orthonormal basis constructed in the paper.


  1. S. Bell, Mapping problems in complex analysis and the $\bar{\partial}$-problem, Bull. Amer. Math. Soc. 22 (1990), 233-259.
  2. S. Bell, Solving the Dirichlet problem in the plane by means of the Cauchy integral, Indiana Univ. Math. J. 39(4) (1990), 1355-1371.
  3. Steve Bell, The Szego projection and the classical objects of potential theory in the plane, Duke Math. J. 64(1) (1991), 1-26.
  4. Steven R. Bell, Complexity of the classical kernel functions of potential theory, Indiana Univ. Math. J. 44(4) (1995), 1337-1369. MR1386771 (97g:30009)
  5. Steven R. Bell, Ahlfors maps, the double of a domain, and complexity in potential theory and conformal mapping, J. Anal. Math. 78 (1999), 329-344. MR1714417 (2000m:30012)
  6. Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89-102. MR0160136 (28 #3350)
  7. Y.-B. Chung, An expression of the Bergman kernel function in terms of the Szego kernel, J. Math. Pures Appl. 75 (1996), 1-7.
  8. Y.-B. Chung, Classi cation of toeplitz operators on hardy spaces of bounded do-mains in the plane, in submission (2014).
  9. P. R. Garabedian, Schwarz's lemma and the Szeg}o kernel function, Trans. Amer. Math. Soc. 67 (1949), 1-35.