# UNIQUENESS OF TOEPLITZ OPERATOR IN THE COMPLEX PLANE

Chung, Young-Bok

• Accepted : 2009.12.14
• Published : 2009.12.25
• 29 2

#### Abstract

We prove using the Szeg$\H{o}$ kernel and the Garabedian kernel that a Toeplitz operator on the boundary of $C^{\infty}$ smoothly bounded domain associated to a smooth symbol vanishes only when the symbol vanishes identically. This gives a generalization of previous results on the unit disk to more general domains in the plane.

#### Keywords

Szeg$\H{o}$ kernel;Toeplitz operator;Garabedian kernel

#### 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. Steven R. Bell, The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR MR1228442 (94k:30013)
3. Steve Bell, The Szego projection and the classical objects of potential theory in the plane, Duke Math. J. 64 (1991), no. 1, 1-26. MR MR1131391 (93e:30018) https://doi.org/10.1215/S0012-7094-91-06401-X
4. P. R. Garabedian, Schwarz's lemma and the Szego kernel function, Trans. Amer. Math. Soc. 67 (1949), 1-35.
5. Dennis A. Hejhal, Theta functions, kernel functions, and Abelian integrals, American Mathematical Society, Providence, R.I., 1972, Memoirs of the American Mathematical Society, No. 129.
6. N. Kerzman and E. M. Stein, The Cauchy kernel. the Szego kernel, and the Riemann mapping function, Math. Ann. 236 (1978), 85-93. https://doi.org/10.1007/BF01420257
7. N. Keramen and M. Trummer, Numerical conformal mapping via the Szego kernel, (1986), Numerical conformal mapping, 111-123, Trefethen, ed., North Holland, Amsterdam.
8. Menahem Schiffer, Various types of orthogonalization, Duke Math. J. 17 (1950), 329-366. MR MR0039071 (12,491g) https://doi.org/10.1215/S0012-7094-50-01731-5
9. Boo Rim Chce, Hyungwoon Koo, and Young Joo Lee, Zero products of Toeplitz operators with n-harmonic symbols, Integral Equations Operator Theory 57(2007), no. 1, 43-66. MR MR2294274 (2008c:47050) https://doi.org/10.1007/s00020-006-1444-2
10. Young Joo Lee. Commuting Toeplitz operators on the Hardy space of the bidisk, J. Math. Anal. Appl. 341 (2008), no. 1, 738-749. MR MR2394121 (2009c:47037) https://doi.org/10.1016/j.jmaa.2007.11.002
11. Stefan Bergman, The kernel function and conformal mapping. revised ed., American Mathematical Society, Providence, R.I., 1970, Mathematical Surveys, No. V. MR MR0507701 (58 #22502)