FREDHOLM TOEPLITZ OPERATORS ON THE DIRICHLET SPACES OF THE POLYDISK

Abstract

We study the Toeplitz operators on the holomorphic and pluriharmonic Dirichlet spaces of the polydisk in terms of when Toeplitz operator is Fredholm operator there. Consequently, we describe the essential spectrum of Toeplitz operators.

Keywords

• Toeplitz operator;
• Dirichlet space;
• pluriharmonic Dirichlet space;
• Fredholm operator

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Acknowledgement

Supported by : Hanshin University

References

1. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
2. G. Cao, Fredholm properties of Toeplitz operators on Dirichlet spaces, Pacific J. Math. 188 (1999), no. 2, 209-223. https://doi.org/10.2140/pjm.1999.188.209
3. B. R. Choe, H. Koo, and Y. J. Lee, Commuting Toeplitz operators on the polydisk, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1727-1749. https://doi.org/10.1090/S0002-9947-03-03430-5
4. J. B. Conway, A Course in Operator Theory, Graduate Studies in Mathematics, 21, American Mathematical Society, Providence, RI, 2000.
5. Y. J. Lee, Compact sums of Toeplitz products and Toeplitz algebra on the Dirichlet space, Tohoku Math. J. (2) 68 (2016), no. 2, 253-271. http://projecteuclid.org/euclid.tmj/1466172772
6. Y. J. Lee, Fredholm Toeplitz operators on the pluriharmonic Dirichlet space, Honam Math. J. 39 (2017), no. 2, 175-185.
7. Y. J. Lee and K. Na, The essential norm of a sum of Toeplitz products on the Dirichlet space, J. Math. Anal. Appl. 431 (2015), no. 2, 1022-1034. https://doi.org/10.1016/j.jmaa.2015.06.028
8. G. McDonald, Fredholm properties of a class of Toeplitz operators on the ball, Indiana Univ. Math. J. 26 (1977), no. 3, 567-576. https://doi.org/10.1512/iumj.1977.26.26044
9. W. Rudin, Function Theory in the Unit Ball of $C^n$, Springer-Verlag, New York, 1980.
10. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, 226, Springer-Verlag, New York, 2005.