Toeplitz operators on the Dirichlet spaces of planar domains

Lee, Young; Nguyen, Quang

doi:10.1090/S0002-9939-2010-10672-X

Toeplitz operators on the Dirichlet spaces of planar domains
HTML articles powered by AMS MathViewer

by Young Joo Lee and Quang Dieu Nguyen PDF

Proc. Amer. Math. Soc. 139 (2011), 547-558 Request permission

Abstract:

We study some algebraic properties of Toeplitz operators on the Dirichlet spaces of planar domains. On domains with real analytic boundary, we show that Toeplitz operators with symbol vanishing near the boundary have rank at most 1. Moreover, we construct explicit examples of Toeplitz operators having exactly rank 1. This is a sharp contrast to a known result on the unit disk. Also, on simply connected domains we characterize compact Toeplitz operators in terms of the boundary vanishing property of the Berezin transform of the symbol.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Guang Fu Cao and Lu Tao Zhu, The compactness of Toeplitz operators on Dirichlet spaces, Acta Math. Sinica (Chinese Ser.) 44 (2001), no. 2, 241–248 (Chinese, with English and Chinese summaries). MR 1831525
Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762, DOI 10.1090/surv/100
Yong Chen, Commuting Toeplitz operators on the Dirichlet space, J. Math. Anal. Appl. 357 (2009), no. 1, 214–224. MR 2526821, DOI 10.1016/j.jmaa.2009.03.068
Miroslav Engliš, Compact Toeplitz operators via the Berezin transform on bounded symmetric domains, Integral Equations Operator Theory 33 (1999), no. 4, 426–455. MR 1682815, DOI 10.1007/BF01291836
Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
Young Joo Lee, Algebraic properties of Toeplitz operators on the Dirichlet space, J. Math. Anal. Appl. 329 (2007), no. 2, 1316–1329. MR 2296927, DOI 10.1016/j.jmaa.2006.07.041
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
Kehe Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, Providence, RI, 2007. MR 2311536, DOI 10.1090/surv/138