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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Toeplitz operators on the Dirichlet spaces of planar domains


Authors: Young Joo Lee and Quang Dieu Nguyen
Journal: Proc. Amer. Math. Soc. 139 (2011), 547-558
MSC (2010): Primary 47B35; Secondary 32A36
Published electronically: September 17, 2010
MathSciNet review: 2736337
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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.


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Additional Information

Young Joo Lee
Affiliation: Department of Mathematics, Chonnam National University, Gwangju 500-757, Korea
Email: leeyj@chonnam.ac.kr

Quang Dieu Nguyen
Affiliation: Department of Mathematics, Ha Noi National University of Education, 136 Xuan Thuy, Ha Noi, Vietnam
Email: dieu_vn@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10672-X
PII: S 0002-9939(2010)10672-X
Keywords: Dirichlet spaces, Toeplitz operators, rank, compact operators
Received by editor(s): August 11, 2009
Received by editor(s) in revised form: March 10, 2010
Published electronically: September 17, 2010
Communicated by: Marius Junge
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.