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Toeplitz operators with BMO symbols on the Segal-Bargmann space


Authors: L. A. Coburn, J. Isralowitz and Bo Li
Journal: Trans. Amer. Math. Soc. 363 (2011), 3015-3030
MSC (2010): Primary 47B32; Secondary 32A36
DOI: https://doi.org/10.1090/S0002-9947-2011-05278-5
Published electronically: January 20, 2011
MathSciNet review: 2775796
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Abstract: We show that Zorboska's criterion for compactness of Toeplitz operators with BMO$ ^1$ symbols on the Bergman space of the unit disc holds, by a different proof, for the Segal-Bargmann space of Gaussian square-integrable entire functions on $ \mathbb{C}^n$. We establish some basic properties of BMO$ ^p$ for $ p \geq 1$ and complete the characterization of bounded and compact Toeplitz operators with BMO$ ^1$ symbols. Via the Bargmann isometry and results of Lo and Englis, we also give a compactness criterion for the Gabor-Daubechies ``windowed Fourier localization operators'' on $ L^2(\mathbb{R}^n, dv)$ when the symbol is in a BMO$ ^1$ Sobolev-type space. Finally, we discuss examples of the compactness criterion and counterexamples to the unrestricted application of this criterion for the compactness of Toeplitz operators.


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Additional Information

L. A. Coburn
Affiliation: Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14260
Email: lcoburn@buffalo.edu

J. Isralowitz
Affiliation: Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14260
Address at time of publication: Institute of Mathematics, University of Göttingen, Bunsenstrasse 3-5, D-37073 Göttingen, Germany
Email: jbi2@buffalo.edu

Bo Li
Affiliation: Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14260
Address at time of publication: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
Email: boli@buffalo.edu, boli@bgsu.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05278-5
Received by editor(s): September 24, 2008
Received by editor(s) in revised form: March 2, 2009
Published electronically: January 20, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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