Toeplitz operators with BMO symbols on the Segal-Bargmann space
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- by L. A. Coburn, J. Isralowitz and Bo Li PDF
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Abstract:
We show that Zorboska’s criterion for compactness of Toeplitz operators with $\text {BMO}^1$ 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 $\text {BMO}^p$ for $p \geq 1$ and complete the characterization of bounded and compact Toeplitz operators with $\text {BMO}^1$ symbols. Via the Bargmann isometry and results of Lo and Englis̆, 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 $\text {BMO}^1$ 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.References
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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
- Received by editor(s): September 24, 2008
- Received by editor(s) in revised form: March 2, 2009
- Published electronically: January 20, 2011
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2775796