Toeplitz operators with BMO symbols on the Segal-Bargmann space

Coburn, L.; Isralowitz, J.; Li, Bo

doi:10.1090/S0002-9947-2011-05278-5

Toeplitz operators with BMO symbols on the Segal-Bargmann space
HTML articles powered by AMS MathViewer

by L. A. Coburn, J. Isralowitz and Bo Li PDF

Trans. Amer. Math. Soc. 363 (2011), 3015-3030 Request permission

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

Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
Sheldon Axler and Dechao Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J. 47 (1998), no. 2, 387–400. MR 1647896, DOI 10.1512/iumj.1998.47.1407
V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214. MR 157250, DOI 10.1002/cpa.3160140303
Wolfram Bauer, Mean oscillation and Hankel operators on the Segal-Bargmann space, Integral Equations Operator Theory 52 (2005), no. 1, 1–15. MR 2138695, DOI 10.1007/s00020-003-1272-6
D. Békollé, C. A. Berger, L. A. Coburn, and K. H. Zhu, BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal. 93 (1990), no. 2, 310–350. MR 1073289, DOI 10.1016/0022-1236(90)90131-4
C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), no. 2, 813–829. MR 882716, DOI 10.1090/S0002-9947-1987-0882716-4
C. A. Berger and L. A. Coburn, Heat flow and Berezin-Toeplitz estimates, Amer. J. Math. 116 (1994), no. 3, 563–590. MR 1277446, DOI 10.2307/2374991
F. A. Berezin, Wick and anti-Wick symbols of operators, Mat. Sb. (N.S.) 86(128) (1971), 578–610 (Russian). MR 0291839
F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1134–1167 (Russian). MR 0350504
Wolfram Bauer and Kenro Furutani, Compact operators and the pluriharmonic Berezin transform, Internat. J. Math. 19 (2008), no. 6, 645–669. MR 2431632, DOI 10.1142/S0129167X08004832
L. A. Coburn, A Lipschitz estimate for Berezin’s operator calculus, Proc. Amer. Math. Soc. 133 (2005), no. 1, 127–131. MR 2085161, DOI 10.1090/S0002-9939-04-07476-3
Ingrid Daubechies, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory 34 (1988), no. 4, 605–612. MR 966733, DOI 10.1109/18.9761
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
Miroslav Engliš, Toeplitz operators and localization operators, Trans. Amer. Math. Soc. 361 (2009), no. 2, 1039–1052. MR 2452833, DOI 10.1090/S0002-9947-08-04547-9
Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
Paul Richard Halmos and Viakalathur Shankar Sunder, Bounded integral operators on $L^{2}$ spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 96, Springer-Verlag, Berlin-New York, 1978. MR 517709, DOI 10.1007/978-3-642-67016-9
Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
Min-Lin Lo, The Bargmann transform and windowed Fourier localization, Integral Equations Operator Theory 57 (2007), no. 3, 397–412. MR 2307818, DOI 10.1007/s00020-006-1462-0
Jie Miao and Dechao Zheng, Compact operators on Bergman spaces, Integral Equations Operator Theory 48 (2004), no. 1, 61–79. MR 2029944, DOI 10.1007/s00020-002-1176-x
Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1990. Translated from the second French edition by Leo F. Boron; Reprint of the 1955 original. MR 1068530
Nina Zorboska, Toeplitz operators with BMO symbols and the Berezin transform, Int. J. Math. Math. Sci. 46 (2003), 2929–2945. MR 2007108, DOI 10.1155/S0161171203212035
Ke He Zhu, BMO and Hankel operators on Bergman spaces, Pacific J. Math. 155 (1992), no. 2, 377–395. MR 1178032, DOI 10.2140/pjm.1992.155.377