Abstract
Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on ℂn. The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander’s L 2 estimates for the \(\bar{\partial}\) operator are key ingredients in the proof.
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References
Bauer, W.: Mean oscillation and Hankel operators on the Segal–Bargmann space. Integral Equ. Oper. Theory 52, 1–15 (2005)
Bauer, W.: Hilbert-Schmidt Hankel operators on the Segal–Bargmann space. Proc. Am. Math. Soc. 132, 2989–2996 (2005)
Beatrous, F., Li, S.-Y.: Trace ideal criteria for operators of Hankel type. Ill. J. Math. 39, 723–754 (1995)
Békollé, D., Berger, C.A., Coburn, L., Zhu, K.: BMO in the Bergman metric on bounded symmetric domains. J. Funct. Anal. 93, 310–350 (1990)
Berger, C.A., Coburn, L.: Toeplitz operators on the Segal–Bargmann space. Trans. Am. Math. Soc. 301, 813–829 (1987)
Berger, C.A., Coburn, L., Zhu, K.: Toeplitz Operators and Function Theory in n-Dimensions. Lecture Notes in Math., vol. 1256. Springer, Berlin (1987)
Berndtsson, B., Charpentier, P.: A Sobolev mapping property of the Bergman kernel. Math. Z. 235, 1–10 (2000)
Bommier-Hato, H., Youssfi, E.H.: Hankel operators on weighted Fock spaces. Integral Equ. Oper. Theory 59, 1–17 (2007)
Bommier-Hato, H., Youssfi, E.H.: Hankel operators and the Stieltjes moment problem. J. Funct. Anal. 258, 978–998 (2010)
Demailly, J.-P.: Estimations L 2 pour l’opérateur \(\bar{\partial }\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. Ann. Sci. École Norm. Super. 15, 457–511 (1982)
Holland, F., Rochberg, R.: Bergman kernel asymptotics for generalized Fock spaces. J. Anal. Math. 83, 207–242 (2001)
Isralowitz, J., Zhu, K.: Toeplitz operators on the Fock space. Integral Equ. Oper. Theory 66, 593–611 (2010)
Janson, S., Peetre, J., Rochberg, R.: Hankel forms and the Fock space. Rev. Mat. Iberoam. 3, 61–138 (1987)
Kriete, T.L., III: Kernel functions and composition operators in weighted Bergman spaces. In: Studies on Composition Operators, Laramie, WY, 1996. Contemp. Math., vol. 213, pp. 73–91. Amer. Math. Soc., Providence (1998)
Marzo, J., Ortega-Cerdà, J.: Pointwise estimates for the Bergman kernel of the weighted Fock space. J. Geom. Anal. 19, 890–910 (2009)
Stroethoff, K.: Hankel operators in the Fock space. Mich. Math. J. 39, 3–16 (1992)
Xia, J., Zheng, D.: Standard deviation and Schatten class Hankel operators on the Segal–Bargmann space. Indiana Univ. Math. J. 53, 1381–1399 (2004)
Zhu, K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990)
Zhu, K.: Schatten class Hankel operators on the Bergman space of the unit ball. Am. J. Math. 113, 147–167 (1991)
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Communicated by Richard Rochberg.
The first author is supported by the Research Council of Norway grant 185359/V30. The second author is supported by the French ANR DYNOP, Blanc07-198398.
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Seip, K., Youssfi, E.H. Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates. J Geom Anal 23, 170–201 (2013). https://doi.org/10.1007/s12220-011-9241-9
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DOI: https://doi.org/10.1007/s12220-011-9241-9