Covariant derivatives of the Berezin transform

Engliš, Miroslav; Otáhalová, Renata

doi:10.1090/S0002-9947-2011-05111-1

Covariant derivatives of the Berezin transform
HTML articles powered by AMS MathViewer

by Miroslav Engliš and Renata Otáhalová PDF

Trans. Amer. Math. Soc. 363 (2011), 5111-5129 Request permission

Abstract:

Improving upon recent results of Coburn, Xia, Li, Engliš and Zhang, Bommier-Hato, and others, we give estimates for higher-order covariant derivatives of the Berezin transform of bounded linear operators on a reproducing kernel Hilbert space of holomorphic functions. The answer turns out to involve the curvature of the Bergman-type metric associated to the reproducing kernel.

References

Jonathan Arazy and Miroslav Engliš, $Q_p$-spaces on bounded symmetric domains, J. Funct. Spaces Appl. 6 (2008), no. 3, 205–240. MR 2444523, DOI 10.1155/2008/342050
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. Based, in part, on notes left by Harry Bateman. MR 0066496
Michael Beals, Charles Fefferman, and Robert Grossman, Strictly pseudoconvex domains in $\textbf {C}^{n}$, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 2, 125–322. MR 684898, DOI 10.1090/S0273-0979-1983-15087-5
F. A. Berezin, Quantization, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1116–1175 (Russian). MR 0395610
Stefan Bergman, The kernel function and conformal mapping, Second, revised edition, Mathematical Surveys, No. V, American Mathematical Society, Providence, R.I., 1970. MR 0507701
Harold P. Boas, Emil J. Straube, and Ji Ye Yu, Boundary limits of the Bergman kernel and metric, Michigan Math. J. 42 (1995), no. 3, 449–461. MR 1357618, DOI 10.1307/mmj/1029005306
K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. MR 0062505
Hélène Bommier-Hato, Lipschitz estimates for the Berezin transform, J. Funct. Spaces Appl. 8 (2010), no. 2, 103–128. MR 2667872, DOI 10.1155/2010/461315
H. Bommier-Hato: Derivatives of the Berezin transform, preprint (2008).
Hélène Bommier-Hato and El Hassan Youssfi, Hankel operators on weighted Fock spaces, Integral Equations Operator Theory 59 (2007), no. 1, 1–17. MR 2351271, DOI 10.1007/s00020-007-1513-1
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
L. A. Coburn, Sharp Berezin Lipschitz estimates, Proc. Amer. Math. Soc. 135 (2007), no. 4, 1163–1168. MR 2262921, DOI 10.1090/S0002-9939-06-08569-8
L. A. Coburn and Bo Li, Directional derivative estimates for Berezin’s operator calculus, Proc. Amer. Math. Soc. 136 (2008), no. 2, 641–649. MR 2358506, DOI 10.1090/S0002-9939-07-09081-8
Miroslav Engliš, A characterization of symmetric domains, J. Math. Kyoto Univ. 46 (2006), no. 1, 123–146. MR 2260820, DOI 10.1215/kjm/1250283003
Miroslav Engliš and Genkai Zhang, On the derivatives of the Berezin transform, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2285–2294. MR 2213701, DOI 10.1090/S0002-9939-06-08238-4
Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018
Kyong T. Hahn, On completeness of the Bergman metric and its subordinate metric, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 12, 4294. MR 417459, DOI 10.1073/pnas.73.12.4294
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR 1242120, DOI 10.1515/9783110870312
Paul F. Klembeck, Kähler metrics of negative curvature, the Bergmann metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets, Indiana Univ. Math. J. 27 (1978), no. 2, 275–282. MR 463506, DOI 10.1512/iumj.1978.27.27020
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
Steven G. Krantz and Jiye Yu, On the Bergman invariant and curvatures of the Bergman metric, Illinois J. Math. 40 (1996), no. 2, 226–244. MR 1398092
T. Mazur, P. Pflug, and M. Skwarczyński, Invariant distances related to the Bergman function, Proc. Amer. Math. Soc. 94 (1985), no. 1, 72–76. MR 781059, DOI 10.1090/S0002-9939-1985-0781059-0
Kyesook Nam, Dechao Zheng, and Changyong Zhong, $m$-Berezin transform and compact operators, Rev. Mat. Iberoam. 22 (2006), no. 3, 867–892. MR 2320405, DOI 10.4171/RMI/477
Jaak Peetre, The Berezin transform and Ha-plitz operators, J. Operator Theory 24 (1990), no. 1, 165–186. MR 1086552
Takashi Sakai, Riemannian geometry, Translations of Mathematical Monographs, vol. 149, American Mathematical Society, Providence, RI, 1996. Translated from the 1992 Japanese original by the author. MR 1390760, DOI 10.1090/mmono/149
Daniel Suárez, Approximation and symbolic calculus for Toeplitz algebras on the Bergman space, Rev. Mat. Iberoamericana 20 (2004), no. 2, 563–610. MR 2073132, DOI 10.4171/RMI/401
Richard M. Timoney, Bloch functions in several complex variables. I, Bull. London Math. Soc. 12 (1980), no. 4, 241–267. MR 576974, DOI 10.1112/blms/12.4.241
Fangyang Zheng, Complex differential geometry, AMS/IP Studies in Advanced Mathematics, vol. 18, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2000. MR 1777835, DOI 10.1090/amsip/018