Journal of the Mathematical Society of Japan

Positive Toeplitz operators on weighted Bergman spaces of a minimal bounded homogeneous domain

Satoshi YAMAJI

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We give criteria for the boundedness of positive Toeplitz operators on weighted Bergman spaces of a minimal bounded homogeneous domain in terms of the Berezin symbol or the averaging function of the symbol. Moreover, we estimate the essential norm of positive Toeplitz operators assuming that they are bounded. As an application of these estimates, we also give necessary and sufficient conditions for the positive Toeplitz operators to be compact.

Article information

J. Math. Soc. Japan, Volume 65, Number 4 (2013), 1101-1115.

First available in Project Euclid: 24 October 2013

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Zentralblatt MATH identifier

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
Secondary: 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.)

Toeplitz operator essential norm Bergman space bounded homogeneous domain minimal domain


YAMAJI, Satoshi. Positive Toeplitz operators on weighted Bergman spaces of a minimal bounded homogeneous domain. J. Math. Soc. Japan 65 (2013), no. 4, 1101--1115. doi:10.2969/jmsj/06541101.

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  • D. Békollé and A. T. Kagou, Reproducing properties and $L^p$-estimates for Bergman projections in Siegel domains of type II, Studia. Math., 115 (1995), 219–239.
  • B. R. Choe and Y. J. Lee, Norm and essential norm estimates of Toeplitz operators on the Bergman space, Commun. Korean Math. Soc., 11 (1996), 937–958.
  • Ž. Čučković and R. Zhao, Essential norm estimates of weighted composition operators between Bergman spaces on strongly pseudoconvex domains, Math. Proc. Cambridge Philos. Soc., 142 (2007), 525–533.
  • C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Stad. Adv. Math., CRC Press, Boca Raton, 1995.
  • M. Engliš, Compact Toeplitz operators via the Berezin transform on bounded symmetric domains, Integral Equations Operator Theory, 33 (1999), 426–455.
  • H. Ishi and C. Kai, The representative domain of a homogeneous bounded domain, Kyushu J. Math., 64 (2010), 35–47.
  • H. Ishi and S. Yamaji, Some estimates of the Bergman kernel of minimal bounded homogeneous domains, J. Lie Theory, 21 (2011), 755–769.
  • S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings An introduction, 2nd ed., World Sci., 2005.
  • M. Maschler, Minimal domains and their Bergman kernel function, Pacific J. Math., 6 (1956), 501–516.
  • È. B. Vinberg, S. G. Gindikin and I. I. Pjateckiĭ-Šapiro, Classification and canonical realization of complex homogeneous bounded domains, Trans. Moscow Math. Soc., 12 (1963), 404–437.
  • S. Yamaji, Positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain, Hokkaido Math. J., 41 (2012), 257–274.
  • S. Yamaji, Composition operators on the Bergman spaces of a minimal bounded homogeneous domain, Hiroshima Math. J., 43 (2013), 107–128.
  • D. C. Zheng, Toeplitz operators and Hankel operators, Integral Equations Operator Theory, 12 (1989), 280–299.
  • K. H. Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, J. Operator Theory, 20 (1988), 329–357.
  • K. H. Zhu, Operator Theory in Function Spaces, 2nd ed., Math. Surveys Monogr., 138, Amer. Math. Soc., Providence RI, 2007.