Hiroshima Mathematical Journal

Composition operators on the Bergman spaces of a minimal bounded homogeneous domain

Satoshi Yamaji

Full-text: Open access


Using an integral formula on a homogeneous Siegel domain, we give a necessary and sufficient condition for composition operators on the weighted Bergman space of a minimal bounded homogeneous domain to be compact in terms of a boundary behavior of the Bergman kernel.

Article information

Hiroshima Math. J., Volume 43, Number 1 (2013), 107-128.

First available in Project Euclid: 10 May 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B33: Composition operators
Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.)

Composition operator Bergman space bounded homogeneous domain minimal domain Carleson measure


Yamaji, Satoshi. Composition operators on the Bergman spaces of a minimal bounded homogeneous domain. Hiroshima Math. J. 43 (2013), no. 1, 107--128. doi:10.32917/hmj/1368217952. https://projecteuclid.org/euclid.hmj/1368217952

Export citation


  • 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.
  • D. Békollé and C. Nana, $L^{p}$-boundedness of Bergman projections in the tube domain over Vinberg's cone, J. Lie Theory 17 (2007), 115–144.
  • C. Cowen and B. MacCluer, Composition operators on spaces of analytic function, CRC Press, Boca Raton, 1994.
  • M. Engliš, Compact Toeplitz operators via the Berezin transform on bounded symmetric domains, Integr. Equ. Oper. Theory 20 (1999), 426–455.
  • J. Fraut and A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), no.1, 64–89.
  • S. G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys 19-4 (1964), 1–89.
  • 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.
  • X. Lv and Z. Hu, Compact composition operators on weighted Bergman spaces on bounded symmetric domains, Acta Math. Scientia 31B(2) (2011), 468–476.
  • M. Maschler, Minimal domains and their Bergman kernel function, Pacific J. Math. 6 (1956), 501–516.
  • È. B. Vinberg, Homogeneous cones, Soviet Math. Dokl. 1 (1960), 787–790.
  • È. B. Vinberg, S. G. Gindikin, I. I. Pjateckiĭ-Šapiro, Classification and canonical realization of complex bounded homogeneous domains, Trans. Moscow Math. Soc. 12 (1963) 404–437.
  • S. Yamaji, Positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain, to appear in Hokkaido Math. J.
  • K. H. Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, J. Oper. Theory 20 (1988), 329–357.
  • K. H. Zhu, Operator theory in function spaces, second edition, Amer. Math. Soc., Mathematical Surveys and Monographs Vol.138, 2007.
  • K. H. Zhu, Compact composition operators on weighted Bergman spaces of the unit ball, Houston J. Math. 33 (2007), 273–283.