JournalsrmiVol. 22, No. 3pp. 867–892

mm-Berezin transform and compact operators

  • Kyesook Nam

    Hanshin University, Gyeonggi, South Korea
  • Dechao Zheng

    Vanderbilt University, Nashville, USA
  • Changyong Zhong

    Vanderbilt University, Nashville, USA
$m$-Berezin transform and compact operators cover
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Abstract

mm-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the mm-Berezin transform as a linear operator from the space of bounded operators to LL^{\infty} is found. We show that the mm-Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the mm-Berezin transforms we show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the unit ball.

Cite this article

Kyesook Nam, Dechao Zheng, Changyong Zhong, mm-Berezin transform and compact operators. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 867–892

DOI 10.4171/RMI/477