Revista Matemática Iberoamericana

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Volume 22, Issue 3, 2006, pp. 867–892
DOI: 10.4171/RMI/477

Published online: 2006-12-31

$m$-Berezin transform and compact operators

Kyesook Nam[1], Dechao Zheng[2] and Changyong Zhong[3]

(1) Hanshin University, Gyeonggi, South Korea
(2) Vanderbilt University, Nashville, USA
(3) Vanderbilt University, Nashville, USA

$m$-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the $m$-Berezin transform as a linear operator from the space of bounded operators to $L^{\infty}$ is found. We show that the $m$-Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the $m$-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.

Keywords: $m$-Berezin transforms, Toeplitz operators

Nam Kyesook, Zheng Dechao, Zhong Changyong: $m$-Berezin transform and compact operators. Rev. Mat. Iberoam. 22 (2006), 867-892. doi: 10.4171/RMI/477