Abstract
$m$-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.
Citation
Kyesook Nam . Dechao Zheng . Changyong Zhong . "$m$-Berezin transform and compact operators." Rev. Mat. Iberoamericana 22 (3) 867 - 892, December, 2006.
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