-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the -Berezin transform as a linear operator from the space of bounded operators to is found. We show that the -Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the -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.
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Kyesook Nam, Dechao Zheng, Changyong Zhong, -Berezin transform and compact operators. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 867–892DOI 10.4171/RMI/477