COMPACT OPERATORS AND THE PLURIHARMONIC BEREZIN TRANSFORM

Abstract

For a series of weighted Bergman spaces over bounded symmetric domains in ℂⁿ, it has been shown by Axler and Zheng [1]; Englis [10] that the compactness of Toeplitz operators with bounded symbols can be characterized via the boundary behavior of its Berezin transform B _a . In case of the pluriharmonic Bergman space, the pluriharmonic Berezin transform B _ph fails to be one-to-one in general and even has non-compact operators in its kernel. From this point of view, perhaps surprisingly we show that via B _ph the same characterization of compactness holds for Toeplitz operators on the pluriharmonic Fock space.