Approximation and the n-Berezin transform of operators on the Bergman space

Abstract

To any bounded operator S on the Bergman space La2 we associate a sequence of linear transforms Bn(S ) ∈ L∞(D), where n ≧ 0, and prove that the Toeplitz operators article image tend to S for some especial classes of operators S. In particular, this holds for every radial operator in the Toeplitz algebra. Finally, we show that the inclusion of the Toeplitz algebra into the essential commutant of the Bergman shift is proper.