Berezin transform and Weyl-type unitary operators on the Bergman space

For D the open complex unit disc with normalized area measure, we consider the Bergman space L ² _a (D) of square-integrable holomorphic functions on D. Induced by the group Aut(D) of biholomorphic automorphisms of D, there is a standard family of Weyl-type unitary operators on L ² _a(D). For all bounded operators X on L ² _a(D), the Berezin transform X is a smooth, bounded function on D. The range of the mapping Ber: X → X is invariant under Aut(D). The "mixing properties" of the elements of Aut(D) are visible in the Berezin transforms of the induced unitary operators. Computations involving these operators show that there is no real number M>0 with; M||X||∞geq ||X|| for all bounded operators X; and are used to check other possible properties of X. Extensions to other domains are discussed.