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 Lₐ²(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 Lₐ²(D). For all bounded operators X on Lₐ²(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 ∞ ≥ X for all bounded operators X and are used to check other possible properties of X. Extensions to other domains are discussed.