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

Abstract:

For $\mathbf {D}$ the open complex unit disc with normalized area measure, we consider the Bergman space $L_{a}^{2}(\mathbf {D})$ of square-integrable holomorphic functions on $\mathbf {D}$. Induced by the group $Aut(\mathbf {D})$ of biholomorphic automorphisms of $\mathbf {D}$, there is a standard family of Weyl-type unitary operators on $L_{a}^{2}(\mathbf {D})$. For all bounded operators $X$ on $L_{a}^{2}(\mathbf {D})$, the Berezin transform $\widetilde X$ is a smooth, bounded function on $\mathbf {D}$. The range of the mapping Ber: $X \rightarrow \widetilde X$ is invariant under $Aut(\mathbf {D} )$. The “mixing properties” of the elements of $Aut(\mathbf {D} )$ 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 $M\Vert \widetilde X \Vert _{\infty } \geq \Vert X \Vert$ for all bounded operators $X$ and are used to check other possible properties of $\widetilde X$. Extensions to other domains are discussed.