The Berezin Transform of Bounded Linear Operators

Let D = {z ∈ C : |z| < 1} and σ be the map from £(L²_a(D)) into L^∞(D) defined as σ(T)(z) = {Tk_z, k_z}, where {k_z}_z∈D are the normalized reproducing kernels for the Bergman space L²_a(D) into itself. The function σ(T) is called the Berezin transform of T. In this paper we have shown that Range(σ) is not a closed subspace of L^∞(D) and give a characterization of functions in L^∞(D) that are in Range(σ).