The Berezin Transform of Bounded Linear Operators

Jump To References Section

Authors

  • P. G. Dept. of Mathematics, Utkal University, Vanivihar, Bhubaneswar, 751 004, Orissa ,IN

Keywords:

Berezin Transform, Bounded Linear Operators, Bergman Space.

Abstract

Let D = {z ∈ C : |z| < 1} and σ be the map from £(L2a(D)) into L∞(D) defined as σ(T)(z) = {Tkz, kz}, where {kz}z∈D are the normalized reproducing kernels for the Bergman space L2a(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).