The Journal of the Indian Mathematical Society

Namita Das ¹

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

Abstract

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).

Keywords

Berezin Transform, Bounded Linear Operators, Bergman Space.

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Namita Das ¹

Affiliations
1 P.G. Dept. of Mathematics, Sambalpur University, Jyotivihar, Burla - 768019, Sambalpur, Orissa, IN

Abstract

Let D = {z ∈ C : |z| < 1} be the open unit disc in the complex plane C. Let L²_h(ID) be the subspace of functions in L²(D) that are harmonic. Let Q be the orthogonal projection of L² onto L²_h. For Φ ∈ L^∞;(D), define L_Φ from L²_h into itself such that LΦf= Q(Φf). In this paper some algebraic properties of the Toeplitz-like operator L_Φ are derived.

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