SCHATTEN CLASS OPERATORS ON THE BERGMAN SPACE OVER BOUNDED SYMMETRIC DOMAIN

摘要

Let Ω be a bounded symmetric domain in C~n with Bergman kernel K(z,w). Let dV_λ(z) =K(z,z)dV(z)/C_λ, where C_λ = ∫_? K(z, z)~λdV(z), λ ∈ R, dV(z) is the volume measure of Ω normalized so that K(z, 0) = K(0, w) = 1. In this paper we have shown that if the Toeplitz operator T_φ defined on L_a~2 (?,dV/C_0) belongs to the Schatten p-class, 1 ≤ p < oo, then φ ∈ Lp(Ω,dη), where dη(z) = K(z,z)dV(z)/C_0 and φ is the Berezin transform of φ. Further if ф ∈ L~p(?,dη_λ), then φ_λ ∈ L~P(Ω, dη_λ) and T_φ~λ belongs to Schatten p-class. Here dη_λ = K(z,z)dV(z)/C_λ, the function φ_λ is the Berezin transform of φ in L_a~2(?,dV_λ) and T_Φ~λ is the Toeplitz operator defined on L_a~2(Ω,dV_λ). We also find conditions on bounded linear operator C defined from L_a~2(Ω,dV_λ) into itself such that C belongs to the Schatten p-class by comparing it with positive Toeplitz operators defined on L_a~2(Ω,dV_λ). Applications of these results are obtained and we also present Schatten class characterization of little Hankel operators defined on L_a~2(Ω,dV_λ).