Schatten Class Toeplitz Operators on the Bergman Space

We have shown that if the Toeplitz operator Tϕ on theBergman space La2(𝔻) belongs to the Schatten class Sp,1≤p<∞, then ϕ˜∈Lp(𝔻,dλ), where ϕ˜ is the Berezin transform of ϕ,dλ(z)=dA(z)/(1−|z|2)2, and dA(z) is the normalized area measure on the open unit disk 𝔻. Further, if ϕ∈Lp(𝔻,dλ) then ϕ˜∈Lp(𝔻,dλ) and Tϕ∈Sp. For certainsubclasses of L∞(𝔻), necessary and sufficient conditions characterizingSchatten class Toeplitz operators are also obtained