Schatten Class Toeplitz Operators on the Bergman Space

Abstract

We have shown that if the Toeplitz operatorTϕon the Bergman spaceLa2(𝔻)belongs to the Schatten classSp,1≤p<∞, thenϕ˜∈Lp(𝔻,dλ), whereϕ˜is the Berezin transform ofϕ,dλ(z)=dA(z)/(1−|z|2)2, anddA(z)is the normalized area measure on the open unit disk𝔻. Further, ifϕ∈Lp(𝔻,dλ)thenϕ˜∈Lp(𝔻,dλ)andTϕ∈Sp. For certain subclasses ofL∞(𝔻), necessary and sufficient conditions characterizing Schatten class Toeplitz operators are also obtained.