Schatten Class Toeplitz Operators on the Bergman Space

Abstract EN

We have shown that if the Toeplitz operator Tϕ on the Bergman space La2(?) belongs to the Schatten class Sp,1≤p<∞, then ϕ˜∈Lp(?,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 ?.

Further, if ϕ∈Lp(?,dλ) then ϕ˜∈Lp(?,dλ) and Tϕ∈Sp.

For certain subclasses of L∞(?), necessary and sufficient conditions characterizing Schatten class Toeplitz operators are also obtained.