International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 921324, 16 pages
doi:10.1155/2009/921324

Schatten class Toeplitz operators on the Bergman space

Namita Das

P. G. Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar, Orissa 751004, India

Abstract

We have shown that if the Toeplitz operator Tϕ on the Bergman 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 certain subclasses of L∞(𝔻), necessary and sufficient conditions characterizing Schatten class Toeplitz operators are also obtained.