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

Schatten Class Toeplitz Operators on the Bergman Space

Namita Das

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

Received 23 July 2009; Revised 7 September 2009; Accepted 14 October 2009

Academic Editor: Palle Jorgensen

Copyright © 2009 Namita Das. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.