Schatten Class Toeplitz Operators on the Bergman Space

Namita Das P. G. Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar, Orissa 751004, India Correspondence should be addressed to Namita Das, [email protected] Received 23 July 2009; Revised 7 September 2009; Accepted 14 October 2009 Recommended by Palle Jorgensen We have shown that if the Toeplitz operator Tφ on the Bergman space La D belongs to the Schatten class Sp, 1 ≤ p < ∞, then ̃ φ ∈ L D, dλ , where ̃ φ is the Berezin transform of φ, dλ z dA z / 1 − |z| , and dA z is the normalized area measure on the open unit disk D. Further, if φ ∈ L D, dλ then ̃ φ ∈ L D, dλ and Tφ ∈ Sp. For certain subclasses of L∞ D , necessary and sufficient conditions characterizing Schatten class Toeplitz operators are also obtained. Copyright q 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.