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Journal of Function Spaces and Applications
Volume 2012, Article ID 160808, 15 pages
http://dx.doi.org/10.1155/2012/160808
Research Article

Derivatives of the Berezin Transform

LATP, UMR CNRS 6632, CMI, Université de Provence, 39 rue Fjoliot-Curie, 13453 Marseille Cedex 13, France

Received 3 July 2009; Accepted 21 December 2011

Academic Editor: Miroslav Englis

Copyright © 2012 Hélène Bommier-Hato. 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

For a rotation invariant domain Ξ©, we consider 𝐴2(Ξ©,πœ‡) the Bergman space and we investigate some properties of the rank one projection 𝐴(𝑧)∢=βŸ¨β‹…,π‘˜π‘§βŸ©π‘˜π‘§. We prove that the trace of all the strong derivatives of A(z) is zero. We also focus on the generalized Fock space 𝐴2(πœ‡π‘š), where πœ‡π‘š is the measure with weight π‘’βˆ’|𝑧|π‘š, π‘š>0, with respect to the Lebesgue measure on ℂ𝑛 and establish estimations of derivatives of the Berezin transform of a bounded operator T on 𝐴2(πœ‡π‘š).