Full paper in PDF:
$% M. Engliš, Berezin and Berezin-Toeplitz quantizations for general function spaces, Rev. Mat. Complut. 19 (2006), no. 2, 385–430.%$

The standard Berezin and Berezin-Toeplitz quantizations on a Kähler manifold are based on operator symbols and on Toeplitz operators, respectively, on weighted -spaces of holomorphic functions (weighted Bergman spaces). In both cases, the construction basically uses only the fact that these spaces have a reproducing kernel. We explore the possibilities of using other function spaces with reproducing kernels instead, such as -spaces of harmonic functions, Sobolev spaces, Sobolev spaces of holomorphic functions, and so on. Both positive and negative results are obtained.

Key words: Berezin quantization, Berezin-Toeplitz quantization, star product, harmonic Bergman space, Sobolev-Bergman space, reproducing kernel.
2000 Mathematics Subject Classication: primary 53D55; secondary 46E22, 47B35, 46E35.