Berezin and Berezin-Toeplitz Quantizations for General Function Spaces

The standard Berezin and Berezin-Toeplitz quantizations on a K¨ahler manifold are based on operator symbols and on Toeplitz operators, respectively, on weighted L2-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 L2-spaces of harmonic functions, Sobolev spaces, Sobolev spaces of holomorphic functions, and so on. Both positive and negative results are obtained.