On the Berezin Transform for Compact Kähler Manifolds

For quantizable compact Kähler manifolds $(M,\omega)$ with associated quantum line bundle $(L,h,\nabla)$ the Berezin transform $I$ for $C^{\infty}$ functions is introduced. This is a generalization of the original transform between contravariant and covariant Berezin symbols. If one considers all positve tensor powers $L^{\otimes m}$ of $L$ and the Berezin transform $I^{(m)}$ then it admits a complete asymptotic expansion in powers of $1/m$ , e.g. $$I^{(m)}f(x)\sim \sum_{k=0}^\infty (1/m)^kI_kf(x)$$ with differential operators $I_k$. It turns out that $I_0=id$ and $I_1=\Delta$, the Laplace-Beltrami operator. Consequences of this expansion for the Berezin-Toeplitz operator quantization and the Berezin-Toeplitz deformation quantization are discussed.