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

CONNECTION AND CURVATURE ON BUNDLES OF BERGMAN AND HARDY SPACES

We consider a complex domain D x V in the space C-m x C-n and a family of weighted Bergman spaces on V defined by a weight e(-k phi(z , w)) for a pluri-subharmonic function phi(z, w) with a quantization parameter k. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain D. We consider the natural covariant differentiation del(z) on the sections, namely the unitary Chern connections preserving the Bergman norm. We prove a Dixmier trace formula for the curvature of the unitary connection and we find the asymptotic expansion for the curvatures R-(k)(Z,Z) for large k and for the induced connection [del((k))(Z), T-f((k))] on Toeplitz operators T-f. In the special case when the domain D is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for [del((k))(Z), T-f((k))] as Toeplitz operators. This generalizes earlier work of J.E. Andersen in Comm. Math. Phys. 255 (2005), 727-745. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D x V replaced by a general strictly pseudoconvex domain V subset of C-m x C-n fibered over a domain D subset of C-m. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed

Toeplitz Quantization and Asymptotic Expansions: Geometric Construction

For a real symmetric domain $G_{\mathbb R}/K_{\mathbb R}$, with complexification $G_{\mathbb C}/K_{\mathbb C}$, we introduce the concept of "star-restriction" (a real analogue of the "star-products" for quantization of K\"ahler manifolds) and give a geometric construction of the $G_{\mathbb R}$-invariant differential operators yielding its asymptotic expansion

Dixmier trace and the Fock space

We give criteria for products of Toeplitz and Hankel operators on the Fock (Segal-Bargmann) space to belong to the Dixmier class, and compute their Dixmier trace. At the same time, analogous results for the Weyl pseudodifferential operators are also obtained.Comment: 23 pages, no figure