Related papers: Toeplitz Quantization and Asymptotic Expansions: G…

For compact quantizable K\"ahler manifolds certain naturally defined star products and their constructions are reviewed. The presentation centers around the Berezin-Toeplitz quantization scheme which is explained. As star products the…

Let $D$ be an irreducible bounded symmetric domain with biholomorphism group $G$ with maximal compact subgroup $K$. For the Toeplitz operators with $K$-invariant symbols we provide explicit simultaneous diagonalization formulas on every…

On a compact K\"ahler manifold $X$, Toeplitz operators determine a deformation quantization $(\operatorname{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$ with separation of variables [10] with respect to transversal complex polarizations…

We generalize several recent results concerning the asymptotic expansions of Bergman kernels to the framework of geometric quantization and establish an asymptotic symplectic identification property. More precisely, we study the asymptotic…

Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $T_P$ be the Toeplitz operator on $X$ associated with some first order pseudodifferential operator $P$. We consider $\chi_k(T_P)$ the functional calculus of $T_P$ by…

For arbitrary compact quantizable Kaehler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via Berezin-Toeplitz operators. Results on their semi-classical behaviour (their asymptotic…

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion.…

An approach to the construction of index formulas for elliptic operators on singular manifolds is suggested on the basis of K-theory of algebras and cyclic cohomology. The equivalence of Toeplitz and pseudodifferential quantizations, well…

We give a new construction of symbols of the differential operators on the sections of a quantum line bundle $L$ over a Kaehler manifold $M$ using the natural contravariant connection on $L$. These symbols are the functions on the tangent…

We use Toeplitz operators to define a star-product on Poisson manifolds whose Poisson structure is induced by a symplectic Lie algebroid. The Toeplitz operators we consider are defined on groupoids whose algebroid can be endowed with a…

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean…

On a relatively compact strictly pseudoconvex domain with smooth boundary in a complex manifold of dimension $n$ we consider a Toeplitz operator $T_R$ with symbol a Reeb-like vector field $R$ near the boundary. We show that the kernel of a…

Let G/H be a pseudo-Riemannian semisimple symmetric space. The tangent bundle T(G/H) contains a maximal G-invariant neighbourhood of the zero section where the adapted complex structure exists. Such neighbourhood is endowed with a canonical…

We study the asymptotic expansion of the product of two Toeplitz operators on the Fock space. In comparison to earlier results we require significantly less derivatives and get the expansion to arbitrary order. This, in particular, improves…

For a real bounded symmetric domain, G/K, we construct various natural enlargements to which several aspects of harmonic analysis on G/K and G have extensions. Our starting point is the realization of G/K as a totally real submanifold in a…

This text reviews certain notions in metric geometry that may have further applications to problems in complex geometry and holomorphic dynamics in several variables. The discussion contains a few unrecorded results and formulates a number…

We study the distribution kernel of a Toeplitz operator associated with a classical pseudodifferential operator on a compact, embeddable, strictly pseudoconvex CR manifold. The main result consists of a formula for the values at the…

Let $M$ be an arbitrary complex manifold and let $L$ be a Hermitian holomorphic line bundle over $M$. We introduce the Berezin-Toeplitz quantization of the open set of $M$ where the curvature on $L$ is non-degenerate. The quantum spaces are…

We use the theory of abstract Wiener spaces to construct a probabilistic model for Berezin-Toeplitz quantization on a complete Hermitian complex manifold endowed with a positive line bundle. We associate to a function with compact support…

This talk reports on results on the deformation quantization (star products) and on approximative operator representations for quantizable compact K"ahler manifolds obtained via Berezin-Toeplitz operators. After choosing a holomorphic…